# An exact estimate result for p-biharmonic equations with Hardy potential and negative exponents

## Abstract

In this paper, p-biharmonic equations involving Hardy potential and negative exponents with a parameter λ are considered. By means of the structure and properties of Nehari manifold, we give uniform lower bounds for $$\varLambda >0$$, which is the supremum of the set of λ. When $$\lambda \in (0, \varLambda )$$, the above problems admit at least two positive solutions.

## Introduction and preliminaries

In this paper, we consider a p-biharmonic equation with Hardy potential and negative exponents:

$$\textstyle\begin{cases} \Delta ^{2}_{p}u-\mu \frac{ \vert u \vert ^{p-2}u}{ \vert x \vert ^{2p}}= f(x)u^{-q}+\lambda g(x)u ^{\gamma } & \text{in } \varOmega \setminus \{0\}, \\ u(x)>0 &\text{in } \varOmega \setminus \{0\}, \\ u=\Delta u=0 &\text{on } \partial \varOmega , \end{cases}$$
(1.1)

where $$0\in \varOmega \subset \mathbb{R}^{N}$$ is a bounded smooth domain with $$1< p<\frac{N}{2}$$, $$\Delta ^{2}_{p}u=\Delta (\vert \Delta u\vert ^{p-2} \Delta u)$$ is the p-biharmonic operator. $$\lambda >0$$ is a parameter, $$0<\mu <\mu _{N,p}=(\frac{(p-1)N(N-2p)}{p^{2}})^{p}$$, $$0< q<1$$ and $$p-1<\gamma <p^{*}-1$$, where $$p^{*}=\frac{Np}{N-2p}$$ is called the critical Sobolev exponent. $$f(x)\geq 0$$, $$f(x)\not \equiv 0$$, $$g(x)$$ satisfies the requirement that the set $$\{x\in \varOmega : g(x)>0 \}$$ has positive measures, $$\operatorname{supp}f \cap \{x\in \varOmega : g(x)>0 \} \neq \emptyset$$ and $$f, g\in C(\overline{\varOmega })$$. Biharmonic equations describe the sport of a rigid body and the deformations of an elastic beam. For example, this type of equation provides a model for considering traveling wave in suspension bridges [5, 16, 27, 30, 36]. Various methods and tools have been adopted to deal with singular problems, such that fixed point theorems , topological methods , Fourier and Laurent transformation [18, 19], monotone iterative methods , global bifurcation theory , and degree theory [22, 31].

In recent years, there was much attention focused on the existence, multiplicity and qualitative properties of solutions for p-biharmonic equations under Dirichlet boundary conditions or Navier boundary conditions with Hardy terms [4, 15, 17, 32, 34]. Xie and Wang  studied the following p-biharmonic equation with Dirichlet boundary conditions:

$$\textstyle\begin{cases} \Delta ^{2}_{p}u-\mu \frac{ \vert u \vert ^{p-2}u}{ \vert x \vert ^{2p}}= f(x,u) & \text{in } \varOmega , \\ u=\frac{\partial u}{\partial n}=0 & \text{on } \partial \varOmega , \end{cases}$$
(1.2)

where $$\frac{\partial }{\partial n}$$ is the outer normal derivative. By using the variational method, the existence of infinitely many solutions with positive energy levels for (1.2) was established. Huang and Liu  considered the following p-biharmonic equation with Navier boundary conditions:

$$\textstyle\begin{cases} \Delta ^{2}_{p}u-\mu \frac{ \vert u \vert ^{p-2}u}{ \vert x \vert ^{2p}}= f(x,u) & \text{in } \varOmega , \\ u=\Delta u=0 & \text{on } \partial \varOmega , \end{cases}$$
(1.3)

where $$1< p<\frac{N}{2}$$. By using invariant sets of gradient flows, the authors proved that (1.3) possesses a sign-changing solution. Furthermore, Yang, Zhang and Liu  showed that (1.3) has a positive solution, a negative solution and a sequence of sign-changing solutions when f satisfies appropriate conditions. Bhakta  established the qualitative properties of entire solutions for a noncompact problem related to p-biharmonic type equations with Hardy terms.

On the other hand, nonlinear biharmonic equations with negative exponents have been studied expensively [1, 6, 8, 13, 20]. Guerra  gave a complete description of entire radially symmetric solutions for the following biharmonic equation:

$$\Delta ^{2} u=-u^{-q}, \qquad u>0 \quad \text{in } \mathbb{R}^{3},$$

where $$q>1$$. Moreover, Cowan et al.  dealt with the regularity of the extremal solution of the following fourth order boundary value problems:

$$\textstyle\begin{cases} \Delta ^{2}u=\frac{\lambda }{(1-u)^{2}} &\text{in } \varOmega , \\ 0< u< 1 &\text{in } \varOmega , \\ u=\frac{\partial u}{\partial n}=0 &\text{on } \partial \varOmega . \end{cases}$$

Very recently, Ansari, Vaezpour and Hesaaraki  considered fourth order elliptic problem with the combinations of Hardy term and negative exponents,

$$\textstyle\begin{cases} \Delta ^{2}u-\lambda M( \Vert \nabla u \Vert ^{2})\Delta u -\frac{\mu }{ \vert x \vert ^{4}}u= \frac{h(x)}{u^{\gamma }}+k(x) u^{\alpha } & \text{in }\varOmega , \\ u=\Delta u=0 & \text{on } \partial \varOmega , \end{cases}$$
(1.4)

where $$\varOmega \subset \mathbb{R}^{N}$$ ($$N\geq 1$$) is a bounded $$C^{4}$$-domain, λ and μ are positive parameters and $$0<\alpha <1$$, $$0<\gamma <1$$ are constants. Here M, h and k are given continuous functions satisfying suitable hypotheses. By using the Galerkin method and the sharp angle lemma, the authors proved that problem (1.4) has a positive solution for $$0<\mu < (\frac{N(N-4)}{4} ) ^{2}$$.

We say that $$u\in W:=W^{2,p}(\varOmega )\cap W_{0}^{1,p}(\varOmega )$$ is a weak solution of (1.1), if for every $$\varphi \in W$$, there holds

$$\int _{\varOmega } \vert \Delta u \vert ^{p-2}\Delta u \Delta \varphi \,dx- \int _{\varOmega }\frac{\mu }{ \vert x \vert ^{2p}} \vert u \vert ^{p-2}u \varphi \,dx= \int _{\varOmega }f(x)u^{-q}\varphi \,dx+\lambda \int _{\varOmega } g(x)u^{\gamma }\varphi \,dx.$$
(1.5)

The following Rellich inequality will be used in this paper:

$$\int _{\varOmega } \vert \Delta u \vert ^{p}\,dx\geq \mu _{N,p} \int _{\varOmega }\frac{ \vert u \vert ^{p}}{ \vert x \vert ^{2p}}\,dx, \quad \forall u\in W,$$

and it is not achieved [9, 24]. For any $$u\in W$$, and $$0<\mu <\mu _{N,p}$$. The energy functional corresponding to (1.1) is defined by

\begin{aligned}[b] I_{\lambda ,\mu }(u)={}&\frac{1}{p} \int _{\varOmega } \biggl( \vert \Delta u \vert ^{p}- \frac{\mu }{ \vert x \vert ^{2p}} \vert u \vert ^{p} \biggr)\,dx- \frac{1}{1-q} \int _{\varOmega }f(x) \vert u \vert ^{1-q}\,dx \\ &{}- \frac{\lambda }{\gamma +1} \int _{\varOmega }g(x) \vert u \vert ^{\gamma +1}\,dx. \end{aligned}
(1.6)

For $$\mu \in [0,\mu _{N,p})$$, W is equipped with the following norm:

$$\Vert u \Vert ^{p}_{\mu }= \int _{\varOmega } \biggl( \vert \Delta u \vert ^{p}- \frac{\mu }{ \vert x \vert ^{2p}} \vert u \vert ^{p} \biggr)\,dx.$$

Negative exponent term $$u^{-q}$$ implies that $$I_{\lambda ,\mu }$$ is not differential on W, therefore, critical point theory cannot be applied to the problem (1.1) directly. We consider the following manifold:

$$\mathcal{M}= \biggl\{ u\in W: \Vert u \Vert ^{p}_{\mu } = \int _{\varOmega }f(x) \vert u \vert ^{1-q}\,dx+\lambda \int _{\varOmega } g(x) \vert u \vert ^{\gamma +1}\,dx \biggr\} ,$$

and make the following splitting for $$\mathcal{M}$$:

\begin{aligned}& \mathcal{M}^{+}= \biggl\{ u\in \mathcal{M}: (p+q-1 ) \Vert u \Vert ^{p}_{ \mu } >\lambda (\gamma +q ) \int _{\varOmega }g(x) \vert u \vert ^{\gamma +1}\,dx \biggr\} , \end{aligned}
(1.7)
\begin{aligned}& \mathcal{M}^{0}= \biggl\{ u\in \mathcal{M}: (p+q-1 ) \Vert u \Vert ^{p}_{ \mu } =\lambda (\gamma +q ) \int _{\varOmega }g(x) \vert u \vert ^{\gamma +1}\,dx \biggr\} , \end{aligned}
(1.8)
\begin{aligned}& \mathcal{M}^{-}= \biggl\{ u\in \mathcal{M}: (p+q-1 ) \Vert u \Vert ^{p}_{ \mu } < \lambda (\gamma +q ) \int _{\varOmega }g(x) \vert u \vert ^{\gamma +1}\,dx \biggr\} . \end{aligned}
(1.9)

In this paper, we will study the dependence of problem (1.1) on q, γ, f, g and Ω and evaluate the extremal value of λ related to multiplicity of positive solutions for problem (1.1). Our idea comes from [7, 28, 29]. Our results improve and complement previous ones obtained in [23, 25]. Denote $$\Vert u\Vert _{t}^{t}= \int _{\varOmega }\vert u\vert ^{t}\,dx$$ and $$D^{2, p}(\mathbb{R}^{N})$$ be the closure of $$C_{0}^{\infty }(\mathbb{R}^{N})$$ with respect to the norm $$(\int _{\mathbb{R}^{N}}\vert \Delta u\vert ^{p} \,dx )^{\frac{1}{p}}$$.

$$\lambda _{1}$$ denotes the smallest eigenvalue for

$$\Delta _{p}^{2}u-\frac{\mu }{ \vert x \vert ^{2p}} \vert u \vert ^{p-2}u=\lambda _{1} \vert u \vert ^{p-2}u, \quad x\in \varOmega \setminus \{0\}, u\in W,$$
(1.10)

and $$\varphi _{1}$$ denotes the corresponding eigenfunction with $$\varphi _{1}>0$$ in Ω [3, 10, 26, 33, 35]. The following minimization problem will be useful in the following discussions:

$$S_{\mu }=\inf \biggl\{ \int _{\mathbb{R}^{N}} \biggl( \vert \Delta u \vert ^{p}- \frac{\mu }{ \vert x \vert ^{2p}} \vert u \vert ^{p} \biggr)\,dx, u\in D^{2,p} \bigl(\mathbb{R}^{N} \bigr), \int _{\mathbb{R}^{N}} \vert u \vert ^{p^{*}}\,dx=1 \biggr\} >0,$$
(1.11)

and $$S_{\mu }$$ is achieved by a family of functions [4, 11]. Thus, for every $$u\in W\setminus \{0\}$$, $$\Vert u\Vert _{p^{*}}\leq \frac{ \Vert u\Vert _{\mu }}{\sqrt[p]{S_{\mu }}}$$. Therefore, combining with the Hölder inequality, we deduce that

\begin{aligned}& \begin{aligned}[b] \int _{\varOmega } \vert u \vert ^{\gamma +1}\,dx&\leq \biggl[ \int _{\varOmega } \vert u \vert ^{(\gamma +1)\frac{p^{*}}{\gamma +1}}\,dx \biggr]^{\frac{ \gamma +1}{p^{*}}} \biggl( \int _{\varOmega }1\,dx \biggr)^{\frac{p^{*}-\gamma -1}{p^{*}}} \\ &= \vert \varOmega \vert ^{\frac{p^{*}-\gamma -1}{p^{*}}} \Vert u \Vert ^{\gamma +1} _{p^{*}} \\ &\leq \vert \varOmega \vert ^{\frac{p^{*}-\gamma -1}{p^{*}}} \biggl( \frac{ \Vert u \Vert _{\mu }}{\sqrt[p]{S_{\mu }}} \biggr)^{\gamma +1}, \end{aligned} \end{aligned}
(1.12)
\begin{aligned}& \begin{aligned}[b] \int _{\varOmega } \vert u \vert ^{1-q}\,dx&\leq \biggl[ \int _{\varOmega } \vert u \vert ^{(1-q)\frac{p^{*}}{1-q}}\,dx \biggr]^{\frac{1-q}{p ^{*}}} \biggl( \int _{\varOmega }1\,dx \biggr)^{\frac{p^{*}-1+q}{p^{*}}} \\ &= \vert \varOmega \vert ^{\frac{p^{*}-1+q}{p^{*}}} \Vert u \Vert ^{1-q} _{p^{*}} \\ &\leq \vert \varOmega \vert ^{\frac{p^{*}-1+q}{p^{*}}} \biggl( \frac{ \Vert u \Vert _{\mu }}{\sqrt[p]{S_{\mu }}} \biggr)^{1-q}, \end{aligned} \end{aligned}
(1.13)

and

\begin{aligned}[b] \int _{\varOmega } \vert u \vert ^{1-q}\,dx&\leq \biggl[ \int _{\varOmega } \vert u \vert ^{(1-q)\frac{\gamma +1}{1-q}}\,dx \biggr]^{\frac{1-q}{ \gamma +1}} \biggl( \int _{\varOmega }1\,dx \biggr)^{\frac{\gamma +q}{\gamma +1}} \\ &= \vert \varOmega \vert ^{\frac{\gamma +q}{\gamma +1}} \Vert u \Vert ^{1-q} _{\gamma +1}. \end{aligned}
(1.14)

Our main results are stated in the following theorems.

### Theorem 1.1

Assume that $$\lambda \in (0,\varLambda )$$, where

\begin{aligned} \begin{aligned}[b] \varLambda \geq{}& T_{\mu }= \biggl(\frac{q+p-1}{q+\gamma } \biggr) \biggl(\frac{ \gamma -p+1}{q+\gamma } \biggr)^{\frac{p-\gamma -1}{1-q-p}} \biggl(\frac{1}{ \Vert f \Vert _{\infty }} \biggr)^{\frac{p-\gamma -1}{1-q-p}} \biggl(\frac{1}{ \Vert g \Vert _{\infty }} \biggr) \biggl( \frac{S_{\mu }}{ \vert \varOmega \vert ^{ \frac{p}{N}}} \biggr)^{\frac{q+\gamma }{p+q-1}} \\ >{}&0. \end{aligned} \end{aligned}
(1.15)

Then problem (1.1) admits at least two solutions $$u_{0}\in \mathcal{M}^{+}$$, $$U_{0}\in \mathcal{M}^{-}$$, with $$\Vert U_{0}\Vert _{\mu }> \Vert u_{0}\Vert _{\mu }$$.

### Corollary 1.2

Let $$U_{\lambda , \mu ,\varepsilon } \in \mathcal{M}^{-}$$ be the solution of problem (1.1) with $$\gamma = \varepsilon +p-1$$, where $$\lambda \in (0,T_{\mu })$$. Then

\begin{aligned}& \Vert U_{\lambda , \mu ,\varepsilon } \Vert _{\mu }>C_{\mu ,\varepsilon } \biggl( \frac{T_{\mu }}{\lambda } \biggr)^{\frac{1}{\varepsilon }} \end{aligned}

with

$$C_{\mu ,\varepsilon }= \vert \varOmega \vert ^{\frac{1}{p}} \bigl( \Vert f \Vert _{\infty } \bigr) ^{\frac{1}{p+q-1}} \biggl(1+ \frac{p+q-1}{\varepsilon } \biggr)^{ \frac{1}{p+q-1}} \biggl(\frac{ \vert \varOmega \vert ^{\frac{2}{N}}}{\sqrt[p]{S _{\mu }}} \biggr)^{\frac{1-q}{p+q-1}} \rightarrow \infty , \quad \textit{as }\varepsilon \rightarrow 0.$$
(1.16)

### Theorem 1.3

There exists $$\lambda ^{*} =\lambda ^{*} (N, \varOmega , \mu , q, \gamma )>0$$ such that problem (1.1) with $$f=g=1$$ admits at least a positive solution for every $$0<\lambda <\lambda ^{*}$$ and has no solution for every $$\lambda >\lambda ^{*}$$.

## Some lemmas

### Lemma 2.1

Assume that $$\lambda \in (0,T_{\mu })$$, where $$T_{\mu }$$ is defined in (1.15). Then $$\mathcal{M}^{\pm }\neq \emptyset$$ and $$\mathcal{M}^{0}=\{0\}$$.

### Proof

(i) We can choose $$u^{*}\in \mathcal{M}\setminus \{0 \}$$ such that $$\int _{\varOmega }f(x)\vert u^{*}\vert ^{1-q}\,dx>0$$ and $$\int _{\varOmega }g(x) \vert u^{*}\vert ^{\gamma +1}\,dx>0$$ from the conditions imposed on f and g. Denote

\begin{aligned} \varphi _{\mu }(t) :=&\frac{1}{t^{\gamma }} \frac{d}{dt}I_{\lambda , \mu } \bigl(tu^{*} \bigr) \\ =&t^{p-1-\gamma } \bigl\Vert u^{*} \bigr\Vert _{\mu }^{p}-t^{-q- \gamma } \int _{\varOmega }f(x) \bigl\vert u^{*} \bigr\vert ^{1-q}\,dx- \lambda \int _{\varOmega }g(x) \bigl\vert u^{*} \bigr\vert ^{\gamma +1}\,dx, \quad t>0. \end{aligned}

Note that $$\varphi '_{\mu }(t)=(p-1-\gamma )t^{p-2-\gamma }\Vert u^{*}\Vert _{\mu }^{p}+(q+\gamma ) t^{-1-q-\gamma } \int _{\varOmega }f(x)\vert u^{*}\vert ^{1-q}\,dx$$. Let $$\varphi '_{\mu }(t)=0$$, we have

$$t:=t_{\max }= \biggl[\frac{(\gamma -p+1) \Vert u^{*} \Vert _{\mu }^{p}}{(q+ \gamma )\int _{\varOmega }f(x) \vert u^{*} \vert ^{1-q}\,dx} \biggr]^{\frac{1}{1-q-p}}.$$
(2.1)

It is easy to check that $$\varphi _{\mu }(t)\rightarrow -\infty$$ as $$t\rightarrow 0^{+}$$ and $$\varphi _{\mu }(t)\rightarrow -\lambda \int _{\varOmega } g(x)\vert u^{*}\vert ^{\gamma +1}\,dx<0$$ as $$t\rightarrow \infty$$. Furthermore, $$\varphi _{\mu }(t)$$ attains its maximum at $$t_{\max }$$. By (1.12) and (1.13), we obtain

\begin{aligned}& \varphi _{\mu }(t_{\max }) \\& \quad = \biggl[\frac{(\gamma -p+1) \Vert u^{*} \Vert _{\mu }^{p}}{(q+\gamma ) \int _{\varOmega }f(x) \vert u^{*} \vert ^{1-q}\,dx} \biggr]^{\frac{p-\gamma -1}{1-q-p}} \bigl\Vert u^{*} \bigr\Vert _{\mu }^{p} \\& \quad\quad{} - \biggl[\frac{(\gamma -p+1) \Vert u^{*} \Vert _{\mu }^{p}}{(q+ \gamma )\int _{\varOmega }f(x) \vert u^{*} \vert ^{1-q}\,dx} \biggr]^{ \frac{-q-\gamma }{1-q-p}} \int _{\varOmega }f(x) \bigl\vert u^{*} \bigr\vert ^{1-q}\,dx \\& \quad \quad {} -\lambda \int _{\varOmega } g(x) \bigl\vert u^{*} \bigr\vert ^{\gamma +1}\,dx \\& \quad = \biggl(\frac{\gamma -p+1}{q+\gamma } \biggr)^{ \frac{p-\gamma -1}{1-q-p}}\frac{( \Vert u^{*} \Vert _{\mu }^{p})^{\frac{- \gamma -q}{1-q-p}}}{(\int _{\varOmega }f(x) \vert u^{*} \vert ^{1-q}\,dx)^{\frac{p- \gamma -1}{1-q-p}}} \\& \quad\quad{} - \biggl( \frac{\gamma -p+1}{q+\gamma } \biggr) ^{\frac{-q- \gamma }{1-q-p}}\frac{( \Vert u^{*} \Vert _{\mu }^{p})^{ \frac{-\gamma -q}{1-q-p}}}{(\int _{\varOmega }f(x) \vert u^{*} \vert ^{1-q}\,dx)^{\frac{p- \gamma -1}{1-q-p}}} \\& \quad \quad {} -\lambda \int _{\varOmega } g(x) \bigl\vert u^{*} \bigr\vert ^{\gamma +1}\,dx \\& \quad = \biggl(\frac{q+p-1}{q+\gamma } \biggr) \biggl(\frac{\gamma -p+1}{q+ \gamma } \biggr) ^{\frac{p-\gamma -1}{1-q-p}}\frac{( \Vert u^{*} \Vert _{ \mu }^{p})^{\frac{-\gamma -q}{1-q-p}}}{(\int _{\varOmega }f(x) \vert u^{*} \vert ^{1-q}\,dx)^{\frac{p- \gamma -1}{1-q-p}}}-\lambda \int _{\varOmega }g(x) \bigl\vert u^{*} \bigr\vert ^{\gamma +1}\,dx \\& \quad \geq \biggl(\frac{q+p-1}{q+\gamma } \biggr) \biggl(\frac{\gamma -p+1}{q+ \gamma } \biggr) ^{\frac{p-\gamma -1}{1-q-p}}\frac{( \Vert u^{*} \Vert _{ \mu }^{p})^{\frac{-\gamma -q}{1-q-p}}}{ [ \Vert f \Vert _{\infty } \vert \varOmega \vert ^{\frac{p^{*}-1+q}{p^{*}}}(\frac{ \Vert u^{*} \Vert _{\mu }}{\sqrt[p]{S_{ \mu }}})^{1-q} ]^{\frac{p-\gamma -1}{1-q-p}}} \\& \quad \quad {} -\lambda \Vert g \Vert _{\infty } \vert \varOmega \vert ^{\frac{p^{*}-\gamma -1}{p^{*}}} \biggl( \frac{ \Vert u^{*} \Vert _{\mu }}{\sqrt[p]{S_{\mu }}} \biggr)^{\gamma +1} \\& \quad = \biggl(\frac{q+p-1}{q+\gamma } \biggr) \biggl(\frac{\gamma -p+1}{q+ \gamma } \biggr) ^{\frac{p-\gamma -1}{1-q-p}} \biggl(\frac{1}{ \Vert f \Vert _{\infty }} \biggr)^{\frac{p-\gamma -1}{1-q-p}} \frac{(\sqrt[p]{S_{ \mu }})^{\frac{(1-q)(p-\gamma -1)}{1-q-p}}}{ \vert \varOmega \vert ^{\frac{p^{*}-1+q}{p ^{*}}\frac{p-\gamma -1}{1-q-p}}} \bigl\Vert u^{*} \bigr\Vert _{\mu } ^{\gamma +1} \\& \quad \quad {} -\lambda \Vert g \Vert _{\infty } \vert \varOmega \vert ^{\frac{p^{*}-\gamma -1}{p ^{*}}} \biggl(\frac{ \Vert u^{*} \Vert _{\mu }}{\sqrt[p]{S_{\mu }}} \biggr) ^{\gamma +1} \\& \quad = \biggl[ \biggl(\frac{q+p-1}{q+\gamma } \biggr) \biggl(\frac{\gamma -p+1}{q+ \gamma } \biggr) ^{\frac{p-\gamma -1}{1-q-p}} \biggl(\frac{1}{ \Vert f \Vert _{\infty }} \biggr)^{\frac{p-\gamma -1}{1-q-p}} \frac{(\sqrt[p]{S_{ \mu }})^{\frac{(1-q)(p-\gamma -1)}{1-q-p}}}{ \vert \varOmega \vert ^{\frac{p^{*}-1+q}{p ^{*}}\frac{p-\gamma -1}{1-q-p}}} \\& \quad \quad {}-\lambda \Vert g \Vert _{\infty } \frac{ \vert \varOmega \vert ^{\frac{p^{*}-\gamma -1}{p^{*}}}}{({\sqrt[p]{S_{\mu }}})^{ \gamma +1}} \biggr] \bigl\Vert u^{*} \bigr\Vert _{\mu }^{\gamma +1} \\& \quad :=A(\mu ,\lambda ) \bigl\Vert u^{*} \bigr\Vert _{\mu }^{\gamma +1} \\& \quad >0. \end{aligned}
(2.2)

When $$A(\mu ,\lambda )=0$$, we get

\begin{aligned} \lambda &= \biggl(\frac{q+p-1}{q+\gamma } \biggr) \biggl( \frac{\gamma -p+1}{q+ \gamma } \biggr)^{\frac{p-\gamma -1}{1-q-p}} \biggl(\frac{1}{ \Vert f \Vert _{\infty }} \biggr)^{\frac{p-\gamma -1}{1-q-p}} \biggl(\frac{1}{ \Vert g \Vert _{\infty }} \biggr) \frac{(\sqrt[p]{S_{\mu }})^{\frac{(1-q)(p-\gamma -1)}{1-q-p}+\gamma +1}}{ \vert \varOmega \vert ^{\frac{p^{*}-1+q}{p^{*}}\frac{p- \gamma -1}{1-q-p} +\frac{p^{*}-\gamma -1}{p^{*}}}} \\ &= \biggl(\frac{q+p-1}{q+\gamma } \biggr) \biggl(\frac{\gamma -p+1}{q+ \gamma } \biggr)^{\frac{p-\gamma -1}{1-q-p}} \biggl(\frac{1}{ \Vert f \Vert _{\infty }} \biggr)^{\frac{p-\gamma -1}{1-q-p}} \biggl( \frac{1}{ \Vert g \Vert _{\infty }} \biggr) \biggl[\frac{S_{\mu }}{ \vert \varOmega \vert ^{\frac{2p}{N}}} \biggr] ^{\frac{q+\gamma }{p+q-1}} =T_{\mu }, \end{aligned}

where we use the following two equalities:

$$\frac{(1-q)(p-\gamma -1)}{1-q-p}+\gamma +1= \frac{p(q+\gamma )}{q+p-1},$$

and

$$\frac{(p^{*}-1+q)(p-\gamma -1)}{p^{*}(1-q-p)}+\frac{p^{*}-\gamma -1}{p ^{*}} =\frac{2p(q+\gamma )}{N(q+p-1)}.$$

In turn, this is also true. Hence $$A(\mu ,\lambda )=0$$ if and only if $$\lambda =T_{\mu }$$. Thus for $$\lambda \in (0,T_{\mu })$$, we have $$A(\mu ,\lambda )>0$$. Moreover, by (2.2), we derive that $$\varphi _{ \mu }(t_{\max })>0$$. Consequently, there exist two numbers $$t_{\mu } ^{-}$$ and $$t_{\mu }^{+}$$ such that $$0< t_{\mu }^{-}< t_{\max }< t_{ \mu }^{+}$$, and

$$\varphi _{\mu } \bigl(t^{-}_{\mu } \bigr)=0=\varphi _{\mu } \bigl(t^{+}_{\mu } \bigr), \quad\quad \varphi '_{\mu } \bigl(t^{-}_{\mu } \bigr)>0> \varphi '_{\mu } \bigl(t^{+}_{\mu } \bigr).$$

It follows that $$t_{\mu }^{-}u^{*}\in \mathcal{M}^{+}$$, and $$t_{\mu }^{+}u^{*}\in \mathcal{M}^{-}$$. In fact, if $$\varphi _{\mu }(t)=0$$, then

$$\varphi _{\mu }(t)=t^{p-1-\gamma } \Vert u \Vert _{\mu }^{p}-t^{-q-\gamma } \int _{\varOmega }f(x) \vert u \vert ^{1-q}\,dx-\lambda \int _{\varOmega } g(x) \vert u \vert ^{\gamma +1}\,dx=0,$$

namely

$$\Vert tu \Vert _{\mu }^{p}= \int _{\varOmega }f(x) \vert tu \vert ^{1-q}\,dx +\lambda \int _{\varOmega }g(x) \vert tu \vert ^{\gamma +1}\,dx.$$

Hence $$tu\in \mathcal{M}$$. Furthermore, if $$\varphi '_{\mu }(t)>0$$, then

$$(p-1-\gamma )t^{p-2-\gamma } \Vert u \Vert _{\mu }^{p} +(q+ \gamma )t^{-1-q- \gamma } \int _{\varOmega }f(x) \vert u \vert ^{1-q}\,dx>0.$$

That is

$$(p-1-\gamma ) \Vert tu \Vert _{\mu }^{p}+(q+\gamma ) \int _{\varOmega }f(x) \vert tu \vert ^{1-q}\,dx>0,$$

i.e.,

$$(p-1-\gamma ) \Vert tu \Vert _{\mu }^{p}+(q+\gamma ) \biggl[ \Vert tu \Vert _{\mu }^{p} - \lambda \int _{\varOmega }g(x) \vert tu \vert ^{\gamma +1}\,dx \biggr]>0.$$

Note that $$tu\in \mathcal{M}$$, we have

$$(p+q-1) \Vert tu \Vert _{\mu }^{p}-\lambda (q+\gamma ) \int _{\varOmega }g(x) \vert tu \vert ^{\gamma +1}\,dx>0.$$

Thus $$tu\in \mathcal{M}^{+}$$. By a similar argument, if $$\varphi _{ \mu }(t)=0$$ and $$\varphi '_{\mu }(t)<0$$, then $$tu\in \mathcal{M}^{-}$$. Therefore, both $$\mathcal{M}^{+}$$ and $$\mathcal{M}^{-}$$ are non-empty sets for every $$\lambda \in (0,T_{\mu })$$.

(ii) We claim that $$\mathcal{M}^{0}=\{0\}$$. Otherwise, we suppose that there exists $$u_{*}\in \mathcal{M}^{0}$$ and $$u_{*}\neq 0$$. Since $$u_{*}\in \mathcal{M}^{0}$$, we have

\begin{aligned}& (p+q-1 ) \Vert u_{*} \Vert ^{p}_{\mu }=\lambda ( \gamma +q ) \int _{\varOmega }g(x) \vert u_{*} \vert ^{\gamma +1} \,dx, \end{aligned}

moreover

\begin{aligned} 0&= \Vert u_{*} \Vert _{\mu }^{p}- \int _{\varOmega }f(x)u_{*}^{1-q}\,dx -\lambda \int _{\varOmega }g(x)u_{*}^{\gamma +1}\,dx \\ &= \Vert u_{*} \Vert _{\mu }^{p}- \int _{\varOmega }f(x)u_{*}^{1-q}\,dx - \frac{p+q-1}{\gamma +q} \Vert u_{*} \Vert _{\mu }^{p} \\ &=\frac{\gamma -p+1}{\gamma +q} \Vert u_{*} \Vert _{\mu } ^{p}- \int _{\varOmega } f(x)u_{*}^{1-q}\,dx. \end{aligned}

For $$\lambda \in (0,T_{\mu })$$ and $$u_{*}\neq 0$$, combining with (2.2), we deduce that

\begin{aligned} 0&< A(\mu ,\lambda ) \Vert u_{*} \Vert _{\mu }^{\gamma +1} \\ &\leq \biggl(\frac{q+p-1}{q+\gamma } \biggr) \biggl(\frac{\gamma -p+1}{q+ \gamma } \biggr) ^{\frac{p-\gamma -1}{1-q-p}}\frac{( \Vert u_{*} \Vert _{ \mu }^{p})^{\frac{-\gamma -q}{1-q-p}}}{(\frac{\gamma -p+1}{q+\gamma } \Vert u_{*} \Vert _{\mu }^{p})^{\frac{p-\gamma -1}{1-q-p}}} - \biggl(\frac{q+p-1}{q+ \gamma } \biggr) \Vert u_{*} \Vert _{\mu }^{p}=0, \end{aligned}

which is a contradiction, Thus $$u_{*}=0$$. That is, $$\mathcal{M}^{0}= \{0\}$$. □

The gap structure in $$\mathcal{M}$$ is embodied in the following lemma.

### Lemma 2.2

Assume that $$\lambda \in (0,T_{\mu })$$, then

\begin{aligned}& \Vert U \Vert _{\mu }>M_{\mu }(\lambda )>M_{\mu ,0}> \Vert u \Vert _{\mu }, \\& \Vert U \Vert _{\gamma +1}>N_{\mu }(\lambda )>N_{\mu ,0}> \Vert u \Vert _{\gamma +1}, \quad \forall u\in \mathcal{M}^{+}, U\in \mathcal{M}^{-}, \end{aligned}

where

\begin{aligned}& M_{\mu ,0}= \biggl[\frac{\gamma +q}{\gamma -p+1} \Vert f \Vert _{\infty } \frac{ \vert \varOmega \vert ^{\frac{p^{*}-1+q}{p^{*}}}}{(\sqrt[p]{S_{\mu }})^{1-q}} \biggr] ^{\frac{1}{p+q-1}}, \\& M_{\mu }(\lambda )= \biggl[\frac{p+q-1}{\lambda (\gamma +q)}\frac{1}{ \Vert g \Vert _{\infty }} \frac{(\sqrt[p]{S_{\mu }})^{\gamma +1}}{ \vert \varOmega \vert ^{\frac{p ^{*}-1-\gamma }{p^{*}}}} \biggr]^{\frac{1}{\gamma +1-p}}, \\& N_{\mu ,0}= \biggl[\frac{\gamma +q}{\gamma -p+1} \Vert f \Vert _{\infty } \frac{ \vert \varOmega \vert ^{\frac{\gamma +q}{\gamma +1}+\frac{(p^{*}-1-\gamma )p}{p^{*}( \gamma +1)}}}{S_{\mu }} \biggr]^{\frac{1}{p+q-1}}, \\& N_{\mu }(\lambda )= \biggl[\frac{p+q-1}{\lambda (\gamma +q)}\frac{1}{ \Vert g \Vert _{\infty }} \frac{S_{\mu }}{ \vert \varOmega \vert ^{p(\frac{p^{*}-1-\gamma }{p ^{*}(\gamma +1)})}} \biggr] ^{\frac{1}{\gamma +1-p}}. \end{aligned}

### Proof

If $$u\in \mathcal{M}^{+}\subset \mathcal{M}$$, then

\begin{aligned} 0&< (p+q-1 ) \Vert u \Vert ^{p}_{\mu }- \lambda (\gamma +q ) \int _{\varOmega }g(x) \vert u \vert ^{\gamma +1}\,dx \\ &= (p+q-1 ) \Vert u \Vert ^{p}_{\mu }- (\gamma +q ) \biggl[ \Vert u \Vert ^{p}_{\mu } - \int _{\varOmega }f(x) \vert u \vert ^{1-q}\,dx \biggr] \\ &= (p-\gamma -1 ) \Vert u \Vert ^{p}_{\mu }+ (\gamma +q ) \int _{\varOmega }f(x) \vert u \vert ^{1-q}\,dx. \end{aligned}

We obtain from (1.13) that

\begin{aligned} (\gamma -p+1 ) \Vert u \Vert ^{p}_{\mu }&< (\gamma +q ) \int _{\varOmega }f(x) \vert u \vert ^{1-q}\,dx \\ &\leq (\gamma +q ) \Vert f \Vert _{\infty } \vert \varOmega \vert ^{\frac{p^{*}-1+q}{p^{*}}} \biggl( \frac{ \Vert u \Vert _{\mu }}{\sqrt[p]{S _{\mu }}} \biggr)^{1-q}, \end{aligned}

\begin{aligned}& \Vert u \Vert _{\mu }< \biggl[\frac{\gamma +q}{\gamma -p+1} \Vert f \Vert _{\infty }\frac{ \vert \varOmega \vert ^{\frac{p^{*}-1+q}{p^{*}}}}{(\sqrt[p]{S_{\mu }})^{1-q}} \biggr] ^{\frac{1}{p+q-1}}=M_{\mu ,0}. \end{aligned}

By (1.12) and (1.14), we have

\begin{aligned} & (\gamma -p+1 ) \Vert u \Vert ^{p}_{\gamma +1} \frac{S_{\mu }}{ \vert \varOmega \vert ^{p(\frac{p^{*}-1-\gamma }{p^{*}(\gamma +1)})}} \\ &\quad \leq (\gamma -p+1 )\frac{S_{\mu }}{ \vert \varOmega \vert ^{p\frac{p^{*}-1-\gamma }{p^{*}(\gamma +1)}}} \biggl[ \vert \varOmega \vert ^{\frac{p ^{*}-1-\gamma }{p^{*}}} \biggl(\frac{ \Vert u \Vert _{\mu }}{\sqrt[p]{S_{\mu }}} \biggr) ^{\gamma +1} \biggr] ^{\frac{p}{\gamma +1}} \\ &\quad = (\gamma -p+1 ) \Vert u \Vert ^{p}_{\mu } \\ &\quad < (\gamma +q ) \int _{\varOmega }f(x) \vert u \vert ^{1-q}\,dx \\ &\quad \leq (\gamma +q ) \Vert f \Vert _{\infty } \vert \varOmega \vert ^{\frac{\gamma +q}{\gamma +1}} \Vert u \Vert ^{1-q} _{\gamma +1}, \end{aligned}

which implies that

\begin{aligned}& \Vert u \Vert _{\gamma +1}< \biggl[\frac{\gamma +q}{\gamma -p+1} \Vert f \Vert _{\infty }\frac{ \vert \varOmega \vert ^{\frac{\gamma +q}{\gamma +1}+\frac{(p^{*}-1-\gamma )p}{p^{*}(\gamma +1)}}}{S_{\mu }} \biggr]^{\frac{1}{p+q-1}}=N_{\mu ,0}. \end{aligned}

If $$U\in \mathcal{M}^{-}\subset \mathcal{M}$$, combining with (1.12), we derive that

\begin{aligned} (p+q-1 ) \Vert U \Vert ^{p}_{\mu }&< \lambda (\gamma +q ) \int _{\varOmega }g(x) \vert U \vert ^{\gamma +1}\,dx \\ &\leq \lambda (\gamma +q ) \Vert g \Vert _{ \infty } \vert \varOmega \vert ^{\frac{p^{*}-\gamma -1}{p^{*}}} \biggl( \frac{ \Vert U \Vert _{\mu }}{\sqrt[p]{S_{\mu }}} \biggr)^{\gamma +1}, \end{aligned}

\begin{aligned}& \Vert U \Vert _{\mu }> \biggl[\frac{p+q-1}{\lambda (\gamma +q)}\frac{1}{ \Vert g \Vert _{\infty }} \frac{(\sqrt[p]{S_{\mu }})^{\gamma +1}}{ \vert \varOmega \vert ^{\frac{p ^{*}-1-\gamma }{p^{*}}}} \biggr]^{\frac{1}{\gamma +1-p}}=M_{\mu }( \lambda ). \end{aligned}

Furthermore

\begin{aligned}& (p+q-1 ) \Vert U \Vert ^{p}_{\gamma +1} \frac{S_{\mu }}{ \vert \varOmega \vert ^{p(\frac{p ^{*}-1-\gamma }{p^{*}(\gamma +1)})}} \\& \quad \leq (p+q-1 )\frac{S_{\mu }}{ \vert \varOmega \vert ^{p\frac{p^{*}-1-\gamma }{p^{*}(\gamma +1)}}} \biggl[ \vert \varOmega \vert ^{\frac{p ^{*}-1-\gamma }{p^{*}}} \biggl(\frac{ \Vert U \Vert _{\mu }}{\sqrt[p]{S_{ \mu }}} \biggr) \biggr]^{\frac{p}{\gamma +1}} \\& \quad = (p+q-1 ) \Vert U \Vert ^{p}_{\mu } \\& \quad < \lambda (\gamma +q ) \int _{\varOmega }g(x) \vert U \vert ^{\gamma +1}\,dx \\& \quad \leq \lambda (\gamma +q ) \Vert g \Vert _{ \infty } \Vert U \Vert ^{\gamma +1} _{\gamma +1}, \end{aligned}

which means that

\begin{aligned}& \Vert U \Vert _{\gamma +1}> \biggl[\frac{p+q-1}{\lambda (\gamma +q)}\frac{1}{ \Vert g \Vert _{\infty }} \frac{S_{\mu }}{ \vert \varOmega \vert ^{p(\frac{p^{*}-1-\gamma }{p ^{*}(\gamma +1)})}} \biggr] ^{\frac{1}{\gamma +1-p}}=N_{\mu }(\lambda ). \end{aligned}

Therefore

\begin{aligned}& \lambda =T_{\mu }= \biggl( \frac{q+p-1}{q+\gamma } \biggr) \biggl(\frac{ \gamma -p+1}{q+\gamma } \biggr)^{\frac{p-\gamma -1}{1-q-p}} \biggl(\frac{1}{ \Vert f \Vert _{\infty }} \biggr)^{\frac{p-\gamma -1}{1-q-p}} \biggl(\frac{1}{ \Vert g \Vert _{\infty }} \biggr) \biggl(\frac{S_{\mu }}{ \vert \varOmega \vert ^{ \frac{2p}{N}}} \biggr)^{\frac{q+\gamma }{p+q-1}} \\& \begin{aligned} \Leftrightarrow \quad M_{\mu }(\lambda )&= \biggl[\frac{p+q-1}{\lambda ( \gamma +q)} \frac{1}{ \Vert g \Vert _{\infty }} \frac{(\sqrt[p]{S_{\mu }})^{ \gamma +1}}{ \vert \varOmega \vert ^{\frac{p^{*}-1-\gamma }{p^{*}}}} \biggr]^{\frac{1}{ \gamma +1-p}} \\ & =\lambda ^{-\frac{1}{\gamma +1-p}} \biggl[\frac{p+q-1}{\gamma +q}\frac{1}{ \Vert g \Vert _{\infty }} \frac{(\sqrt[p]{S_{\mu }}) ^{\gamma +1}}{ \vert \varOmega \vert ^{\frac{p ^{*}-1-\gamma }{p^{*}}}} \biggr]^{\frac{1}{\gamma +1-p}} \\ & = \biggl(\frac{q+\gamma }{q+p-1} \biggr) ^{ \frac{1}{\gamma +1-p}} \biggl(\frac{q+\gamma }{\gamma -p+1} \biggr) ^{\frac{1}{p+q-1}} \bigl( \Vert f \Vert _{\infty } \bigr)^{\frac{1}{p+q-1}} \bigl( \Vert g \Vert _{\infty } \bigr)^{\frac{1}{\gamma +1-p}} \\ & \quad{}\times \frac{ \vert \varOmega \vert ^{\frac{2p}{N}\frac{q+\gamma }{(q+p-1)(\gamma +1-p)}}}{(S_{\mu })^{\frac{q+ \gamma }{(p+q-1) (\gamma +1-p)}}} \biggl[\frac{p+q-1}{\gamma +q}\frac{1}{ \Vert g \Vert _{\infty }}\frac{(\sqrt[p]{S _{\mu }}) ^{\gamma +1}}{ \vert \varOmega \vert ^{\frac{p^{*}-1-\gamma }{p^{*}}}} \biggr] ^{\frac{1}{\gamma +1-p}} \\ & = \biggl(\frac{q+\gamma }{\gamma -p+1} \biggr)^{\frac{1}{p+q-1}} \bigl( \Vert f \Vert _{\infty } \bigr)^{\frac{1}{p+q-1}}\frac{ \vert \varOmega \vert ^{ \frac{2p}{N} \frac{q+\gamma }{{(\gamma -p+1)(p+q-1)}}-\frac{p^{*}-1- \gamma }{p^{*}}}}{(\sqrt[p]{S_{\mu }})^{p \frac{q+\gamma }{(\gamma -p+1)(p+q-1)} -\frac{\gamma +1}{\gamma +1-p}}} \\ & = \biggl[\frac{\gamma +q}{\gamma -p+1} \Vert f \Vert _{\infty } \frac{ \vert \varOmega \vert ^{\frac{p^{*}-1+q}{p^{*}}}}{(\sqrt[p]{S_{\mu }})^{1-q}} \biggr] ^{ \frac{1}{p+q-1}}=M_{\mu ,0}, \end{aligned} \end{aligned}

where we have used the following facts:

\begin{aligned} &\frac{2p}{N}\frac{q+\gamma }{{(\gamma -p+1)(p+q-1)}}- \frac{p^{*}-1- \gamma }{p^{*} (\gamma -p+1)} \\ &\quad =\frac{2p(p^{*}-p)}{2pp^{*}}\frac{q+\gamma }{ {(\gamma -p+1)(p+q-1)}} -\frac{p^{*}-1-\gamma }{p^{*}(\gamma -p+1)} \\ &\quad = \frac{(\gamma -p+1)(p^{*}+q-1)}{p^{*}(\gamma -p+1)(p+q-1)}, \end{aligned}

and

$$p\frac{q+\gamma }{(\gamma -p+1)(p+q-1)}- \frac{\gamma +1}{\gamma +1-p} =\frac{pq-q\gamma +\gamma -p-q+1}{( \gamma -p+1)(p+q-1)}= \frac{1-q}{p+q-1}.$$

Similarly

\begin{aligned}& \lambda =T_{\mu }= \biggl( \frac{q+p-1}{q+\gamma } \biggr) \biggl(\frac{ \gamma -p+1}{q+\gamma } \biggr)^{\frac{p-\gamma -1}{1-q-p}} \biggl(\frac{1}{ \Vert f \Vert _{\infty }} \biggr)^{\frac{p-\gamma -1}{1-q-p}} \biggl(\frac{1}{ \Vert g \Vert _{\infty }} \biggr) \biggl[\frac{S_{\mu }}{ \vert \varOmega \vert ^{ \frac{2p}{N}}} \biggr]^{\frac{q+\gamma }{p+q-1}}. \\& \Leftrightarrow \quad N_{\mu }(\lambda )= \biggl[\frac{p+q-1}{\lambda ( \gamma +q)} \frac{1}{ \Vert g \Vert _{\infty }} \frac{S_{\mu }}{ \vert \varOmega \vert ^{p(\frac{p ^{*}-1-\gamma }{p^{*}(\gamma +1)})}} \biggr] ^{\frac{1}{\gamma +1-p}} \\& \hphantom{\Leftrightarrow \quad N_{\mu }(\lambda )} =\lambda ^{-\frac{1}{\gamma +1-p}} \biggl[\frac{p+q-1}{\lambda ( \gamma +q)}\frac{1}{ \Vert g \Vert _{\infty }} \frac{S_{\mu }}{ \vert \varOmega \vert ^{p(\frac{p ^{*}-1-\gamma }{p^{*}(\gamma +1)})}} \biggr] ^{\frac{1}{\gamma +1-p}} \\& \hphantom{\Leftrightarrow \quad N_{\mu }(\lambda )} = \biggl(\frac{q+\gamma }{q+p-1} \biggr)^{ \frac{1}{\gamma +1-p}} \biggl(\frac{q+\gamma }{\gamma -p+1} \biggr) ^{\frac{1}{p+q-1}} \bigl( \Vert f \Vert _{\infty } \bigr)^{\frac{1}{p+q-1}} \bigl( \Vert g \Vert _{\infty } \bigr)^{\frac{1}{\gamma +1-p}} \\& \hphantom{\Leftrightarrow \quad N_{\mu }(\lambda )} \quad{}\times \frac{ \vert \varOmega \vert ^{\frac{2p}{N}\frac{q+\gamma }{(q+p-1)(\gamma +1-p)}}}{(S_{\mu })^{\frac{q+ \gamma }{(p+q-1)(\gamma +1-p)}}} \biggl[\frac{p+q-1}{\lambda (\gamma +q)}\frac{1}{ \Vert W \Vert _{ \infty }} \frac{S_{\mu }}{ \vert \varOmega \vert ^{p(\frac{p^{*}-1-\gamma }{p^{*}( \gamma +1)})}} \biggr] ^{\frac{1}{\gamma +1-p}} \\& \hphantom{\Leftrightarrow \quad N_{\mu }(\lambda )} = \biggl(\frac{q+\gamma }{\gamma -p+1} \biggr)^{\frac{1}{p+q-1}} \bigl( \Vert f \Vert _{\infty } \bigr)^{\frac{1}{p+q-1}}\frac{ \vert \varOmega \vert ^{ \frac{2p}{N} \frac{q+\gamma }{(\gamma -p+1)(p+q-1)}-p\frac{p^{*}-1- \gamma }{p^{*}(\gamma +1) (\gamma +1-p)}}}{(S_{\mu })^{\frac{q+\gamma }{(\gamma -p+1)(p+q-1)} -\frac{1}{\gamma +1-p}}} \\& \hphantom{\Leftrightarrow \quad N_{\mu }(\lambda )} = \biggl[\frac{\gamma +q}{\gamma -p+1} \Vert f \Vert _{\infty }\frac{ \vert \varOmega \vert ^{\frac{\gamma +q}{\gamma +1}+\frac{(p^{*}-1-\gamma )p}{p^{*}(\gamma +1)}}}{S_{\mu }} \biggr]^{\frac{1}{p+q-1}}=N_{\mu ,0}, \end{aligned}

where we have applied the following equalities:

\begin{aligned} &\frac{2p}{N}\frac{q+\gamma }{(\gamma -p+1)(p+q-1)}-p \frac{p^{*}-1- \gamma }{p^{*}(\gamma +1)(\gamma +1-p)} \\ &\quad =\frac{2p(p^{*}-p)}{2pp^{*}}\frac{q+\gamma }{ {(\gamma -p+1)(p+q-1)}} -\frac{p^{*}-1-\gamma }{p^{*}(\gamma -p+1)} \\ &\quad =\frac{\gamma +q}{\gamma +1}+p\frac{p^{*}-1- \gamma }{p^{*}(\gamma +1)}, \end{aligned}

and

$$\frac{q+\gamma }{(\gamma -p+1)(p+q-1)}-\frac{1}{\gamma +1-p} =\frac{q+ \gamma -(p+q-1)}{(\gamma -p+1)(p+q-1)}= \frac{1}{p+q-1}.$$

Consequently, $$M_{\mu }(\lambda )=M_{\mu ,0}$$ if and only if $$\lambda =T_{\mu }$$ and $$N_{\mu }(\lambda )=N_{\mu ,0}$$ if and only if $$\lambda =T_{\mu }$$ respectively. This completes the proof of Lemma 2.2. □

### Lemma 2.3

Assume that $$\lambda \in (0,T_{\mu })$$. Then $$\mathcal{M}^{-}$$ is a closed set in W-topology.

### Proof

We choose a sequence $$\{U_{n}\}$$ such that $$\{U_{n}\} \subset \mathcal{M}^{-}$$ and $$U_{n}\rightarrow U_{0}$$ with $$U_{0} \in W$$. Then

\begin{aligned} \Vert U_{0} \Vert _{\mu }^{p}&= \lim_{n\rightarrow \infty } \Vert U_{n} \Vert _{ \mu }^{p} \\ &=\lim_{n\rightarrow \infty } \biggl[ \int _{\varOmega }f(x) \vert U_{n} \vert ^{1-q}\,dx+ \lambda \int _{\varOmega } g(x) \vert U_{n} \vert ^{\gamma +1}\,dx \biggr] \\ &= \int _{\varOmega }f(x) \vert U_{0} \vert ^{1-q}\,dx+ \lambda \int _{\varOmega } g(x) \vert U_{0} \vert ^{\gamma +1}\,dx, \end{aligned}

and

\begin{aligned} & (p+q-1 ) \Vert U_{0} \Vert ^{p}_{\mu }-\lambda (\gamma +q ) \int _{\varOmega }g(x) \vert U_{0} \vert ^{\gamma +1}\,dx \\ &\quad =\lim_{n\rightarrow \infty } \biggl[ (p+q-1 ) \Vert U _{n} \Vert ^{p}_{\mu }-\lambda (\gamma +q ) \int _{\varOmega }g(x) \vert U_{n} \vert ^{\gamma +1}\,dx \biggr]\leq 0. \end{aligned}

Hence $$U_{0} \in \mathcal{M}^{-} \cup \mathcal{M}^{0}$$. By Lemma 2.2, we have

$$\Vert U_{0} \Vert _{\mu }=\lim_{n\rightarrow \infty } \Vert U_{n} \Vert _{\mu } \geq M_{\mu ,0}>0,$$

that is, $$U_{0}\neq 0$$. Combining with Lemma 2.1, we obtain $$U_{0}\notin \mathcal{M}^{0}$$. Thus $$U_{0}\in \mathcal{M}^{-}$$. Therefore $$\mathcal{M}^{-}$$ is a closed set in W-topology for every $$\lambda \in (0,T_{\mu })$$. □

### Lemma 2.4

For $$u\in \mathcal{M}^{\pm }$$, there exist a number $$\varepsilon >0$$ and a continuous function $$\widetilde{g}(h)>0$$ with $$h\in W$$ and $$\Vert h\Vert <\varepsilon$$ such that

$$\widetilde{g}(0)=1, \quad\quad \widetilde{g}(h) (u+h)\in \mathcal{M}^{\pm }, \quad \forall h\in W, \Vert h \Vert < \varepsilon .$$

### Proof

We only prove the case that $$\mathcal{M}^{+}$$. Define a function $$\widetilde{F}: W\times \mathbb{R}^{+}\rightarrow \mathbb{R}$$ by:

\begin{aligned}& \widetilde{F}(h,s)=s^{p-1+q} \Vert u+h \Vert _{\mu }^{p}- \int _{\varOmega }f(x) \vert u+h \vert ^{1-q}\,dx-\lambda s^{\gamma +q} \int _{\varOmega }g(x) \vert u+h \vert ^{\gamma +1}\,dx. \end{aligned}

Note that $$u\in \mathcal{M}^{+}$$, we obtain

\begin{aligned}& \widetilde{F}(0,1)= \Vert u \Vert _{\mu }^{p}- \int _{\varOmega }f(x) \vert u \vert ^{1-q}\,dx -\lambda \int _{\varOmega }g(x) \vert u \vert ^{\gamma +1}\,dx=0, \end{aligned}

and

$$\widetilde{F}_{s}(0,1)=(p-1+q) \Vert u \Vert _{\mu }^{p}-(q+\gamma )\lambda \int _{\varOmega }g(x) \vert u \vert ^{\gamma +1}\,dx>0.$$
(2.3)

At $$(0,1)$$, using the implicit function theorem, we know that there exists $$\overline{\varepsilon }>0$$ such that for $$h\in W$$ and $$\Vert h\Vert <\overline{\varepsilon }$$, the equation $$\widetilde{F}(h,s)=0$$ has a unique continuous solution $$s=\widetilde{g}(h)>0$$. Hence $$\widetilde{g}(0)=1$$ and

\begin{aligned} 0&=\widetilde{g}(h)^{p-1+q} \Vert u+h \Vert _{\mu }^{p}- \int _{\varOmega }f(x) \vert u+h \vert ^{1-q}\,dx-\lambda \widetilde{g}(h)^{\gamma +q} \int _{\varOmega } g(x) \vert u+h \vert ^{\gamma +1}\,dx \\ &=\frac{ \Vert \widetilde{g}(h)(u+h) \Vert _{\mu }^{p}- \int _{\varOmega }f(x) \vert \widetilde{g}(h)(u+h) \vert ^{1-q}\,dx -\lambda \int _{\varOmega }g(x) \vert \widetilde{g}(h)(u+h) \vert ^{\gamma +1}\,dx}{ \widetilde{g}(h)^{1-q}}, \end{aligned}

i.e.,

\begin{aligned}& \widetilde{g}(h) (u+h)\in \mathcal{M}, \quad \forall h\in W, \Vert h \Vert < \overline{ \varepsilon }, \end{aligned}

and

\begin{aligned} \widetilde{F}_{s} \bigl(h, \widetilde{g}(h) \bigr)&=(p-1+q)\widetilde{g}(h)^{p+q-2} \Vert u+h \Vert _{\mu }^{p}-(q+ \gamma )\lambda \widetilde{g}(h)^{\gamma +q-1} \int _{\varOmega } g(x) \vert u+h \vert ^{\gamma +1}\,dx \\ &=\frac{(p-1+q) \Vert \widetilde{g}(h)(u+h) \Vert _{\mu }^{p}-(q+\gamma ) \lambda \int _{\varOmega } g(x) \vert \widetilde{g}(h)(u+h) \vert ^{\gamma +1}\,dx}{ \widetilde{g}^{2-q}(h)}, \end{aligned}

together with (2.3), these imply that we can choose $$\varepsilon >0$$ small enough ($$\varepsilon <\overline{\varepsilon }$$) such that for every $$h\in W$$ and $$\Vert h\Vert <\varepsilon$$

$$(p-1+q) \bigl\Vert \widetilde{g}(h) (u+h) \bigr\Vert _{\mu }^{p}-(q+ \gamma )\lambda \int _{\varOmega } g(x) \bigl\vert \widetilde{g}(h) (u+h) \bigr\vert ^{\gamma +1}\,dx>0,$$

that is,

$$\widetilde{g}(h) (u+h)\in \mathcal{M}^{+}, \quad \forall h\in W, \Vert h \Vert < \varepsilon .$$

This completes the proof of Lemma 2.3. □

## Proof of Theorem 1.1

For every $$u\in \mathcal{M}$$, by (1.13), we have

\begin{aligned} I_{\lambda ,\mu }(u)&=\frac{1}{p} \Vert u \Vert _{\mu }^{p}-\frac{1}{1-q} \int _{\varOmega }f(x) \vert u \vert ^{1-q}\,dx - \frac{\lambda }{\gamma +1} \int _{\varOmega }g(x) \vert u \vert ^{\gamma +1}\,dx \\ &=\frac{1}{p} \Vert u \Vert _{\mu }^{p}- \frac{1}{1-q} \int _{\varOmega }f(x) \vert u \vert ^{1-q}\,dx- \frac{1}{\gamma +1} \biggl[ \Vert u \Vert _{\mu } ^{p}- \int _{\varOmega }f(x)u^{1-q}\,dx \biggr] \\ &= \biggl(\frac{1}{p}- \frac{1}{\gamma +1} \biggr) \Vert u \Vert _{\mu }^{p}- \biggl(\frac{1}{1-q}-\frac{1}{\gamma +1} \biggr) \int _{\varOmega }f(x)u^{1-q}\,dx \\ &\geq \biggl(\frac{1}{p}-\frac{1}{\gamma +1} \biggr) \Vert u \Vert _{\mu } ^{p}- \biggl(\frac{1}{1-q} -\frac{1}{\gamma +1} \biggr) \Vert f \Vert _{\infty } \frac{ \vert \varOmega \vert ^{\frac{p^{*}-1+q}{p^{*}}}}{(\sqrt[p]{S_{\mu }})^{1-q}} \Vert u \Vert _{\mu }^{1-q} \\ &:=\mathcal{K}\bigl( \Vert u \Vert _{\mu }\bigr). \end{aligned}
(3.1)

Let

$$\mathcal{K}'\bigl( \Vert u \Vert _{\mu }\bigr)= \biggl(1- \frac{p}{\gamma +1} \biggr) \Vert u \Vert _{\mu }^{p-1}- \biggl(1- \frac{1-q}{\gamma +1} \biggr) \Vert f \Vert _{\infty } \frac{ \vert \varOmega \vert ^{\frac{p^{*}-1+q}{p^{*}}}}{(\sqrt[p]{S_{\mu }})^{1-q}} \Vert u \Vert _{\mu }^{-q}=0.$$

We have

$$\Vert u \Vert _{\mu }:=\bigl( \Vert u \Vert _{\mu } \bigr)_{\min }= \biggl[\frac{(1-\frac{1-q}{ \gamma +1}) \Vert f \Vert _{\infty } \frac{ \vert \varOmega \vert ^{\frac{p^{*}-1+q}{p^{*}}}}{(\sqrt[p]{S _{\mu }}) ^{1-q}}}{1-\frac{p}{\gamma +1}} \biggr]^{\frac{1}{p+q-1}}.$$

Since $$\mathcal{K}''(\Vert u\Vert _{\mu })>0$$ for all $$\Vert u\Vert _{\mu }>0$$ with $$\mathcal{K}(\Vert u\Vert _{\mu })\rightarrow 0$$ as $$\Vert u\Vert _{\mu }\rightarrow 0$$ and $$\mathcal{K}(\Vert u\Vert _{\mu })\rightarrow \infty$$ as $$\Vert u\Vert _{ \mu }\rightarrow \infty$$. Therefore $$\mathcal{K}(u)$$ attains its minimum at $$(\Vert u\Vert _{\mu })_{\min }$$, and

\begin{aligned} \mathcal{K} \bigl(\bigl( \Vert u \Vert _{\mu }\bigr)_{\min } \bigr)&= \biggl(\frac{1}{p}- \frac{1}{ \gamma +1} \biggr) \biggl[ \frac{(1-\frac{1-q}{\gamma +1}) \Vert f \Vert _{ \infty }\frac{ \vert \varOmega \vert ^{\frac{p^{*}-1+q}{p^{*}}}}{(\sqrt[p]{S_{ \mu }})^{1-q}}}{1-\frac{p}{\gamma +1}} \biggr]^{\frac{p}{p+q-1}} \\ &\quad {} - \biggl(\frac{1}{1-q}-\frac{1}{\gamma +1} \biggr) \Vert f \Vert _{\infty }\frac{ \vert \varOmega \vert ^{\frac{p^{*}-1+q}{p^{*}}}}{(\sqrt[p]{S _{\mu }})^{1-q}} \biggl[\frac{(1-\frac{1-q}{\gamma +1}) \Vert f \Vert _{\infty } \frac{ \vert \varOmega \vert ^{\frac{p^{*}-1+q}{p^{*}}}}{(\sqrt[p]{S_{\mu }})^{1-q}}}{1-\frac{p}{ \gamma +1}} \biggr] ^{\frac{1-q}{p+q-1}}. \end{aligned}

By (3.1), we deduce that

\begin{aligned}& \lim_{ \Vert u \Vert _{\mu }\rightarrow \infty }I_{\lambda ,\mu }(u) \geq \lim_{ \Vert u \Vert _{\mu }\rightarrow \infty } \mathcal{K}\bigl( \Vert u \Vert _{\mu }\bigr)=\infty , \end{aligned}

namely, $$I_{\lambda ,\mu }(u)$$ is coercive on $$\mathcal{M}$$. Combining with (3.1), we have

$$I_{\lambda ,\mu }(u)\geq \mathcal{K}(u)\geq \mathcal{K} \bigl( \bigl( \Vert u \Vert _{ \mu }\bigr)_{\min } \bigr).$$
(3.2)

Thus $$I_{\lambda ,\mu }(u)$$ is bounded below on $$\mathcal{M}$$. According to Lemma 2.3, if $$\lambda \in (0,T_{\mu })$$, then $$\mathcal{M}^{+} \cup \mathcal{M}^{0}$$ and $$\mathcal{M}^{-}$$ are two closed sets in $$\mathcal{M}$$. Therefore, we apply the Ekeland variational principle  to derive a minimizing sequence $$\{u_{n}\}\subset \mathcal{M}^{+}\cup \mathcal{M}^{0}$$ satisfying:

\begin{aligned}& (\mathrm{i}) \quad I_{\lambda ,\mu }(u_{n})< \inf _{\mathcal{M}^{+}\cup \mathcal{M}^{0}}I_{\lambda ,\mu }(u)+ \frac{1}{n}; \\& (\mathrm{ii}) \quad I_{\lambda ,\mu }(u)\geq I_{\lambda ,\mu }(u_{n})- \frac{1}{n} \Vert u-u _{n} \Vert , \quad \forall u\in \mathcal{M}^{+}\cup \mathcal{M}^{0}. \end{aligned}

Assume that $$u_{n}\geq 0$$ on $$\varOmega \setminus \{0\}$$. Note that $$I_{\lambda ,\mu }(u)$$ is bounded below on $$\mathcal{M}$$. By (3.2), we get

$$\mathcal{K} \bigl(\bigl( \Vert u_{n} \Vert _{\mu }\bigr)_{\min } \bigr)\leq I_{\lambda ,\mu }(u_{n})< \inf_{\mathcal{M}^{+}\cup \mathcal{M}^{0}}I_{\lambda ,\mu }(u)+ \frac{1}{n}\leq C_{1},$$
(3.3)

for n large enough and a positive constant $$C_{1}$$. Hence $$\{u_{n}\}$$ is bounded in $$\mathcal{M}$$. Let us, for a subsequence, suppose that

$$\textstyle\begin{cases} u_{n}\rightharpoonup u_{0} &\text{in }W, \\ u_{n}(x)\rightarrow u_{0}(x) & \text{a.e. in } \varOmega , \\ u_{n}\rightarrow u_{0} &\text{in } L^{1-q}(\varOmega ) \text{ and } L^{\gamma +1}(\varOmega ). \end{cases}$$

For every $$u\in \mathcal{M}^{+}$$, we deduce from $$p>1$$ that

\begin{aligned} I_{\lambda ,\mu }(u)&=\frac{1}{p} \Vert u \Vert _{\mu }^{p}-\frac{1}{1-q} \int _{\varOmega }f(x) \vert u \vert ^{1-q}\,dx - \frac{\lambda }{\gamma +1} \int _{\varOmega }g(x) \vert u \vert ^{\gamma +1}\,dx \\ &=\frac{1}{p} \Vert u \Vert _{\mu }^{p}- \frac{1}{1-q} \biggl[ \Vert u \Vert _{\mu }^{p} - \lambda \int _{\varOmega }g(x) \vert u \vert ^{\gamma +1}\,dx \biggr]- \frac{\lambda }{\gamma +1} \int _{\varOmega }g(x) \vert u \vert ^{\gamma +1}\,dx \\ & = \biggl(\frac{1}{p}-\frac{1}{1-q} \biggr) \Vert u \Vert _{\mu }^{p}+ \biggl(\frac{1}{1-q}- \frac{1}{ \gamma +1} \biggr)\lambda \int _{\varOmega }g(x) \vert u \vert ^{\gamma +1}\,dx \\ &< \biggl(\frac{1}{p}-\frac{1}{1-q} \biggr) \Vert u \Vert _{\mu }^{p} + \biggl(\frac{1}{1-q}-\frac{1}{ \gamma +1} \biggr)\frac{p+q-1}{\gamma +q} \Vert u \Vert _{\mu }^{p} \\ &=\frac{p+q-1}{\gamma +q} \biggl(\frac{1}{\gamma +1}-\frac{1}{p} \biggr) \Vert u \Vert _{\mu }^{p}< 0, \end{aligned}

which implies that $$\inf_{\mathcal{M}^{+}}I_{\lambda ,\mu }(u)<0$$. For $$\lambda \in (0,T_{\mu })$$, it follows from Lemma 2.1 that $$\mathcal{M}^{0}=\{0 \}$$. Thus $$u_{n}\in \mathcal{M}^{+}$$ for n large enough and $$\inf_{\mathcal{M}^{+}\cup \mathcal{M}^{0}}I_{\lambda ,\mu }(u) =\inf_{\mathcal{M}^{+}}I_{\lambda ,\mu }(u)<0$$. Therefore

\begin{aligned}& I_{\lambda ,\mu }(u_{0})\leq \liminf_{n\rightarrow \infty }I _{\lambda ,\mu }(u_{n}) =\inf_{\mathcal{M}^{+}\cup \mathcal{M} ^{0}}I_{\lambda ,\mu }< 0, \end{aligned}

i.e., $$u_{0}\geq 0$$ and $$u_{0}\neq 0$$.

In the following, we prove that, when $$\lambda \in (0,T_{\mu })$$,

$$(p+q-1 ) \int _{\varOmega }f(x)u_{0}^{1-q}\,dx >\lambda ( \gamma -q+1 ) \int _{\varOmega }g(x)u_{0}^{\gamma +1}\,dx.$$
(3.4)

For $$\{u_{n}\}\subset \mathcal{M}^{+}$$, we have

\begin{aligned}& (p+q-1 ) \int _{\varOmega }f(x)u_{0}^{1-q}\,dx -\lambda ( \gamma -p+1 ) \int _{\varOmega }g(x)u_{0}^{\gamma +1}\,dx \\& \quad =\lim_{n\rightarrow \infty } \biggl[ (p+q-1 ) \int _{\varOmega }f(x)u_{n}^{1-q}\,dx-\lambda ( \gamma -p+1 ) \int _{\varOmega }g(x)u_{n}^{\gamma +1}\,dx \biggr] \\& \quad =\lim_{n\rightarrow \infty } \biggl\{ (p+q-1 ) \biggl[ \Vert u_{n} \Vert _{\mu }^{p}-\lambda \int _{\varOmega }g(x)u_{n}^{\gamma +1}\,dx \biggr]- \lambda (\gamma -p+1 ) \int _{\varOmega }g(x)u_{n}^{\gamma +1}\,dx \biggr\} \\& \quad =\lim_{n\rightarrow \infty } \biggl[ (p+q-1 ) \Vert u _{n} \Vert _{\mu }^{p}-\lambda (\gamma +q ) \int _{\varOmega }g(x)u_{n} ^{\gamma +1}\,dx \biggr] \geq 0. \end{aligned}

We suppose that

$$(p+q-1 ) \int _{\varOmega }f(x)u_{0}^{1-q}\,dx-\lambda ( \gamma -p+1 ) \int _{\varOmega }g(x)u_{0}^{\gamma +1}\,dx=0.$$
(3.5)

It follows from $$u_{n}\in \mathcal{M}$$, the weak lower semi-continuity of the norm and (3.5) that

\begin{aligned} 0&=\lim_{n\rightarrow \infty } \biggl[ \Vert u_{n} \Vert _{\mu }^{p}- \int _{\varOmega } f(x)u_{n}^{1-q}\,dx-\lambda \int _{\varOmega }g(x)u_{n}^{\gamma +1}\,dx \biggr] \\ &\geq \Vert u_{0} \Vert _{\mu }^{p}- \int _{\varOmega } f(x)u_{0}^{1-q}\,dx-\lambda \int _{\varOmega }g(x)u_{0}^{\gamma +1}\,dx \\ &= \textstyle\begin{cases} \Vert u_{0} \Vert _{\mu }^{p}-\lambda \frac{\gamma +q}{p+q-1} \int _{\varOmega } g(x)u_{0}^{\gamma +1}\,dx, \\ \Vert u_{0} \Vert _{\mu }^{p}-\lambda \frac{\gamma +q}{ \gamma -p+1} \int _{\varOmega } f(x)u_{0}^{1-q}\,dx. \end{cases}\displaystyle \end{aligned}

Hence, for every $$\lambda \in (0,T_{\mu })$$ and $$u_{0}\neq 0$$, combining with (2.2), we obtain

\begin{aligned} 0&< A(\mu ,\lambda ) \Vert u_{0} \Vert _{\mu }^{\gamma +1} \\ &\leq \biggl(\frac{q+p-1}{q+\gamma } \biggr) \biggl(\frac{\gamma -p+1}{q+ \gamma } \biggr) ^{\frac{p-\gamma -1}{1-q-p}}\frac{( \Vert u_{0} \Vert _{ \mu }^{p})^{\frac{-\gamma -q}{1-q-p}}}{(\int _{\varOmega }f(x) \vert u_{0} \vert ^{1-q}\,dx)^{\frac{p- \gamma -1}{1-q-p}}}-\lambda \int _{\varOmega }g(x) \vert u_{0} \vert ^{\gamma +1}\,dx \\ &\leq \biggl(\frac{q+p-1}{q+\gamma } \biggr) \biggl(\frac{\gamma -p+1}{q+ \gamma } \biggr) ^{\frac{p-\gamma -1}{1-q-p}}\frac{( \Vert u_{0} \Vert _{ \mu }^{p})^{\frac{-\gamma -q}{1-q-p}}}{(\frac{\gamma -p+1}{q+\gamma } \Vert u_{0} \Vert _{\mu }^{p})^{\frac{p-\gamma -1}{1-q-p}}} -\frac{p+q-1}{ \gamma +q} \Vert u_{0} \Vert _{\mu }^{p}=0, \end{aligned}

which is a contradiction. In view of (3.4), we get

$$(p+q-1 ) \int _{\varOmega }f(x)u_{n}^{1-q}\,dx-\lambda ( \gamma -p+1 ) \int _{\varOmega }g(x)u_{n}^{\gamma +1}\,dx\geq C_{2}$$
(3.6)

for n large enough and some positive constant $$C_{2}$$. Since $$u_{n}\in \mathcal{M}$$, we have

$$(p+q-1 ) \Vert u_{n} \Vert _{\mu }^{p}- \lambda (\gamma +q ) \int _{\varOmega }g(x)u_{n}^{\gamma +1}\,dx\geq C_{2}>0.$$
(3.7)

Set $$\phi \in \mathcal{M}$$ with $$\phi \geq 0$$. Using Lemma 2.4, there exists $$\widetilde{g}_{n}(t)$$ such that $$\widetilde{g}_{n}(0)=1$$ and $$\widetilde{g}_{n}(t)(u_{n}+t\phi )\in \mathcal{M}^{+}$$. Thus

\begin{aligned}& \Vert u_{n} \Vert _{\mu }^{p}- \int _{\varOmega }f(x)u_{n}^{1-q}\,dx-\lambda \int _{\varOmega }g(x)u_{n}^{\gamma +1}\,dx=0 \end{aligned}

and

\begin{aligned}& \widetilde{g}_{n}^{p}(t) \Vert u_{n}+t\phi \Vert _{\mu }^{p}-\widetilde{g} _{n}^{1-q}(t) \int _{\varOmega }f(x) (u_{n}+t\phi )^{1-q}\,dx- \lambda \widetilde{g}_{n} ^{\gamma +1}(t) \int _{\varOmega }g(x) (u_{n}+t\phi )^{\gamma +1} \,dx=0. \end{aligned}

Therefore

\begin{aligned} 0&= \bigl[\widetilde{g}_{n}^{p}(t)-1 \bigr] \Vert u_{n}+t\phi \Vert _{\mu }^{p}+ \bigl( \Vert u_{n}+t\phi \Vert _{\mu }^{p}- \Vert u_{n} \Vert _{\mu }^{p} \bigr) \\ &\quad {}- \bigl[ \widetilde{g}_{n} ^{1-q}(t)-1 \bigr] \int _{\varOmega }f(x) (u_{n}+t\phi )^{1-q}\,dx \\ &\quad {} - \int _{\varOmega }f(x) \bigl[(u_{n}+t\phi )^{1-q}-u_{n}^{1-q} \bigr]\,dx- \lambda \bigl[ \widetilde{g}_{n}^{\gamma +1}(t)-1 \bigr] \int _{\varOmega }g(x) (u_{n}+t\phi )^{\gamma +1}\,dx \\ &\quad {} -\lambda \int _{\varOmega }g(x) \bigl[(u_{n}+t\phi )^{\gamma +1}-u_{n}^{\gamma +1} \bigr]\,dx \\ & \leq \bigl[\widetilde{g}_{n}^{p}(t)-1 \bigr] \Vert u_{n}+t\phi \Vert _{\mu }^{p}+ \bigl( \Vert u_{n}+t\phi \Vert _{\mu }^{p}- \Vert u_{n} \Vert _{\mu } ^{p} \bigr) \\ &\quad {}- \bigl[ \widetilde{g}_{n} ^{1-q}(t)-1 \bigr] \int _{\varOmega }f(x) (u_{n}+t\phi )^{1-q}\,dx \\ &\quad {} - \lambda \bigl[\widetilde{g}_{n}^{\gamma +1}(t)-1 \bigr] \int _{\varOmega }g(x) (u_{n}+t\phi )^{\gamma +1}\,dx- \lambda \int _{\varOmega }g(x) \bigl[(u_{n}+t\phi )^{\gamma +1}-u_{n}^{\gamma +1} \bigr]\,dx. \end{aligned}

Dividing by $$t>0$$ and letting $$t\rightarrow 0$$, we have

\begin{aligned}[b] 0& \leq p \widetilde{g}'_{n}(0) \Vert u_{n} \Vert _{\mu }^{p}+p \int _{\varOmega } \biggl( \vert \Delta u_{n} \vert ^{p-2}\Delta u_{n} \Delta \phi - \mu \frac{ \vert u_{n} \vert ^{p-2}u_{n} \phi }{ \vert x \vert ^{2p}} \biggr)\,dx \\ &\quad {}- (1-q ) \widetilde{g}'_{n}(0) \int _{\varOmega }f(x)u_{n}^{1-q}\,dx \\ &\quad {} -\lambda (\gamma +1 )\widetilde{g}'_{n}(0) \int _{\varOmega } g(x)u_{n}^{\gamma +1}\,dx-\lambda ( \gamma +1 ) \int _{\varOmega } g(x)u_{n}^{\gamma }\phi \,dx \\ & =\widetilde{g}'_{n}(0) \biggl[p \Vert u_{n} \Vert _{\mu }^{p}- (1-q ) \int _{\varOmega }g(x)u_{n}^{\gamma +1}\,dx \biggr] \\ &\quad {}+p \int _{\varOmega } \biggl( \vert \Delta u_{n} \vert ^{p-2}\Delta u_{n} \Delta \phi - \mu \frac{ \vert u_{n} \vert ^{p-2}u_{n} \phi }{ \vert x \vert ^{2p}} \biggr)\,dx \\ &\quad {} -\lambda (\gamma +1 ) \int _{\varOmega }g(x)u_{n}^{\gamma }\phi \,dx \\ & =\widetilde{g}'_{n}(0) \biggl[ (p+q-1 ) \Vert u_{n} \Vert _{\mu }^{p}-\lambda (\gamma +q ) \int _{\varOmega }g(x)u_{n}^{\gamma +1}\,dx \biggr] \\ &\quad {} +p \int _{\varOmega } \biggl( \vert \Delta u_{n} \vert ^{p-2}\Delta u_{n} \Delta \phi - \mu \frac{ \vert u_{n} \vert ^{p-2}u_{n} \phi }{ \vert x \vert ^{2p}} \biggr)\,dx-\lambda (\gamma +1 ) \int _{\varOmega }g(x)u_{n}^{\gamma }\phi \,dx, \end{aligned}
(3.8)

where $$\widetilde{g}'_{n}(0)$$ denotes the right derivative of $$\widetilde{g}_{n}(t)$$ at zero. If it does not exist, $$\widetilde{g}'_{n}(0)$$ should be replaced by $$\lim_{k\rightarrow \infty }\frac{\widetilde{g}_{n}(t_{k})- \widetilde{g}_{n}(0)}{t_{k}}$$ for some sequence $$\{t_{k}\}_{k=1}^{ \infty }$$ with $$\lim_{k\rightarrow \infty }t_{k} =0$$ and $$t_{k}>0$$.

Combining with (3.7) and (3.8), we have $$\widetilde{g}'_{n}(0)\neq - \infty$$. Now we prove that $$\widetilde{g}'_{n}(0)\neq +\infty$$. Otherwise, we suppose that $$\widetilde{g}'_{n}(0)=+\infty$$. Note that $$\widetilde{g}_{n}(t)>\widetilde{g}_{n}(0)=1$$ for n large enough, and

\begin{aligned}[b] \bigl\vert \widetilde{g}_{n}(t)-1 \bigr\vert \cdot \Vert u_{n} \Vert +t\widetilde{g}_{n}(t) \Vert \phi \Vert &\geq \bigl\Vert \bigl[ \widetilde{g}_{n}(t)-1 \bigr]u_{n}+t\widetilde{g} _{n}(t)\phi \bigr\Vert \\ &= \bigl\Vert \widetilde{g}_{n}(t) (u_{n}+t\phi )-u_{n} \bigr\Vert . \end{aligned}
(3.9)

Using condition (ii) with $$u=\widetilde{g}_{n}(t)(u_{n}+t\phi ) \in \mathcal{M}^{+}$$, we deduce that

\begin{aligned}& \bigl[\widetilde{g}_{n}(t)-1 \bigr] \cdot \frac{ \Vert u_{n} \Vert }{n}+t \widetilde{g}_{n}(t) \frac{ \Vert \phi \Vert }{n} \\ & \quad \geq \frac{1}{n} \bigl\Vert \widetilde{g}_{n}(t) ( u _{n}+t\phi )-u_{n} \bigr\Vert \\ & \quad \geq I_{\lambda ,\mu }(u_{n})- I_{\lambda ,\mu } \bigl( \widetilde{g}_{n} (t) (u_{n}+t\phi ) \bigr) \\ & \quad =\frac{1}{p} \Vert u_{n} \Vert _{\mu }^{p} -\frac{1}{1-q} \int _{\varOmega }f(x) \vert u_{n} \vert ^{1-q}\,dx- \frac{\lambda }{\gamma +1} \int _{\varOmega }g(x) \vert u_{n} \vert ^{\gamma +1}\,dx- \frac{1}{p}\widetilde{g}_{n} ^{p}(t) \Vert u_{n}+t\phi \Vert _{\mu }^{p} \\ & \quad \quad {} +\frac{1}{1-q} \int _{\varOmega }f(x) \bigl\vert \widetilde{g}_{n} (u_{n}+t\phi ) \bigr\vert ^{1-q}\,dx+\frac{ \lambda }{\gamma +1} \int _{\varOmega }g(x) \bigl\vert \widetilde{g}_{n} (u_{n}+t\phi ) \bigr\vert ^{ \gamma +1}\,dx \\ & \quad =\frac{1}{p} \Vert u_{n} \Vert _{\mu }^{p}- \frac{1}{1-q} \biggl[ \Vert u_{n} \Vert _{\mu }^{p}- \lambda \int _{\varOmega }g(x) \vert u_{n} \vert ^{\gamma +1}\,dx \biggr]-\frac{\lambda }{ \gamma +1} \int _{\varOmega }g(x) \vert u_{n} \vert ^{\gamma +1}\,dx \\ & \quad \quad {} -\frac{1}{p}\widetilde{g}_{n}^{p}(t) \Vert u_{n}+t\phi \Vert _{\mu }^{p}+ \frac{1}{1-q} \biggl[\widetilde{g}_{n}^{p}(t) \Vert u_{n}+t \phi \Vert _{ \mu }^{p}-\lambda \int _{\varOmega }g(x) \vert u_{n}+t\phi \vert ^{\gamma +1}\,dx \biggr] \\ & \quad \quad {} +\frac{\lambda }{\gamma +1} \widetilde{g}_{n}^{\gamma +1}(t) \int _{\varOmega }g(x) \vert u_{n}+t\phi \vert ^{\gamma +1}\,dx \\ & \quad = \biggl(\frac{1}{p}-\frac{1}{1-q} \biggr) \Vert u_{n} \Vert _{\mu } ^{p}+ \biggl( \frac{1}{1-q}- \frac{1}{\gamma +1} \biggr)\lambda \int _{\varOmega }g(x) \vert u_{n} \vert ^{\gamma +1}\,dx \\ & \quad \quad {} + \biggl(\frac{1}{1-q}-\frac{1}{p} \biggr) \widetilde{g}_{n} ^{p}(t) \Vert u_{n}+t\phi \Vert _{\mu }^{p} \\ & \quad\quad{} - \biggl(\frac{1}{1-q}- \frac{1}{ \gamma +1} \biggr)\lambda \widetilde{g}_{n} ^{\gamma +1}(t) \int _{\varOmega }g(x) \vert u_{n}+t\phi \vert ^{\gamma +1}\,dx \\ & \quad = \biggl(\frac{1}{1-q}-\frac{1}{p} \biggr) \bigl( \Vert u_{n}+t \phi \Vert _{\mu }^{p}- \Vert u_{n} \Vert _{\mu }^{p} \bigr)+ \biggl( \frac{1}{1-q} -\frac{1}{p} \biggr) \bigl[\widetilde{g}_{n}^{p}(t)-1 \bigr] \Vert u_{n}+t \phi \Vert _{\mu }^{p} \\ & \quad \quad {} - \biggl(\frac{1}{1-q}-\frac{1}{\gamma +1} \biggr) \lambda \widetilde{g}_{n}^{\gamma +1}(t) \int _{\varOmega }g(x) \bigl[ (u_{n}+t\phi )^{\gamma +1}-u _{n}^{\gamma +1} \bigr]\,dx \\ & \quad \quad {} - \biggl(\frac{1}{1-q}-\frac{1}{\gamma +1} \biggr)\lambda \bigl[ \widetilde{g}_{n}^{\gamma +1}(t)-1 \bigr] \int _{\varOmega }g(x) u_{n}^{\gamma +1}\,dx. \end{aligned}

Dividing by $$t>0$$ and letting $$t\rightarrow 0$$, we obtain

\begin{aligned}& \widetilde{g}'_{n}(0)\cdot \frac{ \Vert u_{n} \Vert }{n}+ \frac{ \Vert \phi \Vert }{n} \\ & \quad \geq \biggl(\frac{1}{1-q}-\frac{1}{p} \biggr)\cdot p \int _{\varOmega } \biggl( \vert \Delta u_{n} \vert ^{p-2}\Delta u_{n} \Delta \phi - \mu \frac{ \vert u_{n} \vert ^{p-2}u_{n} \phi }{ \vert x \vert ^{2p}} \biggr)\,dx \\ & \quad \quad {}+ \biggl(\frac{1}{1-q}- \frac{1}{p} \biggr)\cdot p \widetilde{g}'_{n}(0) \Vert u_{n} \Vert _{\mu }^{p} \\ & \quad \quad {}-\lambda \biggl(\frac{1}{1-q}-\frac{1}{\gamma +1} \biggr) (\gamma +1 ) \int _{\varOmega }g(x)u_{n}^{\gamma }\phi \,dx \\ & \quad \quad {}-\lambda \biggl(\frac{1}{1-q} -\frac{1}{\gamma +1} \biggr) (\gamma +1 ) \widetilde{g}'_{n}(0) \int _{\varOmega }g(x)u_{n}^{\gamma +1}\,dx \\ & \quad = \frac{p-1+q}{1-q} \int _{\varOmega } \biggl( \vert \Delta u_{n} \vert ^{p-2}\Delta u_{n} \Delta \phi - \mu \frac{ \vert u_{n} \vert ^{p-2}u_{n} \phi }{ \vert x \vert ^{2p}} \biggr)\,dx + \frac{p-1+q}{1-q}\widetilde{g}'_{n}(0) \Vert u_{n} \Vert _{\mu }^{p} \\ & \quad \quad {}-\lambda \frac{\gamma +q}{1-q} \int _{\varOmega } g(x)u_{n}^{\gamma }\phi \,dx- \lambda \frac{\gamma +q}{1-q}\widetilde{g}'_{n}(0) \int _{\varOmega }g(x)u_{n}^{\gamma +1}\,dx \\ & \quad = \frac{\widetilde{g}'_{n}(0)}{1-q} \biggl[ (p-1+q ) \Vert u_{n} \Vert _{\mu }^{p}-\lambda (\gamma +q ) \int _{\varOmega } g(x)u_{n}^{\gamma +1}\,dx \biggr] \\ & \quad \quad {}+\frac{p-1+q}{1-q} \int _{\varOmega } \biggl( \vert \Delta u_{n} \vert ^{p-2}\Delta u_{n} \Delta \phi - \mu \frac{ \vert u_{n} \vert ^{p-2}u_{n} \phi }{ \vert x \vert ^{2p}} \biggr)\,dx-\lambda \frac{ \gamma +q}{1-q} \int _{\varOmega } g(x)u_{n}^{\gamma }\phi \,dx, \end{aligned}

that is,

\begin{aligned}[b] \frac{ \Vert \phi \Vert }{n}&\geq \frac{\widetilde{g}'_{n}(0)}{1-q} \biggl[(p-1+q) \Vert u_{n} \Vert _{\mu }^{p}-\lambda (\gamma +q ) \int _{\varOmega } g(x)u_{n}^{\gamma +1}\,dx- \frac{(1-q) \Vert u_{n} \Vert }{n} \biggr] \\ &\quad {} + \frac{p-1+q}{1-q} \int _{\varOmega } \biggl( \vert \Delta u_{n} \vert ^{p-2}\Delta u_{n} \Delta \phi - \mu \frac{ \vert u_{n} \vert ^{p-2}u_{n} \phi }{ \vert x \vert ^{2p}} \biggr)\,dx \\ &\quad {}-\lambda \frac{ \gamma +q}{1-q} \int _{\varOmega }g(x)u_{n}^{\gamma }\phi \,dx, \end{aligned}
(3.10)

which is not true since $$\widetilde{g}'_{n}(0)=+\infty$$ and

\begin{aligned}& (p-1+q ) \Vert u_{n} \Vert _{\mu }^{p}- \lambda ( \gamma +q ) \int _{\varOmega }g(x)u_{n}^{\gamma +1}\,dx- \frac{(1-q) \Vert u_{n} \Vert }{n}\geq C _{2}-\frac{(1-q)C_{3}}{n}>0. \end{aligned}

It follows from (3.7), (3.8) and (3.10) that

\begin{aligned}& \bigl\vert \widetilde{g}_{n}^{\prime }(0) \bigr\vert \leq C_{4} \end{aligned}

for n sufficiently large and a suitable positive constant $$C_{4}$$.

In the following, we prove that $$u_{0}\in \mathcal{M}^{+}$$ is a solution of problem (1.1). By (3.9) and condition (ii) again, we have

\begin{aligned}& \frac{1}{n} \bigl[ \bigl\vert \widetilde{g}_{n}(t)-1 \bigr\vert \cdot \Vert u_{n} \Vert +t \widetilde{g}_{n}(t) \Vert \phi \Vert \bigr] \\& \quad \geq \frac{1}{n} \bigl\Vert \widetilde{g}_{n}(t) (u_{n}+t\phi )-u_{n} \bigr\Vert \\& \quad \geq I_{\lambda ,\mu }(u_{n})-I_{\lambda ,\mu } \bigl( \widetilde{g}_{n}(t) (u_{n}+t\phi ) \bigr) \\& \quad = \frac{1}{p} \Vert u_{n} \Vert _{\mu }^{p} -\frac{1}{1-q} \int _{\varOmega }f(x) \vert u_{n} \vert ^{1-q}\,dx- \frac{\lambda }{\gamma +1} \int _{\varOmega }g(x) \vert u_{n} \vert ^{\gamma +1}\,dx- \frac{1}{p}\widetilde{g}_{n} ^{p}(t) \Vert u_{n}+t\phi \Vert _{\mu }^{p} \\& \quad \quad {} +\frac{1}{1-q} \int _{\varOmega }f(x) \bigl\vert \widetilde{g}_{n} (u_{n}+t\phi ) \bigr\vert ^{1-q}\,dx +\frac{\lambda }{\gamma +1} \int _{\varOmega }g(x) \bigl\vert \widetilde{g}_{n} (u_{n}+t\phi ) \bigr\vert ^{ \gamma +1}\,dx \\& \quad =-\frac{\widetilde{g}_{n}^{p}(t)-1}{p} \Vert u_{n} \Vert _{\mu }^{p}- \frac{ \widetilde{g}_{n}^{p}(t)}{p} \bigl( \Vert u_{n}+t\phi \Vert _{\mu }^{p}- \Vert u _{n} \Vert _{\mu }^{p} \bigr) \\& \quad \quad {}+\frac{\widetilde{g}_{n}^{1-q}(t)-1}{1-q} \int _{\varOmega } f(x) (u_{n}+t\phi )^{1-q}\,dx \\& \quad \quad {} +\frac{1}{1-q} \int _{\varOmega }f(x) \bigl[(u_{n}+t\phi )^{1-q}-u_{n}^{1-q} \bigr]\,dx +\frac{ \lambda (\widetilde{g}_{n}^{\gamma +1}(t)-1)}{\gamma +1} \int _{\varOmega }g(x) (u_{n}+t\phi )^{\gamma +1}\,dx \\& \quad \quad {} +\frac{\lambda }{\gamma +1} \int _{\varOmega }g(x) \bigl[(u_{n}+t\phi )^{\gamma +1} -u_{n}^{\gamma +1} \bigr]\,dx. \end{aligned}

Dividing by $$t>0$$ and letting $$t\rightarrow 0^{+}$$, we derive that

\begin{aligned}& \frac{1}{n} \bigl[ \bigl\vert \widetilde{g}'_{n}(0) \bigr\vert \cdot \Vert u_{n} \Vert + \Vert \phi \Vert \bigr] \\ & \quad \geq -\widetilde{g}'_{n}(0) \Vert u_{n} \Vert _{\mu }^{p}- \int _{\varOmega } \biggl( \vert \Delta u_{n} \vert ^{p-2}\Delta u_{n} \Delta \phi - \mu \frac{ \vert u_{n} \vert ^{p-2}u_{n} \phi }{ \vert x \vert ^{2p}} \biggr)\,dx+ \widetilde{g}_{n}^{\prime }(0) \int _{\varOmega }f(x)u_{n}^{1-q}\,dx \\ & \quad \quad {} +\lambda \widetilde{g}'_{n}(0) \int _{\varOmega }g(x)u_{n} ^{\gamma +1}\,dx+\lambda \int _{\varOmega }g(x)u_{n}^{\gamma }\phi \,dx \\ & \quad \quad {}+\liminf _{t\rightarrow 0^{+}} \frac{1}{1-q} \int _{\varOmega }\frac{f(x)[(u_{n}+t\phi )^{1-q} -u_{n}^{1-q}]}{t}\,dx \\ & \quad =-\widetilde{g}'_{n}(0) \biggl[ \Vert u_{n} \Vert _{\mu }^{p}- \int _{\varOmega }f(x)u_{n}^{1-q}\,dx-\lambda \int _{\varOmega }g(x)u_{n}^{\gamma +1}\,dx \biggr] \\ & \quad \quad {} - \int _{\varOmega } \biggl( \vert \Delta u_{n} \vert ^{p-2}\Delta u_{n} \Delta \phi - \mu \frac{ \vert u_{n} \vert ^{p-2}u_{n} \phi }{ \vert x \vert ^{2p}} \biggr)\,dx +\lambda \int _{\varOmega }g(x) u_{n}^{\gamma }\phi \,dx \\ & \quad \quad {} +\liminf_{t\rightarrow 0^{+}}\frac{1}{1-q} \int _{\varOmega }\frac{f(x)[(u_{n}+t\phi )^{1-q}-u_{n}^{1-q}]}{t}\,dx. \end{aligned}

Noting $$f(x) [(u_{n}+t\phi )^{1-q}-u_{n}^{1-q} ]\geq 0$$, for every $$x \in \varOmega$$ and $$t>0$$, together with the Fatou lemma, we find that

$$\liminf_{t\rightarrow 0^{+}} \biggl[\frac{f(x)[(u_{n}+t\phi )^{1-q}-u _{n}^{1-q}]}{t} \biggr]$$

is integrable, and

\begin{aligned} & \int _{\varOmega }f(x)u_{n}^{-q}\phi \,dx \\ &\quad \leq \liminf_{t\rightarrow 0^{+}}\frac{1}{1-q} \int _{\varOmega } \frac{f(x)[(u_{n}+t\phi )^{1-q}-u_{n}^{1-q}]}{t}\,dx \\ &\quad \leq \frac{ \vert \widetilde{g}'_{n}(0) \vert \Vert u_{n} \Vert + \Vert \phi \Vert }{n}+ \int _{\varOmega } \biggl( \vert \Delta u_{n} \vert ^{p-2}\Delta u_{n} \Delta \phi - \mu \frac{ \vert u_{n} \vert ^{p-2}u_{n} \phi }{ \vert x \vert ^{2p}} \biggr)\,dx \\ &\quad\quad{} -\lambda \int _{\varOmega }g(x)u_{n}^{\gamma }\phi \,dx \\ &\quad \leq \frac{C_{3}C_{4}+ \Vert \phi \Vert }{n}+ \int _{\varOmega } \biggl( \vert \Delta u_{n} \vert ^{p-2}\Delta u_{n} \Delta \phi - \mu \frac{ \vert u_{n} \vert ^{p-2}u_{n} \phi }{ \vert x \vert ^{2p}} \biggr)\,dx -\lambda \int _{\varOmega }g(x)u_{n}^{\gamma }\phi \,dx. \end{aligned}

Applying the Fatou lemma again, we have

\begin{aligned} & \int _{\varOmega }f(x)u_{0}^{-q}\phi \,dx \\ &\quad = \int _{\varOmega } \Bigl[\liminf_{n\rightarrow \infty }f(x)u_{n} ^{-q}\phi \Bigr]\,dx \leq \liminf_{n\rightarrow \infty } \int _{\varOmega } f(x)u_{n}^{-q}\phi \,dx \\ &\quad \leq \liminf_{n\rightarrow \infty } \biggl[\frac{C_{3}C_{4}+ \Vert \phi \Vert }{n}+ \int _{\varOmega } \biggl( \vert \Delta u_{n} \vert ^{p-2}\Delta u_{n} \Delta \phi - \mu \frac{ \vert u_{n} \vert ^{p-2}u_{n} \phi }{ \vert x \vert ^{2p}} \biggr)\,dx \\ &\quad\quad{} -\lambda \int _{\varOmega }g(x)u_{n}^{\gamma }\phi \,dx \biggr] \\ &\quad = \int _{\varOmega } \biggl( \vert \Delta u_{0} \vert ^{p-2}\Delta u_{0} \Delta \phi - \mu \frac{ \vert u_{0} \vert ^{p-2}u_{0} \phi }{ \vert x \vert ^{2p}} \biggr)\,dx -\lambda \int _{\varOmega }g(x)u_{0}^{\gamma }\phi \,dx. \end{aligned}

Since $$\int _{\varOmega }u_{0}^{-q}\varphi _{1}\,dx<\infty$$, we have $$u_{0}>0$$ a.e. in Ω. For every $$\phi \in \mathcal{M}$$ and $$\phi \geq 0$$, we have

\begin{aligned}[b] & \int _{\varOmega } \biggl( \vert \Delta u_{0} \vert ^{p-2}\Delta u_{0} \Delta \phi - \mu \frac{ \vert u_{0} \vert ^{p-2}u_{0} \phi }{ \vert x \vert ^{2p}} \biggr)\,dx- \int _{\varOmega }f(x) u_{0}^{-q}\phi \,dx \\ &\quad{} -\lambda \int _{\varOmega }g(x)u_{0}^{\gamma }\phi \,dx\geq 0. \end{aligned}
(3.11)

Set $$\phi =u_{0}$$ in (3.11), we derive that

\begin{aligned}& \Vert u_{0} \Vert _{\mu }^{p}= \int _{\varOmega } \biggl( \vert \Delta u_{0} \vert ^{p}-\mu \frac{ \vert u_{0} \vert ^{p}}{ \vert x \vert ^{2p}} \biggr)\,dx\geq \int _{\varOmega }f(x) u_{0}^{1-q}\,dx+\lambda \int _{\varOmega }g(x)u_{0}^{\gamma +1}\,dx. \end{aligned}

Furthermore

\begin{aligned}[b] \Vert u_{0} \Vert _{\mu }^{p}&\leq \liminf_{n\rightarrow \infty } \Vert u _{n} \Vert _{\mu }^{p}\leq \limsup _{n\rightarrow \infty } \Vert u_{n} \Vert _{\mu }^{p} \\ & =\limsup_{n\rightarrow \infty } \biggl[ \int _{\varOmega }f(x)u_{n}^{1-q}\,dx +\lambda \int _{\varOmega }g(x)u_{n}^{\gamma +1}\,dx \biggr] \\ &= \int _{\varOmega }f(x)u_{0}^{1-q}\,dx +\lambda \int _{\varOmega }g(x)u_{0}^{\gamma +1}\,dx. \end{aligned}
(3.12)

Hence

$$\Vert u_{0} \Vert _{\mu }^{p}= \int _{\varOmega }f(x)u_{0}^{1-q}\,dx +\lambda \int _{\varOmega }g(x)u_{0}^{\gamma +1}\,dx.$$
(3.13)

Therefore $$u_{n}\rightarrow u_{0}$$ in $$\mathcal{M}$$ and $$u_{0}\in \mathcal{M}$$. By (3.4), we have

\begin{aligned} & (p+q-1 ) \Vert u_{0} \Vert _{\mu }^{p}-\lambda (\gamma +q ) \int _{\varOmega }g(x)u_{0}^{\gamma +1}\,dx \\ &\quad = (p+q-1 ) \biggl[ \int _{\varOmega }f(x)u_{0}^{1-q}\,dx+\lambda \int _{\varOmega }g(x)u_{0}^{\gamma +1}\,dx \biggr]- \lambda (\gamma +q ) \int _{\varOmega }g(x)u_{0}^{\gamma +1}\,dx \\ &\quad = (p+q-1 ) \int _{\varOmega }f(x)u_{0}^{1-q}\,dx-\lambda ( \gamma -1 ) \int _{\varOmega }g(x)u_{0}^{\gamma +1}\,dx>0, \end{aligned}

i.e., $$u_{0}\in \mathcal{M}^{+}$$.

Next, we only need to show that $$u_{0}$$ is a positive weak solution of problem (1.1). Define

$$\varPhi =(u_{0}+\varepsilon \phi )^{+}, \quad \phi \in W, \varepsilon >0.$$

Substituting Φ into (3.11), combining with (3.12), we deduce that

\begin{aligned} 0&\leq \int _{\varOmega } \biggl[ \vert \Delta u_{0} \vert ^{p-2}\Delta u_{0} \Delta \varPhi - \mu \frac{ \vert u_{0} \vert ^{p-2}u_{0} \varPhi }{ \vert x \vert ^{2p}} -f(x)u_{0}^{-q}\varPhi - \lambda g(x)u_{0} ^{\gamma }\varPhi \biggr]\,dx \\ &= \int _{\varOmega _{1}} \biggl[ \vert \Delta u_{0} \vert ^{p-2}\Delta u_{0} \Delta \varPhi -\mu \frac{ \vert u_{0} \vert ^{p-2}u_{0} \varPhi }{ \vert x \vert ^{2p}} -f(x)u_{0}^{-q} \varPhi -\lambda g(x)u_{0} ^{\gamma }\varPhi \biggr]\,dx \\ &\quad {} + \int _{\varOmega _{2}} \biggl[ \vert \Delta u_{0} \vert ^{p-2}\Delta u_{0} \Delta \varPhi -\mu \frac{ \vert u_{0} \vert ^{p-2}u_{0} \varPhi }{ \vert x \vert ^{2p}} -f(x)u_{0}^{-q} \varPhi -\lambda g(x)u_{0} ^{\gamma }\varPhi \biggr]\,dx \\ &= \int _{\varOmega } \biggl[ \vert \Delta u_{0} \vert ^{p-2}\Delta u_{0} \Delta (u_{0}+ \varepsilon \phi )-\mu \frac{ \vert u_{0} \vert ^{p-2}u_{0} (u_{0}+\varepsilon \phi )}{ \vert x \vert ^{2p}} -f(x)u_{0}^{-q} (u_{0}+ \varepsilon \phi ) \\ &\quad {} -\lambda g(x)u_{0}^{\gamma }(u_{0}+ \varepsilon \phi ) \biggr]\,dx \\ &\quad {} - \int _{\varOmega _{2}} \biggl[ \vert \Delta u_{0} \vert ^{p-2}\Delta u_{0} \Delta (u _{0}+\varepsilon \phi )-\mu \frac{ \vert u_{0} \vert ^{p-2}u_{0} (u_{0}+\varepsilon \phi )}{ \vert x \vert ^{2p}} -f(x)u_{0}^{-q} (u_{0}+ \varepsilon \phi ) \\ &\quad {} -\lambda g(x)u_{0}^{\gamma }(u_{0}+ \varepsilon \phi ) \biggr]\,dx \\ &= \int _{\varOmega } \biggl[ \vert \Delta u_{0} \vert ^{p}-\mu \frac{ \vert u_{0} \vert ^{p}}{ \vert x \vert ^{2p}}-f(x)u_{0}^{1-q}- \lambda g(x)u_{0}^{ \gamma +1} \biggr]\,dx \\ &\quad {} +\varepsilon \int _{\varOmega } \biggl[ \vert \Delta u_{0} \vert ^{p-2}\Delta u_{0} \Delta \phi - \mu \frac{ \vert u_{0} \vert ^{p-2}u_{0} \phi }{ \vert x \vert ^{2p}}-f(x)u_{0}^{-q} \phi - \lambda g(x)u_{0}^{\gamma }\phi \biggr]\,dx \\ &\quad {} - \int _{\varOmega _{2}} \biggl[ \vert \Delta u_{0} \vert ^{p}+\varepsilon \vert \Delta u_{0} \vert ^{p-2} \Delta u_{0} \Delta \phi -\mu \frac{ \vert u_{0} \vert ^{p-2}u_{0} (u_{0}+ \varepsilon \phi )}{ \vert x \vert ^{2p}} \biggr]\,dx \\ &\quad {} - \int _{\varOmega _{2}} \bigl[-f(x) u_{0}^{-q}(u_{0}+ \varepsilon \phi )- \lambda g(x)u_{0}^{\gamma +1}-\varepsilon \lambda g(x)u_{0}^{\gamma } \phi \bigr]\,dx \\ &\leq \varepsilon \int _{\varOmega } \biggl[ \vert \Delta u_{0} \vert ^{p-2}\Delta u_{0} \Delta \phi - \mu \frac{ \vert u_{0} \vert ^{p-2}u_{0} \phi }{ \vert x \vert ^{2p}}-f(x)u_{0}^{-q} \phi - \lambda g(x)u_{0}^{\gamma } \phi \biggr]\,dx \\ &\quad {} -\varepsilon \int _{\varOmega _{2}} \vert \Delta u_{0} \vert ^{p-2} \Delta u_{0} \Delta \phi \,dx+ \lambda \Vert g \Vert _{\infty } \int _{\varOmega _{2}} \vert \varepsilon \phi \vert ^{\gamma +1} \,dx+ \varepsilon \lambda \int _{\varOmega _{2}}g(x)u_{0}^{\gamma }\phi \,dx \\ &=\varepsilon \int _{\varOmega } \biggl[ \vert \Delta u_{0} \vert ^{p-2}\Delta u_{0} \Delta \phi - \mu \frac{ \vert u_{0} \vert ^{p-2}u_{0} \phi }{ \vert x \vert ^{2p}}-f(x)u_{0}^{-q} \phi - \lambda g(x)u_{0}^{\gamma }\phi \biggr]\,dx \\ &\quad {} -\varepsilon \int _{\varOmega _{2}} \vert \Delta u_{0} \vert ^{p-2} \Delta u_{0} \Delta \phi \,dx+ \varepsilon \lambda \varepsilon ^{\gamma } \Vert g \Vert _{\infty } \int _{\varOmega _{2}} \vert \phi \vert ^{\gamma +1}\,dx+ \varepsilon \lambda \int _{\varOmega _{2}}g(x)u_{0}^{\gamma } \phi \,dx, \end{aligned}

where $$\varOmega _{1}=\{x\vert u_{0}(x)+\varepsilon \phi (x)>0, x\in \varOmega \}$$ and $$\varOmega _{2}=\{x\vert u_{0}(x)+\varepsilon \phi (x)\leq 0, x\in \varOmega \}$$. Since the measure of $$\varOmega _{2}$$ tends to zero as $$\varepsilon \rightarrow 0$$, we have $$\int _{\varOmega _{2}} \vert \Delta u_{0}\vert ^{p-2}\Delta u_{0} \Delta \phi \,dx \rightarrow 0$$ as $$\varepsilon \rightarrow 0$$. By the same arguments, we have $$\lambda \varepsilon ^{\gamma }\Vert g\Vert _{\infty } \int _{\varOmega _{2}}\vert \phi \vert ^{\gamma +1}\,dx \longrightarrow 0$$ and $$\lambda \int _{\varOmega _{2}}g(x)u_{0}^{\gamma }\phi \,dx\longrightarrow 0$$ as $$\varepsilon \rightarrow 0$$. Dividing by ε and taking the limit for $$\varepsilon \rightarrow 0$$, we deduce that

\begin{aligned}& \int _{\varOmega } \biggl[ \vert \Delta u_{0} \vert ^{p-2}\Delta u_{0} \Delta \phi - \mu \frac{ \vert u_{0} \vert ^{p-2}u_{0} \phi }{ \vert x \vert ^{2p}}-f(x)u_{0}^{-q} \phi - \lambda g(x)u_{0}^{\gamma }\phi \biggr]\,dx\geq 0. \end{aligned}

Therefore $$u_{0}$$ is a positive weak solution of problem (1.1).

We adopt the Ekeland variational principle again to derive a minimizing sequence $$U_{n}\subset \mathcal{M}^{-}$$ for the minimization problem $$\inf_{\mathcal{M}^{-}} I_{\lambda ,\mu }$$ such that for $$U_{n}\in \mathcal{M}$$, $$U_{n} \rightharpoonup U_{0}$$ weakly in $$\mathcal{M}$$ and pointwise a.e. in Ω. By similar arguments to those in (3.4) and (3.6), for $$\lambda \in (0,T_{\mu })$$, we have

$$(p+q-1 ) \int _{\varOmega }f(x) \vert U_{0} \vert ^{1-q}\,dx- \lambda (\gamma -p+1 ) \int _{\varOmega }g(x) \vert U_{0} \vert ^{\gamma +1}\,dx< 0,$$
(3.14)

\begin{aligned}& (p+q-1 ) \int _{\varOmega }f(x) \vert U_{n} \vert ^{1-q}\,dx- \lambda (\gamma -p+1 ) \int _{\varOmega }g(x) \vert U_{n} \vert ^{\gamma +1}\,dx \leq -C_{5}, \end{aligned}

for n large enough and a positive constant $$C_{5}$$. Therefore $$U_{0}>0$$ is the positive weak solution of problem (1.1). Furthermore $$U_{0}\in \mathcal{M}$$. By (3.14), we obtain

\begin{aligned} & (p+q-1 ) \Vert U_{0} \Vert _{\mu }^{p}- (q+\gamma ) \lambda \int _{\varOmega }g(x)U_{0}^{\gamma +1}\,dx \\ & \quad = (p+q-1 ) \biggl[ \int _{\varOmega }f(x)U_{0}^{1-q}\,dx+\lambda \int _{\varOmega }g(x)U_{0}^{\gamma +1}\,dx \biggr]- \lambda (\gamma +q ) \int _{\varOmega }g(x)U_{0}^{\gamma +1}\,dx \\ & \quad = (p+q-1 ) \int _{\varOmega }f(x)U_{0}^{1-q}\,dx-\lambda ( \gamma -p+1 ) \int _{\varOmega }g(x)U_{0}^{\gamma +1}\,dx< 0, \end{aligned}

i.e., $$U_{0}\in \mathcal{M}^{-}$$. According to Lemma 2.2, we know that problem (1.1) has at least two positive weak solutions $$u_{0}\in \mathcal{M}^{+}$$ and $$U_{0}\in \mathcal{M}^{-}$$ with $$\Vert U_{0}\Vert _{ \mu }>\Vert u_{0}\Vert _{\mu }$$ for every $$\lambda \in (0,T_{\mu })$$. This completes the proof of Theorem 1.1.

## Proof of Corollary 1.2

For every $$U\in \mathcal{M}^{-}$$, by Lemma 2.2, we deduce that

\begin{aligned} \Vert U \Vert _{\mu }&>M_{\mu }( \lambda ) \\ & = \biggl[\frac{p+q-1}{\lambda (\gamma +q)}\frac{1}{ \Vert g \Vert _{\infty }}\frac{(\sqrt[p]{S_{\mu }})^{\gamma +1}}{ \vert \varOmega \vert ^{\frac{p ^{*}-1-\gamma }{p^{*}}}} \biggr]^{\frac{1}{\gamma +1-p}} \\ &= \biggl(\frac{1}{\lambda } \biggr)^{\frac{1}{\gamma +1-p}} \biggl(\frac{p+q-1}{ \gamma +q} \biggr)^{\frac{1}{\gamma +1-p}} \biggl(\frac{1}{ \Vert g \Vert _{ \infty }} \biggr)^{\frac{1}{\gamma +1-p}} \frac{(\sqrt[p]{S_{\mu }}) ^{\frac{\gamma +1}{\gamma +1-p}}}{ \vert \varOmega \vert ^{\frac{p^{*}-1-\gamma }{p ^{*}(\gamma +1-p)}}} \\ &= (T_{\mu } )^{-\frac{1}{\gamma +1-p}} \biggl(\frac{p+q-1}{ \gamma +q} \biggr)^{\frac{1}{\gamma +1-p}} \biggl(\frac{1}{ \Vert g \Vert _{ \infty }} \biggr)^{\frac{1}{\gamma +1-p}} \frac{(\sqrt[p]{S_{\mu }}) ^{\frac{\gamma +1}{\gamma +1-p}}}{ \vert \varOmega \vert ^{\frac{p^{*}-1-\gamma }{p ^{*}(\gamma +1-p)}}} \biggl(\frac{T_{\mu }}{\lambda } \biggr)^{\frac{1}{ \gamma +1-p}}. \end{aligned}

Combining with the definition of $$T_{\mu }$$, we have

\begin{aligned} \Vert U \Vert _{\mu }&> \biggl(\frac{q+\gamma }{q+p-1} \biggr)^{\frac{1}{ \gamma +1-p}} \biggl(\frac{q+\gamma }{\gamma -p+1} \biggr)^{ \frac{1}{p+q-1}} \bigl( \Vert f \Vert _{\infty } \bigr) ^{\frac{1}{p+q-1}} \bigl( \Vert g \Vert _{\infty } \bigr)^{\gamma -p+1}\frac{ \vert \varOmega \vert ^{ \frac{2p}{N}\frac{q+\gamma }{p+q-1}\frac{1}{\gamma +1-p}}}{S_{\mu } ^{\frac{q+\gamma }{p+q-1}\frac{1}{\gamma +1-p}}} \\ & \quad{}\times \biggl(\frac{p+q-1}{\gamma +q} \biggr)^{ \frac{1}{\gamma +1-p}} \biggl(\frac{1}{ \Vert g \Vert _{\infty }} \biggr) ^{\frac{1}{ \gamma +1-p}}\frac{(\sqrt[p]{S_{\mu }})^{ \frac{\gamma +1}{\gamma +1-p}}}{ \vert \varOmega \vert ^{\frac{p^{*}-1-\gamma }{p ^{*}(\gamma +1-p)}}} \biggl(\frac{T_{\mu }}{\lambda } \biggr)^{\frac{1}{ \gamma +1-p}} \\ & = \biggl( \frac{q+\gamma }{\gamma -p+1} \biggr)^{ \frac{1}{p+q-1}} \bigl( \Vert f \Vert _{\infty } \bigr) ^{\frac{1}{p+q-1}} \biggl(\frac{ \vert \varOmega \vert ^{\frac{2p}{N}\frac{q+\gamma }{p+q-1} \frac{1}{ \gamma +1-p}-\frac{p^{*}-1-\gamma }{p^{*}(\gamma +1-p)}}}{(\sqrt[p]{S _{\mu }})^{p\cdot \frac{q+\gamma }{p+q-1}\frac{1}{\gamma +1-p}- \frac{ \gamma +1}{\gamma +1-p}}} \biggr) \biggl( \frac{T_{\mu }}{\lambda } \biggr) ^{\frac{1}{\gamma +1-p}} \\ &= \vert \varOmega \vert ^{\frac{1}{p}} \biggl( \frac{q+\gamma }{ \gamma -p+1} \biggr)^{\frac{1}{p+q-1}} \bigl( \Vert f \Vert _{\infty } \bigr) ^{\frac{1}{p+q-1}} \biggl( \frac{ \vert \varOmega \vert ^{\frac{2}{N}}}{\sqrt[p]{S _{\mu }}} \biggr) ^{\frac{1-q}{p+q-1}} \biggl( \frac{T_{\mu }}{\lambda } \biggr)^{\frac{1}{\gamma +1-p}} \\ &= \vert \varOmega \vert ^{\frac{1}{p}} \bigl( \Vert f \Vert _{\infty } \bigr) ^{\frac{1}{p+q-1}} \biggl(1+\frac{p+q-1}{\gamma -p+1} \biggr)^{ \frac{1}{p+q-1}} \biggl( \frac{ \vert \varOmega \vert ^{\frac{2}{N}}}{\sqrt[p]{S _{\mu }}} \biggr) ^{\frac{1-q}{p+q-1}} \biggl( \frac{T_{\mu }}{\lambda } \biggr)^{\frac{1}{\gamma +1-p}}, \end{aligned}

where we adopted the following facts:

\begin{aligned}& \begin{aligned} &\frac{2p}{N}\frac{q+\gamma }{p+q-1} \frac{1}{\gamma +1-p}-\frac{p ^{*}-1-\gamma }{p^{*}(\gamma +1-p)} \\ &\quad =\frac{p^{*}-1+q}{p^{*}(p+q-1)}=\frac{\frac{Np}{N-2p}+q-1}{ \frac{Np}{N-2p}(p+q-1)} \\ &\quad =\frac{N(p+q-1)+2p(1-q)}{Np(p+q-1)}=\frac{1}{p}+\frac{2}{N}\cdot \frac{1-q}{p+q-1}, \end{aligned} \\& p\cdot \frac{q+\gamma }{p+q-1}\frac{1}{\gamma +1-p}-\frac{\gamma +1}{\gamma +1-p}= \frac{(1-q)(\gamma +1-p)}{(p+q-1)(\gamma +1-p)}= \frac{1-q}{p+q-1}. \end{aligned}

Let $$U_{\lambda , \mu ,\varepsilon }\in \mathcal{M}^{-}$$ be the solution of problem (1.1) with $$\gamma =\varepsilon +p-1$$, where $$\lambda \in (0,T_{\mu })$$. Then

\begin{aligned}& \Vert U_{\lambda , \mu ,\varepsilon } \Vert _{\mu }>C_{\mu ,\varepsilon } \biggl( \frac{T_{\mu }}{\lambda } \biggr)^{\frac{1}{\varepsilon }}, \end{aligned}

where $$C_{\mu , \varepsilon }$$ is given in (1.16). This completes the proof of Corollary 1.2.

## Proof of Theorem 1.3

For simplicity, we consider problem (1.1) with $$f=g=1$$,

$$\textstyle\begin{cases} \Delta ^{2}_{p}u-\mu \frac{ \vert u \vert ^{p-2}u}{ \vert x \vert ^{2p}}=u^{-q}+\lambda u^{ \gamma } &\text{in } \varOmega \setminus \{0\}, \\ u(x)>0 &\text{in }\varOmega \setminus \{0\}, \\ u=\Delta u=0 &\text{on } \partial \varOmega . \end{cases}$$
(5.1)

Let us define

\begin{aligned}& \lambda ^{*}=\lambda ^{*}(N,\varOmega ,\mu ,q,\gamma )=\sup \bigl\{ \lambda >0: \text{problem (5.1) has a positive solution} \bigr\} . \end{aligned}

Using Theorem 1.1, we provide uniform estimates for $$\lambda ^{*}(N, \varOmega ,\mu ,q,\gamma )$$.

### Lemma 5.1

For $$1< p<\frac{N}{2}$$, $$0<\mu <\mu _{N,p}$$, $$0< q<1<\gamma <p^{*}-1$$ and $$\varOmega \in \mathbb{U}$$, where $$\mathbb{U}=\{\varOmega \in \mathbb{R}^{N}: \varOmega \textit{ is an open and bounded domain}\}$$, we have

\begin{aligned}& 0< \lambda ^{-}\leq \lambda ^{*}\leq \lambda ^{+}< \infty , \end{aligned}

where

$$\lambda ^{-}= \biggl(\frac{q+p-1}{q+\gamma } \biggr) \biggl( \frac{ \gamma -p+1}{q+\gamma } \biggr)^{\frac{p-\gamma -1}{1-q-p}} \biggl[\frac{S _{\mu }}{ \vert \varOmega \vert ^{\frac{2p}{N}}} \biggr]^{\frac{q+\gamma }{p+q-1}}$$

and

$$\lambda ^{+}=\lambda _{1}^{\frac{\gamma +q}{q-1+p}} \biggl( \frac{\gamma -p+1}{\gamma +q} \biggr)^{\frac{\gamma -p+1}{q+p-1}}\frac{-1+p+q}{ \gamma +q}+ \frac{1}{2}.$$

### Proof

(1) Assume that $$\lambda \in (0,\lambda ^{-})$$, then problem (5.1) has at least two solutions. By the definition of $$\lambda ^{*}$$, we have $$\lambda ^{*}\geq \lambda ^{-}>0$$.

(2) Assume that (5.1) has a positive solution u. Integrating over Ω by multiplying (5.1) by $$\varphi _{1}$$, we obtain

$$\lambda _{1} \int _{\varOmega } \vert u \vert ^{p-2}u \varphi _{1}\,dx= \int _{\varOmega } \biggl(\Delta ^{2}_{p}u-\mu \frac{ \vert u \vert ^{p-2}u}{ \vert x \vert ^{2p}} \biggr) \varphi _{1}\,dx= \int _{\varOmega }u^{-q}\varphi _{1}\,dx+\lambda \int _{\varOmega }u^{\gamma } \varphi _{1}\,dx.$$
(5.2)

We claim that there exists $$\lambda ^{+}>0$$ such that

$$t^{-q}+\lambda ^{+}t^{\gamma }>\lambda _{1}t^{p-1}, \quad \forall t>0.$$
(5.3)

In fact, letting

$$F_{\lambda }(t)=t^{-q}+\lambda t^{\gamma }- \lambda _{1}t^{p-1}=t^{ \gamma } \bigl(t^{-q-\gamma }+ \lambda -\lambda _{1}t^{-\gamma +p-1} \bigr) :=t^{ \gamma } \cdot G_{\lambda }(t), \quad t>0.$$
(5.4)

We have

$$G'_{\lambda }(t)=(-\gamma -q)t^{-\gamma -q-1}+\lambda _{1} (\gamma -p+1)t ^{-\gamma +p-2}=0,$$

i.e.,

$$t:=t_{\min }= \biggl(\frac{\gamma +q}{\lambda _{1} (\gamma -p+1)} \biggr) ^{\frac{1}{q-1+p}}.$$

Then $$G_{\lambda }(t)$$ attains minimum at $$t_{\min }$$, and

$$G_{\lambda }(t_{\min })=\lambda +\lambda _{1}^{\frac{\gamma +q}{q-1+p}} \biggl(\frac{\gamma -p+1}{\gamma +q} \biggr)^{ \frac{\gamma -p+1}{q+p-1}}\frac{1-p-q}{\gamma +q}.$$

We may choose $$\lambda =\lambda _{1}^{\frac{\gamma +q}{q-1+p}} (\frac{ \gamma -p+1}{\gamma +q} )^{\frac{\gamma -p+1}{q+p-1}}\frac{-1+p+q}{ \gamma +q}+\frac{1}{2}=\lambda ^{+} >0$$ such that

$$G_{\lambda ^{+}}(t)\geq G_{\lambda ^{+}}(t_{\min })= \frac{1}{2}>0, \quad \text{for }t>0.$$

Therefore

$$F_{\lambda ^{+}}(t)=t^{\gamma }\cdot G_{\lambda ^{+}}(t)>0 \quad \text{for }t>0.$$

Using (5.3) with $$t=u$$, we have

$$\int _{\varOmega }u^{-q}\varphi _{1}\,dx+\lambda ^{+} \int _{\varOmega } u^{\gamma }\varphi _{1}\,dx\geq \lambda _{1} \int _{\varOmega } \vert u \vert ^{p-2}u \varphi _{1}\,dx.$$
(5.5)

Combining with (5.2) and (5.5), we obtain $$\lambda \leq \lambda ^{+}$$. Since λ is arbitrary, we have $$\lambda ^{*}\leq \lambda ^{+}< \infty$$. □

### Proof of Theorem 1.3

We only prove the case that $$0<\lambda <\lambda ^{*}$$. By the definition of $$\lambda ^{*}$$, there exists $$\overline{\lambda }\in (\lambda , \lambda ^{*})$$ such that the problem

\begin{aligned}& \Delta ^{2}_{p}u-\mu \frac{ \vert u \vert ^{p-2}u}{ \vert x \vert ^{2p}}=u^{-q}+ \overline{ \lambda } u^{\gamma } \end{aligned}

has a positive solution, denoted by $$u_{\overline{\lambda }}$$. It follows that

\begin{aligned}& \Delta ^{2}_{p}u_{\overline{\lambda }}-\mu \frac{ \vert u_{\overline{ \lambda }} \vert ^{p-2}u_{\overline{\lambda }}}{ \vert x \vert ^{2p}} =u_{\overline{ \lambda }}^{-q}+\overline{\lambda } u_{\overline{\lambda }}^{\gamma } \geq u_{\overline{\lambda }}^{-q}+\lambda u_{\overline{\lambda }}^{ \gamma }. \end{aligned}

Hence $$u_{\overline{\lambda }}$$ is an upper solution of (5.1). Note that $$\lim_{t\rightarrow 0^{+}}G_{\lambda }(t)=\infty$$, we can take $$\varepsilon >0$$ small enough with $$\varepsilon \varphi _{1}< u_{\overline{ \lambda }}$$ and $$G_{\lambda }(\varepsilon \varphi _{1})\geq 0$$. Thus

\begin{aligned}& F_{\lambda }(\varepsilon \varphi _{1})= (\varepsilon \varphi _{1})^{ \gamma }G_{\lambda }(\varepsilon \varphi _{1})\geq 0, \quad \text{for all } \lambda >0, \end{aligned}

i.e.,

$$\lambda _{1}(\varepsilon \varphi _{1})^{p-1} \leq (\varepsilon \varphi _{1})^{-q}+\lambda (\varepsilon \varphi _{1})^{\gamma }, \quad \text{for all } \lambda >0.$$
(5.6)

Combining with (1.10) and (5.6), we obtain

\begin{aligned} \Delta ^{2}_{p}(\varepsilon \varphi _{1})-\mu \frac{ \vert (\varepsilon \varphi _{1}) \vert ^{p-2} (\varepsilon \varphi _{1})}{ \vert x \vert ^{2p}}&=\varepsilon ^{p-1} \biggl(\Delta ^{2}_{p}\varphi _{1}- \mu \frac{ \vert \varphi _{1} \vert ^{p-2} \varphi _{1}}{ \vert x \vert ^{2p}} \biggr) \\ & =\varepsilon ^{p-1}\lambda _{1} \vert \varphi _{1} \vert ^{p-1}= \lambda _{1}(\varepsilon \varphi _{1})^{p-1}\leq (\varepsilon \varphi _{1})^{-q}+\lambda (\varepsilon \varphi _{1})^{\gamma }, \end{aligned}

namely, $$\varepsilon \varphi _{1}$$ is a lower solution of (5.1). Note that $$\Delta _{p}^{2}-\frac{\mu }{\vert x\vert ^{2p}}$$ is monotone, then problem (5.1) has a positive solution $$u_{\lambda }$$ with $$\varepsilon \varphi _{1}\leq u_{\lambda }\leq u_{\overline{\lambda }}$$. □

## Conclusions

In this paper, we study a class of p-biharmonic equations with Hardy potential and negative exponents. We establish the dependence of the above problem on q, γ, f, g and Ω and evaluate the extremal value of λ related to the multiplicity of positive solutions for this problem.

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## Availability of data and materials

No data were used to support this study.

## Funding

This project is supported by the Natural Science Foundation of Shanxi Province (201601D011003), and the Natural Science Foundation of Shandong Province of China (ZR2017MA036).

## Author information

Authors

### Contributions

All authors contributed equally to this work. All authors read and approved the final manuscript.

### Corresponding author

Correspondence to Yanbin Sang.

## Ethics declarations

### Competing interests

The authors declare that they have no competing interests. 