Solutions for the quasi-linear elliptic problems involving the critical Sobolev exponent

Abstract

In this article, we study the existence and multiplicity of positive solutions for the quasi-linear elliptic problems involving critical Sobolev exponent and a Hardy term. The main tools adopted in our proofs are the concentration compactness principle and Nehari manifold.

1 Introduction

$$\textstyle\begin{cases} -\Delta_{p}u-\mu\frac{ \vert u \vert ^{p-2}u}{ \vert x \vert ^{p}}= \vert u \vert ^{p^{*}-2}u+\beta \vert x \vert ^{\alpha-p} \vert u \vert ^{p-2}u+\lambda \vert u \vert ^{q-2}u\quad \mbox{in }\Omega, \\ u=0\quad \mbox{on }\partial\Omega, \end{cases}$$
(1.1)

where $$\Omega\subset\mathbb{R}^{N}\ (N\geq3)$$ is a bounded domain with the smooth boundary âˆ‚Î© such that $$0\in\Omega$$. $$\Delta _{p}u=\operatorname{div}( \vert \nabla u \vert ^{p-2}\nabla u)$$ is the p-Laplacian operator of u, $$1< p< N, \lambda>0$$ is a positive real number. $$0\leq\mu<\overline {\mu}$$ ($$\overline{\mu}=\frac{(N-p)^{p}}{p}$$ is the best Hardy constant). $$1< q< p$$ and $$p^{*}=\frac{Np}{N-p}$$ is the critical Sobolev exponent. $$0<\alpha<p-1$$, $$0<\beta<\beta_{1}$$ ($$\beta_{1}$$ is the first eigenvalue that $$-\Delta_{p}u-\mu\frac{ \vert u \vert ^{p-2}u}{ \vert x \vert ^{p}}= \vert x \vert ^{\alpha -p} \vert u \vert ^{p-2}u$$ under Dirichlet boundary condition).

Definition 1.1

The function $$u\in W_{0}^{1,p}(\Omega)$$ is called a weak solution of (1.1) if u satisfies

\begin{aligned} &\int_{\Omega}\biggl( \vert \nabla u \vert ^{p-2}\nabla u\cdot\nabla v-\mu\frac{ \vert u \vert ^{p-2}uv}{ \vert x \vert ^{p}}\biggr)\,dx \\ &\quad = \int_{\Omega}\bigl( \vert u \vert ^{p^{*}-2}uv+\beta \vert x \vert ^{\alpha-p} \vert u \vert ^{p-2}uv+\lambda \vert u \vert ^{q-2}uv\bigr)\,dx \end{aligned}
(1.2)

for all $$v\in W_{0}^{1,p}(\Omega)$$.

In this paper, we use the following norm of $$W_{0}^{1,p}(\Omega)$$:

$$\Vert u \Vert =\biggl( \int_{\Omega}\biggl( \vert \nabla u \vert ^{p}-\mu \frac{ \vert u \vert ^{p}}{ \vert x \vert ^{p}}\biggr)\,dx\biggr)^{\frac{1}{p}}.$$

By the Hardy inequality (see [1, 2])

$$\int_{\Omega}\frac{ \vert u \vert ^{p}}{ \vert x \vert ^{p}}\,dx\leq\frac{1}{\overline{\mu}} \int _{\Omega} \vert \nabla u \vert ^{p}\,dx, \quad\forall u\in W_{0}^{1,p}(\Omega),$$

so this norm is equivalent to $$(\int_{\Omega} \vert \nabla u \vert ^{p}\,dx)^{\frac {1}{p}}$$, the usual norm in $$W_{0}^{1,p}(\Omega)$$.

The norm in $$L^{p}(\Omega)$$ is represented by $$\Vert u \Vert _{p}=(\int_{\Omega } \vert u \vert ^{p}\,dx)^{\frac{1}{p}}$$. According to Hardy inequality, the following best Sobolev constant is well defined for $$1< p< N$$, and $$0\leq\mu<\overline{\mu}$$:

$$S_{\mu,0}=\inf_{u\in W_{0}^{1,p}(\Omega)\backslash\{0\} }\frac{\int_{\Omega}( \vert \nabla u \vert ^{p}-\mu\frac{ \vert u \vert ^{p}}{ \vert x \vert ^{p}})\,dx}{(\int _{\Omega} \vert u \vert ^{p^{*}}\,dx)^{\frac{p}{p^{*}}}}.$$
(1.3)

The quasi-linear problems on Hardy inequality have been studied extensively, either in the smooth bounded domain or in the whole space $$\mathbb{R}^{N}$$. More and more excellent results have been obtained, which provide us opportunities to understand the singular problems. However, compared with the semilinear case, the quasi-linear problems related to Hardy inequality are more complicated [3â€“16]. Abdellaoui, Felli and Peral [3] considered the extremal function which achieves the best constant $$S_{\mu,0}$$, and gave the properties of the extremal functions. The conclusions obtained in [3] can be applied in the problems with critical Sobolev exponent and Hardy term.

Wang, Wei and Kang [10] investigated the following problem:

$$\textstyle\begin{cases} -\Delta_{p} u-\lambda\frac{ \vert u \vert ^{p-2}}{ \vert x \vert ^{p}}u=\mu f(x) \vert u \vert ^{q-2} u+g(x) \vert u \vert ^{p^{*}-2} u,\quad x\in\Omega,\\ u(x)=0,\quad x\in\partial\Omega, \end{cases}$$
(1.4)

where $$1< q< p, \mu>0$$, f and g are non-negative functions and $$p^{*}=\frac{Np}{N-p}$$ is the critical Sobolev exponent. The property of the Nehari manifold was used to prove the existence of multiple positive solutions for (1.4). Furthermore, Hsu [11, 12] improved and complemented the main results obtained in [10]. Recently, Goyal and Sreenadh [13] investigated a class of singular N-Laplacian problems with exponential nonlinearities in $$\mathbb{R}^{N}$$. Very recently, Xiang [14] established the asymptotic estimates of weak solutions for p-Laplacian equation with Hardy term and critical Sobolev exponent.

We should mention that Liu, Guo and Lei [17] studied the existence and multiplicity of positive solutions of Kirchhoff equation with critical exponential nonlinearity. Inspired by [17, 18], we study the problem (1.1) on critical Sobolev exponent. Comparing with the main results obtained in [4, 6, 10â€“12], in this paper, on the one hand, we will analysis the effect of $$\beta \vert x \vert ^{\alpha-p} \vert u \vert ^{p-2}u$$, and the more careful estimates are needed. On the other hand, we establish an lower bound for $$\lambda_{*}$$ ($$\lambda_{*}$$ is defined in TheoremÂ 1.1).

Define the energy functional associated to problem (1.1) as follows:

$$I_{\lambda}(u)=\frac{1}{p} \Vert u \Vert ^{p}-\frac{\beta}{p} \int_{\Omega } \vert u \vert ^{p} \vert x \vert ^{\alpha-p}\,dx- \frac{1}{p^{*}} \int_{\Omega} \vert u \vert ^{p^{*}}\,dx- \frac{\lambda}{q} \int_{\Omega } \vert u \vert ^{q}\,dx.$$
(1.5)

We obtain the following result.

Theorem 1.1

Suppose that $$1< q< p$$, $$0<\alpha <p-1$$. Then there exists $$\lambda_{*}>0$$ such that problem (1.1) admits at least two solutions and one of the solutions is a ground state solution for all $$\lambda\in(0,\lambda_{*})$$.

2 Preliminaries

Firstly, we introduce the Nehari manifold

$$\mathcal{N}_{\lambda}=\bigl\{ u\in W_{0}^{1,p}(\Omega) \backslash\{0\}:\bigl\langle I_{\lambda}^{\prime}(u),u\bigr\rangle =0 \bigr\} .$$

Furthermore $$u\in\mathcal{N}_{\lambda}$$ if and only if

$$\Vert u \Vert ^{p}- \int_{\Omega} \vert u \vert ^{p^{*}}\,dx-\beta \int_{\Omega } \vert u \vert ^{p} \vert x \vert ^{\alpha-p}\,dx -\lambda \int_{\Omega} \vert u \vert ^{q}\,dx=0.$$
(2.1)

Let

$$\psi(u):= \Vert u \Vert ^{p}-\beta \int_{\Omega} \vert u \vert ^{p} \vert x \vert ^{\alpha-p}\,dx - \int_{\Omega} \vert u \vert ^{p^{*}}\,dx -\lambda \int_{\Omega} \vert u \vert ^{q}\,dx,$$

then

$$\bigl\langle \psi^{\prime}(u), u\bigr\rangle =p \Vert u \Vert ^{p}-p\beta \int_{\Omega } \vert u \vert ^{p} \vert x \vert ^{\alpha-p}\,dx -p^{*} \int_{\Omega} \vert u \vert ^{p^{*}}\,dx-q\lambda \int_{\Omega} \vert u \vert ^{q}\,dx.$$

$$\mathcal{N}_{\lambda}$$ can be divided into the following three parts:

\begin{aligned} & \mathcal{N}_{\lambda}^{+}=\biggl\{ u\in \mathcal{N}_{\lambda}:p \Vert u \Vert ^{p}-p\beta \int_{\Omega} \vert x \vert ^{\alpha-p} \vert u \vert ^{p}\,dx \\ &\phantom{\mathcal{N}_{\lambda}^{+}=}{}-p^{*} \int_{\Omega} \vert u \vert ^{p^{*}}\,dx-q\lambda \int_{\Omega} \vert u \vert ^{q}\,dx>0\biggr\} , \end{aligned}
(2.2)
\begin{aligned} &\mathcal{N}_{\lambda}^{0}=\biggl\{ u\in \mathcal{N}_{\lambda}:p \Vert u \Vert ^{p}-p\beta \int_{\Omega} \vert x \vert ^{\alpha-p} \vert u \vert ^{p}\,dx \\ &\phantom{\mathcal{N}_{\lambda}^{0}=}{}-p^{*} \int_{\Omega} \vert u \vert ^{p^{*}}\,dx-q\lambda \int_{\Omega} \vert u \vert ^{q}\,dx=0\biggr\} , \end{aligned}
(2.3)
\begin{aligned} &\mathcal{N}_{\lambda}^{-}=\biggl\{ u\in \mathcal{N}_{\lambda}:p \Vert u \Vert ^{p}-p\beta \int_{\Omega} \vert x \vert ^{\alpha-p} \vert u \vert ^{p}\,dx \\ &\phantom{\mathcal{N}_{\lambda}^{0}=}{} -p^{*} \int_{\Omega} \vert u \vert ^{p^{*}}\,dx-q\lambda \int_{\Omega} \vert u \vert ^{q}\,dx< 0\biggr\} . \end{aligned}
(2.4)

Applying the HÃ¶lder inequality and the Sobolev inequality, for all $$u\in W_{0}^{1,p}(\Omega)\backslash\{0\}$$ we have

$$\int_{\Omega} \vert u \vert ^{q}\,dx\leq\biggl( \int_{\Omega} \vert u \vert ^{q\cdot\frac {p^{*}}{q}}\,dx \biggr)^{\frac{q}{p^{*}}}\biggl( \int_{\Omega}1\,dx\biggr)^{1-\frac{q}{p^{*}}} = \vert \Omega \vert ^{\frac{p^{*}-q}{p^{*}}}\biggl( \int_{\Omega} \vert u \vert ^{p^{*}}\,dx \biggr)^{\frac{q}{p^{*}}}.$$
(2.5)

Lemma 2.1

Assume that $$\lambda\in(0,T_{1})$$ with

$$T_{1}=\frac{(\frac{(\beta_{1} -\beta)(p-p^{*})}{\beta_{1} (q-p^{*})})^{\frac {q-p^{*}}{p-p^{*}}}(\frac{q-p}{p-p^{*}})^{\frac{q-p}{p-p^{*}}}S_{\mu,0}^{\frac {q-p^{*}}{p-p^{*}}}}{ \vert \Omega \vert ^{\frac{p^{*} -q}{p^{*}}}}.$$

Then (i) $$\mathcal{N}_{\lambda}^{\pm}\neq\emptyset$$, and (ii) $$\mathcal {N}_{\lambda}^{0}=\emptyset$$.

Proof

(i) We define a function $$\Phi\in C(\mathbb {R}^{+},\mathbb{R})$$ by

$$\Phi(s)=\biggl(1-\frac{\beta}{\beta_{1}}\biggr)s^{p-p^{*}} \Vert u \Vert ^{p}-\lambda s^{q-p^{*}} \int_{\Omega} \vert u \vert ^{q}\,dx.$$

Let $$\Phi^{\prime}(s)=0$$, that is,

$$\Phi^{\prime}(s)=\biggl(1-\frac{\beta}{\beta_{1}}\biggr) \bigl(p-p^{*}\bigr)s^{p-p^{*}-1} \Vert u \Vert ^{p}- \lambda\bigl(q-p^{*}\bigr)s^{q-p^{*}-1} \int_{\Omega} \vert u \vert ^{q}\,dx=0.$$

We can deduce that

$$s_{\max}:=s=\biggl[\frac{(\beta_{1} -\beta)(p-p^{*}) \Vert u \Vert ^{p}}{\beta_{1} \lambda (q-p^{*})\int_{\Omega} \vert u \vert ^{q}\,dx}\biggr] ^{\frac{1}{q-p}}.$$

It is easy to check that $$\Phi^{\prime}(s)>0$$ for all $$0< s< s_{\max}$$ and $$\Phi^{\prime}(s)<0$$ for all $$s>s_{\max}$$. Consequently, $$\Phi(s)$$ attains its maximum at $$s_{\max}$$, that is,

\begin{aligned} \Phi(s_{\max})={}&\biggl(1-\frac{\beta}{\beta_{1}}\biggr)\biggl\{ \biggl[ \frac {(\beta_{1} -\beta)(p-p^{*}) \Vert u \Vert ^{p}}{\beta_{1} \lambda(q-p^{*}) \int_{\Omega} \vert u \vert ^{q}\,dx}\biggr]^{\frac{1}{q-p}}\biggr\} ^{p-p^{*}} \Vert u \Vert ^{p} \\ &{}-\lambda\biggl\{ \biggl[\frac{(\beta_{1} -\beta)(p-p^{*}) \Vert u \Vert ^{p}}{ \beta_{1} \lambda(q-p^{*})\int_{\Omega} \vert u \vert ^{q}\,dx}\biggr]^{\frac{1}{q-p}}\biggr\} ^{q-p^{*}} \int_{\Omega} \vert u \vert ^{q}\,dx \\ ={}&\biggl(\frac{(\beta_{1} -\beta )(p-p^{*})}{\beta_{1}(q-p^{*})}\biggr)^{\frac{q-p^{*}}{q-p}}\biggl(\frac{q-p}{p-p^{*}}\biggr) \frac{ \Vert u \Vert ^{\frac{p(q-p^{*})}{q-p}}}{ (\lambda\int_{\Omega} \vert u \vert ^{q}\,dx)^{\frac{p-p^{*}}{q-p}}}. \end{aligned}

Since

\begin{aligned} \widetilde{\Phi}(s)&:=s^{p-p^{*}} \Vert u \Vert ^{p}-\beta s^{p-p^{*}} \int_{\Omega} \vert u \vert ^{p} \vert x \vert ^{\alpha-p}\,dx-\lambda s^{q-p^{*}} \int_{\Omega} \vert u \vert ^{q}\,dx \\ &\geq s^{p-p^{*}}\biggl(1-\frac{\beta}{\beta_{1}}\biggr) \Vert u \Vert ^{p}-\lambda s^{q-p^{*}} \int_{\Omega} \vert u \vert ^{q}\,dx. \end{aligned}

By (1.3) and (2.5), we have

\begin{aligned} &\widetilde{\Phi}(s_{\max}) - \int_{\Omega} \vert u \vert ^{p^{*}}\,dx \\ &\quad\geq\Phi(s_{\max}) - \int_{\Omega} \vert u \vert ^{p^{*}}\,dx \\ &\quad=\biggl(\frac{(\beta_{1} -\beta)(p-p^{*})}{\beta_{1} (q-p^{*})}\biggr)^{\frac{q-p^{*}}{q-p}} \biggl(\frac{q-p}{p-p^{*}} \biggr)\frac{ \Vert u \Vert ^{\frac{p(q-p^{*})}{q-p}}}{ (\lambda\int_{\Omega} \vert u \vert ^{q}\,dx) ^{\frac{p-p^{*}}{q-p}}}- \int_{\Omega} \vert u \vert ^{p^{*}}\,dx \\ &\quad>\biggl(\frac{(\beta_{1} -\beta)(p-p^{*})}{\beta_{1} (q-p^{*})}\biggr)^{\frac{q-p^{*}}{q-p}} \biggl(\frac{q-p}{p-p^{*}} \biggr)\frac{ \Vert u \Vert ^{\frac{p(q-p^{*})}{q-p}}}{ [\lambda \vert \Omega \vert ^{\frac{p^{*}-q}{p^{*}}}(\int_{\Omega } \vert u \vert ^{p^{*}}\,dx)^{\frac{q}{p^{*}}}]^{\frac{p-p^{*}}{q-p}} }- \int_{\Omega} \vert u \vert ^{p^{*}}\,dx \\ &\quad= \biggl\{ \biggl(\frac{(\beta_{1} -\beta)(p-p^{*})}{\beta_{1} (q-p^{*})}\biggr)^{\frac{q-p^{*}}{q-p}} \biggl( \frac{q-p}{p-p^{*}}\biggr)\frac{1}{ [\lambda \vert \Omega \vert ^{\frac{p^{*}-q}{p^{*}}}]^{\frac{p-p^{*}}{q-p}}} \biggl(\frac{ \Vert u \Vert ^{p}}{ (\int_{\Omega} \vert u \vert ^{p^{*}}\,dx)^{\frac{p}{p^{*}}}} \biggr)^{\frac {q-p^{*}}{q-p}}-1\biggr\} \\ &\qquad{}\times\int_{\Omega} \vert u \vert ^{p^{*}}\,dx \\ &\quad\geq \biggl\{ \biggl(\frac{(\beta_{1} -\beta)(p-p^{*})}{\beta_{1} (q-p^{*})}\biggr)^{\frac{q-p^{*}}{q-p}} \biggl( \frac{q-p}{p-p^{*}}\biggr)\frac{1}{ [\lambda \vert \Omega \vert ^{\frac{p^{*}-q}{p^{*}}}]^{\frac{p-p^{*}}{q-p}}} S_{\mu,0}^{\frac{q-p^{*}}{q-p}}-1 \biggr\} \int_{\Omega} \vert u \vert ^{p^{*}}\,dx \\ &\quad>0, \end{aligned}

where $$0<\lambda<T_{1}$$. Thus, there exist constants $$s^{+}$$ and $$s^{-}$$ such that

$$0< s^{+}=s^{+}(u)< s_{\max}< s^{-}=s^{-}(u),\quad s^{+}u\in\mathcal {N}_{\lambda}^{+} \mbox{ and } s^{-}u\in\mathcal{N}_{\lambda}^{-}.$$

(ii) We prove that $$\mathcal{N}_{\lambda}^{0}=\emptyset$$ for all $$\lambda\in(0,T_{1})$$. By contradiction, assume that there exists $$u_{0}\neq 0$$ such that $$u_{0}\in\mathcal{N}_{\lambda}^{0}$$. From (2.1), we have

$$\Vert u_{0} \Vert ^{p}- \int_{\Omega} \vert u_{0} \vert ^{p^{*}}\,dx- \beta \int_{\Omega} \vert u_{0} \vert ^{p} \vert x \vert ^{\alpha-p}\,dx-\lambda \int_{\Omega} \vert u_{0} \vert ^{q}\,dx=0,$$
(2.6)

combining with (2.3), we obtain

$$\bigl(p-p^{*}\bigr) \Vert u_{0} \Vert ^{p}=\bigl(p-p^{*}\bigr)\beta \int_{\Omega} \vert u_{0} \vert ^{p} \vert x \vert ^{\alpha-p}\,dx +\bigl(p^{*}-q\bigr)\lambda \int_{\Omega} \vert u_{0} \vert ^{q}\,dx.$$
(2.7)

Equations (2.6) and (2.7) imply that

$$(p-q) \Vert u_{0} \Vert ^{p}-(p-q)\beta \int_{\Omega} \vert u_{0} \vert ^{p} \vert x \vert ^{\alpha-p}\,dx= \bigl(p^{*}-q\bigr) \int_{\Omega} \vert u_{0} \vert ^{p^{*}}\,dx,$$

that is,

$$\int_{\Omega} \vert u_{0} \vert ^{p^{*}}\,dx \geq\frac{p-q}{p^{*}-q} \biggl(1-\frac{\beta}{\beta_{1}}\biggr) \Vert u_{0} \Vert ^{p}.$$
(2.8)

Similarly,

$$\bigl(p-p^{*}\bigr) \Vert u_{0} \Vert ^{p}- \bigl(p-p^{*}\bigr)\beta \int_{\Omega } \vert u_{0} \vert ^{p} \vert x \vert ^{\alpha-p}\,dx =\lambda\bigl(q-p^{*}\bigr) \int_{\Omega} \vert u_{0} \vert ^{q}\,dx,$$

that is,

$$\lambda \int_{\Omega} \vert u_{0} \vert ^{q}\,dx \geq\frac{p-p^{*}}{q-p^{*}}\biggl(1-\frac {\beta}{\beta_{1}}\biggr) \Vert u_{0} \Vert ^{p}.$$
(2.9)

Note that (1.3) holds for $$u\in\mathcal{N}_{\lambda}^{0}\backslash\{0\}$$. Then

\begin{aligned} \Theta&:=\frac{( \vert \Omega \vert ^{\frac{p^{*} -q}{p^{*}}})^{\frac {p-p^{*}}{q-p}}}{S_{\mu,0}^{\frac{q-p^{*}}{q-p}}} \frac{ \Vert u_{0} \Vert ^{\frac{p(q-p^{*})}{q-p}}}{(\int_{\Omega}(u_{0}^{+})^{q} \,dx)^{\frac {p-p^{*}}{q-p}}}- \int_{\Omega} \vert u_{0} \vert ^{p^{*}}\,dx \\ &>\biggl[\frac{1}{S_{\mu,0}^{\frac{q-p^{*}}{q-p}}}\biggl(\frac{ \Vert u_{0} \Vert ^{p}}{(\int _{\Omega} \vert u_{0} \vert ^{p^{*}}\,dx)^{\frac{p}{p^{*}}}}\biggr)^{\frac{q-p^{*}}{q-p}}-1 \biggr] \int _{\Omega} \vert u_{0} \vert ^{p^{*}}\,dx \geq0. \end{aligned}

It follows from (2.8) and (2.9) that

\begin{aligned} \Theta&=\frac{( \vert \Omega \vert ^{\frac{p^{*} -q}{p^{*}}})^{\frac {p-p^{*}}{q-p}}}{S_{\mu,0}^{\frac{q-p^{*}}{q-p}}} \lambda^{\frac{p-p^{*}}{q-p}}\frac{ \Vert u_{0} \Vert ^{\frac {p(q-p^{*})}{q-p}}}{(\lambda\int_{\Omega}(u_{0}^{+})^{q} \,dx)^{\frac {p-p^{*}}{q-p}}}- \int_{\Omega} \vert u_{0} \vert ^{p^{*}}\,dx \\ &\leq\frac{( \vert \Omega \vert ^{\frac{p^{*} -q}{p^{*}}})^{\frac{p-p^{*}}{q-p}}}{S_{\mu ,0}^{\frac{q-p^{*}}{q-p}}} \lambda^{\frac{p-p^{*}}{q-p}}\frac{ \Vert u_{0} \Vert ^{p}}{[(\frac {p-p^{*}}{q-p^{*}})(1-\frac{\beta}{\beta_{1}})]^{\frac{p-p^{*}}{q-p}}}-\biggl( \frac {p-q}{p^{*} -q}\biggr) \biggl(1-\frac{\beta}{\beta_{1}}\biggr) \Vert u_{0} \Vert ^{p} \\ &=\biggl[\frac{( \vert \Omega \vert ^{\frac{p^{*} -q}{p^{*}}})^{\frac{p-p^{*}}{q-p}}}{S_{\mu ,0}^{\frac{q-p^{*}}{q-p}}} \frac{\lambda^{\frac{p-p^{*}}{q-p}}}{[(\frac{p-p^{*}}{q-p^{*}})(1-\frac{\beta }{\beta_{1}})]^{\frac{p-p^{*}}{q-p}}}-\biggl(\frac{p-q}{p^{*} -q}\biggr) \biggl(1-\frac{\beta }{\beta_{1}}\biggr)\biggr] \Vert u_{0} \Vert ^{p} \\ &< 0, \end{aligned}

for $$0<\lambda<T_{1}$$. This is a contradiction.â€ƒâ–¡

Lemma 2.2

$$I_{\lambda}$$ is coercive and bounded below on $$\mathcal{N}_{\lambda}$$.

Proof

For $$u\in\mathcal{N}_{\lambda}$$, we can deduce from (1.3) and (2.5) that

\begin{aligned} I_{\lambda}(u)&=\frac{1}{p} \Vert u \Vert ^{p}- \frac{\beta}{p} \int_{\Omega} \vert u \vert ^{p} \vert x \vert ^{\alpha-p}\,dx- \frac{1}{p^{*}} \int_{\Omega} \vert u \vert ^{p^{*}}\,dx - \frac{\lambda}{q} \int_{\Omega} \vert u \vert ^{q}\,dx \\ &=\biggl(\frac{1}{p}-\frac{1}{p^{*}}\biggr) \Vert u \Vert ^{p} -\biggl(\frac{1}{p}-\frac{1}{p^{*}}\biggr)\beta \int_{\Omega} \vert u \vert ^{p} \vert x \vert ^{\alpha-p}\,dx-\biggl(\frac{1}{q}-\frac{1}{p^{*}}\biggr)\lambda \int_{\Omega} \vert u \vert ^{q}\,dx \\ &\geq\biggl(\frac{1}{p}-\frac{1}{p^{*}}\biggr) \biggl(1- \frac{\beta}{\beta_{1}}\biggr) \Vert u \Vert ^{p}-\lambda \biggl( \frac{1}{q}-\frac{1}{p^{*}}\biggr) \vert \Omega \vert ^{\frac{p^{*} -q}{p^{*}}}S_{\mu,0}^{-\frac{q}{p}} \Vert u \Vert ^{q}. \end{aligned}

Note that $$1< q< p$$ and $$0<\beta<\beta_{1}$$, we see that $$I_{\lambda}$$ is coercive and bounded below on $$\mathcal{N}_{\lambda}$$.â€ƒâ–¡

From LemmaÂ 2.1, we know that $$\mathcal{N}_{\lambda}^{+}$$ and $$\mathcal {N}_{\lambda}^{-}$$ are nonempty. Furthermore, taking into account LemmaÂ 2.2, we define

$$\kappa_{\lambda}=\inf_{u\in\mathcal{N}_{\lambda}}I_{\lambda}(u), \qquad\kappa _{\lambda}^{+}=\inf_{u\in\mathcal{N}_{\lambda}^{+}}I_{\lambda}(u),\qquad \kappa_{\lambda}^{-}=\inf_{u\in\mathcal{N}_{\lambda}^{-}}I_{\lambda }(u).$$

Lemma 2.3

$$\kappa_{\lambda}\leq\kappa_{\lambda}^{+}<0$$.

Proof

For $$u\in\mathcal{N}_{{\lambda}}^{+}$$, using (2.1) and (2.2), we have

$$(p-q) \Vert u \Vert ^{p}-(p-q)\beta \int_{\Omega} \vert u \vert ^{p} \vert x \vert ^{\alpha-p}\,dx >\bigl(p^{*}-q\bigr) \int_{\Omega} \vert u \vert ^{p^{*}}\,dx$$

and

$$(p-q) \Vert u \Vert ^{p}\biggl(1-\frac{\beta}{\beta_{1}}\biggr)> \bigl(p^{*}-q\bigr) \int_{\Omega} \vert u \vert ^{p^{*}}\,dx,$$

that is,

$$\int_{\Omega} \vert u \vert ^{p^{*}}\,dx< \frac{p-q}{p^{*}-q}\biggl(1-\frac{\beta}{\beta _{1}}\biggr) \Vert u \Vert ^{p}.$$
(2.10)

By (2.10), we get

\begin{aligned} I_{\lambda}(u)&=\biggl(\frac{1}{p}-\frac{1}{q}\biggr) \Vert u \Vert ^{p}- \biggl(\frac{1}{p}-\frac{1}{q}\biggr) \beta \int_{\Omega} \vert u \vert ^{p} \vert x \vert ^{\alpha-p}\,dx -\biggl(\frac{1}{p^{*}}-\frac{1}{q}\biggr) \int_{\Omega} \vert u \vert ^{p^{*}}\,dx \\ &< \biggl(\frac{1}{p}-\frac{1}{q}\biggr) \biggl(1- \frac{\beta}{\beta_{1}}\biggr) \Vert u \Vert ^{p}-\biggl( \frac{1}{p^{*}}-\frac{1}{q}\biggr) \biggl(1-\frac{\beta}{\beta_{1}}\biggr) \biggl(\frac{p-q}{p^{*}-q}\biggr) \Vert u \Vert ^{p} \\ &=\biggl(1-\frac{\beta}{\beta_{1}}\biggr) (q-p) \biggl(\frac{1}{qp} - \frac{1}{qp^{*}}\biggr) \Vert u \Vert ^{p} \\ &< 0. \end{aligned}

Therefore, we have $$\kappa_{\lambda}\leq\kappa_{\lambda}^{+}<0$$.â€ƒâ–¡

Lemma 2.4

For $$u\in\mathcal{N}_{\lambda}$$, there exist $$\varepsilon>0$$ and a differentiable function $$\widehat {f}=\widehat{f}(\omega): B(0,\varepsilon)\subset W_{0}^{1,p}(\Omega )\longrightarrow\mathbb{R}^{+}$$ such that

$$\widehat{f}(0)=1,\qquad \widehat{f}(\omega) (u+\omega)\in\mathcal{N}_{\lambda },\quad \forall\omega\in B(0,\varepsilon).$$

Proof

Define

$$\widehat{F}:\mathbb{R}\times W_{0}^{1,p}(\Omega) \longrightarrow\mathbb{R}$$

as follows:

\begin{aligned} \widehat{F}(s,\omega)={}&s^{p-q} \int_{\Omega} \biggl( \bigl\vert \nabla(u+\omega) \bigr\vert ^{p}-\mu\frac{ \vert u+\omega \vert ^{p}}{ \vert x \vert ^{p}}\biggr)\,dx -s^{p-q}\beta \int_{\Omega} \vert u+\omega \vert ^{p} \vert x \vert ^{\alpha-p}\,dx \\ & {}-s^{p^{*}-q} \int_{\Omega} \vert u+\omega \vert ^{p^{*}}\,dx -\lambda \int_{\Omega} \vert u+\omega \vert ^{q}\,dx, \quad u\in \mathcal{N}_{\lambda}. \end{aligned}

It is clear that

$$\widehat{F}(1,0)= \int_{\Omega}\biggl( \vert \nabla u \vert ^{p}-\mu \frac { \vert u \vert ^{p}}{ \vert x \vert ^{p}}\biggr)\,dx-\beta \int_{\Omega} \vert u \vert ^{p} \vert x \vert ^{\alpha-p}\,dx- \int_{\Omega} \vert u \vert ^{p^{*}}\,dx-\lambda \int_{\Omega} \vert u \vert ^{q}\,dx$$

and

\begin{aligned} \widehat{F}_{s}(s,\omega)={}&(p-q)s^{p-q-1} \int_{\Omega}\biggl( \bigl\vert \nabla(u+\omega) \bigr\vert ^{p}- \mu\frac{ \vert u+\omega \vert ^{p}}{ \vert x \vert ^{p}}\biggr)\,dx\\ &{} -(p-q)s^{p-q-1}\beta \int_{\Omega} \vert u+\omega \vert ^{p} \vert x \vert ^{\alpha-p}\,dx \\ &{}-\bigl(p^{*}-q\bigr)s^{p^{*}-q-1} \int_{\Omega} \vert u+\omega \vert ^{p^{*}}\,dx, \end{aligned}

which implies that

\begin{aligned} \widehat{F}_{s}(1,0)={}&(p-q) \int_{\Omega}\biggl( \vert \nabla u \vert ^{p}-\mu \frac{ \vert u \vert ^{p}}{ \vert x \vert ^{p}}\biggr)\,dx-(p-q)\beta \int_{\Omega} \vert u \vert ^{p} \vert x \vert ^{\alpha-p}\,dx \\ &{}-\bigl(p^{*}-q\bigr) \int_{\Omega} \vert u \vert ^{p^{*}}\,dx. \end{aligned}

LemmaÂ 2.1 tells us that $$\widehat{F}_{s}(1,0)\neq0$$. Thus, by the implicit function theorem at the point $$(0,1)$$, there exist $$\varepsilon >0$$, and a differentiable function

$$\widehat{f}:B(0,\varepsilon)\subset W_{0}^{1,p}(\Omega) \longrightarrow \mathbb{R}^{+}$$

such that

$$\widehat{f}(0)=1,\qquad \widehat{f}(\omega)>0\quad \mbox{and} \quad\widehat{f}(\omega ) (u+\omega)\in \mathcal{N}_{\lambda}, \quad\forall\omega\in B(0,\varepsilon).$$

â€ƒâ–¡

Lemma 2.5

For $$u\in\mathcal{N}_{\lambda}^{-}$$, there exist $$\varepsilon>0$$ and a differentiable function $$\widetilde {f}=\widetilde{f}(v): B(0,\varepsilon)\subset W_{0}^{1,p}(\Omega )\longrightarrow\mathbb{R}^{+}$$ such that

$$\widetilde{f}(0)=1\quad \textit{and} \quad\widetilde{f}(v) (u+v)\in\mathcal{N}_{\lambda }^{-},\quad \forall v\in B(0, \varepsilon).$$

Proof

The proof is similar to that of LemmaÂ 2.4, and we omit it here.â€ƒâ–¡

Lemma 2.6

If $$\{u_{n}\}\subset\mathcal {N}_{\lambda}$$ is a minimizing sequence of $$I_{\lambda}$$, for every $$\phi \in W_{0}^{1,p}(\Omega)$$, then

$$-\frac{ \vert f_{n}^{\prime}(0) \vert \Vert u_{n} \Vert + \Vert \phi \Vert }{n}\leq\bigl\langle I_{\lambda }^{\prime}(u_{n}), \phi\bigr\rangle \leq\frac{ \vert f_{n}^{\prime}(0) \vert \Vert u_{n} \Vert + \Vert \phi \Vert }{n}.$$
(2.11)

Proof

It follows from LemmaÂ 2.2 that $$I_{\lambda}$$ is coercive on $$\mathcal{N}_{\lambda}$$. Using the Ekeland variational principle [19], we can find a minimizing sequence $$\{u_{n}\}\subset \mathcal{N}_{\lambda}$$ of $$I_{\lambda}$$ satisfying

$$I_{\lambda}(u_{n})< \kappa_{\lambda}+ \frac{1}{n},\qquad I_{\lambda}(u_{n})\leq I_{\lambda}(w)+ \frac{1}{n} \Vert w-u_{n} \Vert \quad \forall w\in \mathcal{N}_{\lambda}.$$
(2.12)

Without loss of generality, we can assume that $$u_{n}\geq0$$. By LemmaÂ 2.2, we know that $$\{u_{n}\}$$ is bounded in $$W_{0}^{1,p}(\Omega)$$. As a consequence, there exist a subsequence (still denoted by $$\{u_{n}\}$$) and $$u_{*}$$ in $$W_{0}^{1,p}(\Omega)$$ such that

$$\textstyle\begin{cases} u_{n}\rightharpoonup u_{*}\quad \mbox{weakly in }W_{0}^{1,p}(\Omega), \\ u_{n}\rightarrow u_{*}\quad \mbox{strongly in }L^{p}(\Omega)\ (1\leq p< p^{*}), \\ u_{n}(x)\rightarrow u_{*}(x)\quad \mbox{a.e. in }\Omega. \end{cases}$$
(2.13)

From LemmaÂ 2.4, for $$s>0$$ sufficiently small and $$\phi\in W_{0}^{1,p}(\Omega)$$, and set $$u=u_{n}$$, $$\omega=s\phi\in W_{0}^{1,p}(\Omega)$$, we can find that $$f_{n}(s)=f_{n}(s\phi)$$ such that $$f_{n}(0)=1$$ and $$f_{n}(s)(u_{n}+s\phi)\in\mathcal{N}_{\lambda}$$. Since

$$\Vert u_{n} \Vert ^{p}- \int_{\Omega} \vert u_{n} \vert ^{p^{*}}\,dx- \beta \int_{\Omega } \vert u_{n} \vert ^{p} \vert x \vert ^{\alpha-p}\,dx -\lambda \int_{\Omega} \vert u_{n} \vert ^{q}\,dx=0.$$
(2.14)

By (2.12), we obtain

\begin{aligned} \frac{1}{n}\bigl[ \bigl\vert f_{n}(s)-1 \bigr\vert \Vert u_{n} \Vert +sf_{n}(s) \Vert \phi \Vert \bigr]&\geq\frac{1}{n} \bigl\Vert f_{n}(s) (u_{n}+s\phi)-u_{n} \bigr\Vert \\ &\geq I_{\lambda}(u_{n})-I_{\lambda }\bigl[f_{n}(s) (u_{n}+s\phi)\bigr]. \end{aligned}
(2.15)

Notice that

\begin{aligned} I_{\lambda}\bigl[f_{n}(s) (u_{n}+s\phi)\bigr]={}& \frac{1}{p} \bigl\Vert f_{n}(s) (u_{n}+s\phi) \bigr\Vert ^{p} -\frac{\beta}{p} \int_{\Omega} \vert x \vert ^{\alpha-p} \bigl\vert f_{n}(s) (u_{n}+s\phi) \bigr\vert ^{p}\,dx \\ &{}-\frac{1}{p^{*}} \int_{\Omega} \bigl\vert f_{n}(s) (u_{n}+s\phi) \bigr\vert ^{p^{*}}\,dx-\frac{\lambda}{q} \int_{\Omega} \bigl\vert f_{n}(s) (u_{n}+s\phi) \bigr\vert ^{q}\,dx \\ ={}&\frac{f_{n}^{p}(s)}{p} \Vert u_{n}+s\phi \Vert ^{p}- \frac{\beta}{p} f_{n}^{p}(s) \int_{\Omega} \vert x \vert ^{\alpha-p} \bigl\vert (u_{n}+s\phi) \bigr\vert ^{p}\,dx \\ &{}-\frac{f_{n}^{p^{*}}(s)}{p^{*}} \int_{\Omega} \bigl\vert (u_{n}+s\phi) \bigr\vert ^{p^{*}}\,dx- \frac{\lambda}{q}f_{n}^{q}(s) \int_{\Omega} \bigl\vert (u_{n}+s\phi) \bigr\vert ^{q}\,dx. \end{aligned}

Therefore

\begin{aligned} &I_{\lambda}(u_{n})-I_{\lambda}\bigl[f_{n}(s) (u_{n}+s\phi )\bigr] \\ &\quad=\frac{1}{p} \Vert u_{n} \Vert ^{p}- \frac{f_{n}^{p}(s)}{p} \Vert u_{n} \Vert ^{p} + \frac{f_{n}^{p^{*}}(s)}{p^{*}} \int_{\Omega} \vert u_{n}+s\phi \vert ^{p^{*}}\,dx -\frac{1}{p^{*}} \int_{\Omega} \vert u_{n}+s\phi \vert ^{p^{*}}\,dx \\ &\qquad{}+\frac{\lambda}{q}f_{n}^{q}(s) \int_{\Omega} \vert u_{n}+s\phi \vert ^{q}\,dx -\frac{\lambda}{q} \int_{\Omega} \vert u_{n}+s\phi \vert ^{q}\,dx+\frac{\beta}{p} f_{n}^{p}(s) \int_{\Omega} \vert x \vert ^{\alpha-p} \vert u_{n}+s\phi \vert ^{p}\,dx \\ &\qquad{}-\frac{\beta}{p} \int_{\Omega} \vert x \vert ^{\alpha-p} \vert u_{n}+s\phi \vert ^{p}\,dx +\frac{f_{n}^{p}(s)}{p} \Vert u_{n} \Vert ^{p}-\frac{f_{n}^{p}(s)}{p} \Vert u_{n}+s\phi \Vert ^{p} +\frac{1}{p^{*}} \int_{\Omega} \vert u_{n}+s\phi \vert ^{p^{*}}\,dx \\ &\qquad{}-\frac{1}{p^{*}} \int_{\Omega} \vert u_{n} \vert ^{p^{*}}\,dx +\frac{\lambda}{q} \int_{\Omega} \vert u_{n}+s\phi \vert ^{q}\,dx -\frac{\lambda}{q} \int_{\Omega} \vert u_{n} \vert ^{q}\,dx \\ &\qquad{}+\frac{\beta}{p} \int_{\Omega} \vert x \vert ^{\alpha-p} \vert u_{n}+s\phi \vert ^{p}\,dx-\frac{\beta}{p} \int_{\Omega} \vert u_{n} \vert ^{p} \vert x \vert ^{\alpha-p}\,dx \\ &\quad=\frac{1-f_{n}^{p}(s)}{p} \Vert u_{n} \Vert ^{p}+ \frac{f_{n}^{p^{*}}(s)-1}{p^{*}} \int_{\Omega} \vert u_{n}+s\phi \vert ^{p^{*}}\,dx+\frac{\lambda}{q}\bigl(f_{n}^{q}(s)-1 \bigr) \int_{\Omega} \vert u_{n}+s\phi \vert ^{q}\,dx \\ &\qquad{}+\frac{\beta}{p}\bigl(f_{n}^{p}(s)-1\bigr) \int_{\Omega} \vert x \vert ^{\alpha-p} \vert u_{n}+s\phi \vert ^{p}\,dx+\frac{f_{n}^{p}(s)}{p}\bigl( \Vert u_{n} \Vert ^{p}- \Vert u_{n}+s\phi \Vert ^{p}\bigr) \\ &\qquad{}+\frac{1}{p^{*}}\biggl( \int_{\Omega} \vert u_{n}+s\phi \vert ^{p^{*}}\,dx- \int_{\Omega} \vert u_{n} \vert ^{p^{*}}\,dx \biggr)+\frac{\lambda}{q} \int_{\Omega}\bigl( \vert u_{n}+s\phi \vert ^{q}- \vert u_{n} \vert ^{q}\bigr)\,dx \\ &\qquad{}+\frac{\beta}{p} \int_{\Omega}\bigl[ \vert u_{n}+s\phi \vert ^{p} - \vert u_{n} \vert ^{p}\bigr] \vert x \vert ^{\alpha-p}\,dx. \end{aligned}

Dividing by $$s>0$$ and taking the limit for $$s\rightarrow0$$, combining with (2.14) and (2.15), we have

\begin{aligned} &\frac{ \vert f_{n}^{\prime}(0) \vert \Vert u_{n} \Vert + \Vert \phi \Vert }{n} \\ &\quad\geq-f_{n}^{\prime}(0) \Vert u_{n} \Vert ^{p}+f_{n}^{\prime}(0) \int_{\Omega} \vert u_{n} \vert ^{p^{*}}\,dx +\lambda f_{n}^{\prime}(0) \int_{\Omega} \vert u_{n} \vert ^{q}\,dx \\ &\qquad{}+\beta f_{n}^{\prime}(0) \int_{\Omega} \vert u_{n} \vert ^{p} \vert x \vert ^{\alpha-p}\,dx- \int_{\Omega} \vert \nabla u_{n} \vert ^{p-2}\nabla u_{n}\nabla\phi \,dx\\ &\qquad{}+\mu \int_{\Omega}\frac{ \vert u_{n} \vert ^{p-2}u_{n}\phi}{ \vert x \vert ^{p}}\,dx + \int_{\Omega} \vert u_{n} \vert ^{p^{*}-1} \phi \,dx \\ &\qquad{}+\lambda \int_{\Omega} \vert u_{n} \vert ^{q-1} \phi \,dx+ \beta \int_{\Omega} \vert u_{n} \vert ^{p-1} \phi \vert x \vert ^{\alpha-p}\,dx \\ &\quad=-f_{n}^{\prime}(0)\biggl[ \Vert u_{n} \Vert ^{p}- \int_{\Omega} \vert u_{n} \vert ^{p^{*}}\,dx- \lambda \int_{\Omega} \vert u_{n} \vert ^{q}\,dx- \beta \int_{\Omega} \vert u_{n} \vert ^{p} \vert x \vert ^{\alpha-p}\,dx\biggr] -\bigl\langle I_{\lambda}^{\prime}, \phi\bigr\rangle \\ &\quad=-\bigl\langle I_{\lambda}^{\prime},\phi\bigr\rangle . \end{aligned}

Consequently

$$-\frac{ \vert f_{n}^{\prime}(0) \vert \Vert u_{n} \Vert + \Vert \phi \Vert }{n}\leq\bigl\langle I_{\lambda }^{\prime}, \phi\bigr\rangle$$
(2.16)

for every $$\phi\in W_{0}^{1,p}(\Omega)$$. Note that (2.16) holds equally for âˆ’Ï•, we see that (2.11) holds.â€ƒâ–¡

Lemma 2.7

see [8, 10]

Set $$D^{1,p}(\mathbb {R}^{N})=\{u\in L^{p^{*}}(\mathbb{R}^{N}): \vert \nabla u \vert \in L^{p} (\mathbb{R}^{N})\}$$. Assume that $$1< p< N$$ and $$0\leq\mu<\overline{\mu}$$. Then the limiting problem

$$\textstyle\begin{cases} -\Delta_{p}u-\mu\frac{u^{p-1}}{ \vert x \vert ^{p}}=u^{p^{*}-1}\quad \textit{in }\mathbb {R}^{N}\backslash\{0\}, \\ u>0\quad \textit{in }\mathbb{R}^{N}\backslash\{0\},\\ u\in D^{1,p}(\mathbb{R}^{N}) \end{cases}$$
(2.17)

$$V_{\epsilon}(x)=\epsilon^{\frac{p-N}{p}}U_{p,\mu}\biggl( \frac{x}{\epsilon }\biggr)=\epsilon^{\frac{p-N}{p}}U_{p,\mu}\biggl( \frac{ \vert x \vert }{\epsilon}\biggr)\quad \forall \epsilon>0,$$

such that

$$\int_{\mathbb{R}^{N}}\biggl( \bigl\vert \nabla V_{\epsilon}(x) \bigr\vert ^{p}-\mu\frac{ \vert V_{\epsilon }(x) \vert ^{p}}{ \vert x \vert ^{p}}\biggr)\,dx = \int_{\mathbb{R}^{N}} \bigl\vert V_{\epsilon}(x) \bigr\vert ^{p^{*}}\,dx=S_{\mu,0}^{\frac{N}{p}},$$

where the function $$U_{p,\mu}(x)=U_{p,\mu}( \vert x \vert )$$ is the unique radial solution of the above limiting problem with

$$U_{p,\mu}(1)=\biggl(\frac{N(\overline{\mu}-\mu)}{N-p}\biggr)^{\frac{1}{p^{*}-p}}.$$

In the following, we define $$\Lambda=\frac{1}{N}S_{\mu,0}^{\frac{N}{p}}$$.

Lemma 2.8

Let $$\{u_{n}\}\subset\mathcal {N}^{-}_{\lambda}$$ be a minimizing sequence for $$I_{\lambda}$$ with $$\kappa_{\lambda}^{-}<\Lambda-D\lambda^{\frac{p}{p-q}}$$, where

$$D=\frac{p-q}{p}\biggl[\frac{p^{*} -q}{p^{*} q} \vert \Omega \vert ^{\frac{p^{*}-q}{p^{*}}} S_{\mu,0}^{-\frac{q}{p}}\biggl(\frac{\beta_{1} -\beta}{N \beta_{1}} \biggr)^{-\frac {q}{p}}\biggr]^{\frac{p}{p-q}}.$$
(2.18)

Then there exists $$u\in W_{0}^{1,p}(\Omega)$$ such that $$u_{n}\rightarrow u$$ in $$L^{p^{*}}(\Omega)$$.

Proof

Since

$$I_{\lambda}(u_{n})\rightarrow \kappa_{\lambda}^{-}\quad \mbox{as }n\rightarrow+\infty.$$
(2.19)

By LemmaÂ 2.2, we know that $$\{u_{n}\}$$ is bounded in $$W_{0}^{1,p}(\Omega )$$. In fact, we can deduce from (1.3) and (2.19) that

\begin{aligned} &1+\kappa_{\lambda}^{-}+o\bigl( \Vert u_{n} \Vert \bigr) \\ &\quad\geq I_{\lambda }(u_{n})-\frac{1}{p^{*}}\bigl\langle I_{\lambda}^{\prime}(u_{n}),u_{n}\bigr\rangle \\ &\quad=\frac{1}{p} \Vert u_{n} \Vert ^{p}- \frac{\beta}{p} \int_{\Omega} \vert u_{n} \vert ^{p} \vert x \vert ^{\alpha-p}\,dx -\frac{1}{p^{*}} \int_{\Omega} \vert u_{n} \vert ^{p^{*}}\,dx -\frac{\lambda}{q} \int_{\Omega} \vert u_{n} \vert ^{q}\,dx \\ &\qquad{}-\frac{1}{p^{*}}\biggl( \Vert u_{n} \Vert ^{p} - \int_{\Omega} \vert u_{n} \vert ^{p^{*}}\,dx- \lambda \int_{\Omega} \vert u_{n} \vert ^{q}\,dx- \beta \int_{\Omega} \vert u_{n} \vert ^{p} \vert x \vert ^{\alpha-p}\,dx\biggr) \\ &\quad=\biggl(\frac{1}{p}-\frac{1}{p^{*}}\biggr) \Vert u_{n} \Vert ^{p}-\biggl(\frac{1}{p} -\frac{1}{p^{*}}\biggr) \beta \int_{\Omega} \vert u_{n} \vert ^{p} \vert x \vert ^{\alpha-p}\,dx\\ &\qquad{} +\biggl(\frac{1}{p^{*}}-\frac{1}{q} \biggr)\lambda \int_{\Omega} \vert u_{n} \vert ^{q}\,dx \\ &\quad \geq\biggl(\frac{1}{p}-\frac{1}{p^{*}}\biggr) \biggl(1- \frac{\beta}{ \beta_{1}}\biggr) \Vert u_{n} \Vert ^{p} + \biggl(\frac{1}{p^{*}}-\frac{1}{q}\biggr)\lambda \int_{\Omega} \vert u_{n} \vert ^{q}\,dx \\ &\quad\geq \biggl(\frac{1}{p}-\frac{1}{p^{*}}\biggr) \biggl(1- \frac{\beta}{\beta_{1}}\biggr) \Vert u_{n} \Vert ^{p}\\ &\qquad{} + \biggl(\frac{1}{p^{*}}-\frac{1}{q}\biggr)\lambda \vert \Omega \vert ^{\frac{p^{*}-q}{p^{*}}}\biggl( \int_{\Omega} \vert u_{n} \vert ^{p^{*}}\,dx \biggr)^{\frac{q}{p^{*}}} \\ &\quad\geq \biggl(\frac{1}{p}-\frac{1}{p^{*}}\biggr) \biggl(1- \frac{\beta}{\beta_{1}}\biggr) \Vert u_{n} \Vert ^{p} + \biggl(\frac{1}{p^{*}}-\frac{1}{q}\biggr)\lambda \vert \Omega \vert ^{\frac {p^{*}-q}{p^{*}}}S_{\mu,0}^{-\frac{q}{p}} \Vert u_{n} \Vert ^{q}, \end{aligned}

where $$0<\beta<\beta_{1}$$, $$1< q< p$$, we see that $$\{u_{n}\}$$ is bounded in $$W_{0}^{1,p}(\Omega)$$. We can choose a subsequence (still denoted by $$\{u_{n}\}$$) and $$u\in W_{0}^{1,p}(\Omega)$$ satisfying

$$\textstyle\begin{cases} u_{n}\rightharpoonup u\quad \mbox{weakly in }W_{0}^{1,p}(\Omega), \\ u_{n}\rightarrow u \quad\mbox{strongly in }L^{p}(\Omega)\ (1\leq p< p^{*}). \\ u_{n}(x)\rightarrow u(x) \quad\mbox{a.e. in }\Omega. \end{cases}$$
(2.20)

In term of the concentration compactness principle, going if necessary to a subsequence, there exist an at most countable set $$\mathcal{J}$$, a set of points $$\{x_{j}\}_{j\in\mathcal{J}}\subset\Omega\setminus\{0\}$$, and real numbers $$\mu_{j}$$, $$\nu_{j}$$, $$\widetilde{\chi_{0}}$$ such that

\begin{aligned} &\vert \nabla u_{n} \vert ^{p}\rightharpoonup \,d\mu\geq \vert \nabla u \vert ^{p}+\sum_{j\in \mathcal{J}} \mu_{j}\delta_{x_{j}}+\mu_{0}\delta_{0}, \\ &\vert u_{n} \vert ^{p^{*}}\rightharpoonup \,d\nu= \vert u \vert ^{p^{*}}+\sum_{j\in\mathcal {J}} \nu_{j}\delta_{x_{j}}+\nu_{0}\delta_{0}, \\ &\frac{ \vert u_{n} \vert ^{p}}{ \vert x \vert ^{p}}\rightharpoonup \,d\widetilde{\chi}=\frac { \vert u \vert ^{p}}{ \vert x \vert ^{p}}+ \widetilde{\chi_{0}}\delta_{0}, \end{aligned}

where $$\delta_{x_{j}}$$ is the Dirac mass at $$x_{j}$$.

Let Ïµ be sufficient small satisfying $$0\notin B(x_{j}, \epsilon )$$ and $$B(x_{j}, \epsilon)\cap B(x_{i}, \epsilon)=\emptyset$$ for $$i\neq j, i, j=1, 2, \ldots, k$$. Let $$\psi_{\epsilon,j}(x)$$ be a smooth cut-off function centered at $$x_{j}$$ such that $$0\leq\psi_{\epsilon ,j}(x)\leq1$$, $$\psi_{\epsilon,j}(x)=1$$ for $$x\in B(x_{j}, \frac{\epsilon}{2})$$, $$\psi_{\epsilon,j}(x)=0$$ for $$x\in\Omega\backslash B(x_{j},\epsilon)$$ and $$\vert \nabla\psi_{\epsilon,j}(x) \vert \leq\frac{4}{\epsilon}$$. Note that

\begin{aligned} &\bigl\langle I_{\lambda}^{\prime}(u_{n}),u_{n} \psi_{\epsilon ,j}(x)\bigr\rangle \\ &\quad= \int_{\Omega} \vert \nabla u_{n} \vert ^{p}\psi_{\epsilon,j}(x)\,dx+ \int_{\Omega}u_{n} \vert \nabla u_{n} \vert ^{p-2}\nabla u_{n}\nabla\psi_{\epsilon,j}(x)\,dx -\mu \int_{\Omega}\frac{ \vert u_{n} \vert ^{p}}{ \vert x \vert ^{p}}\psi_{\epsilon,j}(x)\,dx \\ &\qquad{}- \int_{\Omega} \vert u_{n} \vert ^{p^{*}} \psi_{\epsilon,j}(x)\,dx-\lambda \int_{\Omega} \vert u_{n} \vert ^{q} \psi_{\epsilon,j}(x)\,dx-\beta \int_{\Omega} \vert u_{n} \vert ^{p} \vert x \vert ^{\alpha-p} \psi_{\epsilon,j}(x)\,dx. \end{aligned}

Furthermore, we have

\begin{aligned} &\lim_{n\rightarrow\infty} \int_{\Omega} \vert \nabla u_{n} \vert ^{p}\psi_{\epsilon,j}(x)\,dx= \int_{\Omega} \psi_{\epsilon,j}(x)\,d\mu\geq \int_{\Omega} \vert \nabla u \vert ^{p} \psi_{\epsilon ,j}(x)\,dx+\mu_{j}, \\ &\lim_{n\rightarrow\infty} \int_{\Omega } \vert u_{n} \vert ^{p^{*}} \psi_{\epsilon,j}(x)\,dx = \int_{\Omega}\psi_{\epsilon,j}(x)\,d\nu= \int_{\Omega} \vert u \vert ^{p^{*}} \psi_{\epsilon,j}(x)\,dx+\nu_{j}, \\ &\lim_{\epsilon\rightarrow0}\lim_{n\rightarrow\infty} \biggl\vert \int_{\Omega }u_{n} \vert \nabla u_{n} \vert ^{p-2}\nabla u_{n}\cdot\nabla\psi_{\epsilon,j}(x) \biggr\vert =0, \\ &\lim_{\epsilon\rightarrow0}\lim_{n\rightarrow \infty} \biggl\vert \int_{\Omega}\frac{ \vert u_{n} \vert ^{p}}{ \vert x \vert ^{p}}\psi_{\epsilon,j} (x) \biggr\vert =0. \end{aligned}

By (1.3), we deduce that

\begin{aligned} \biggl\vert \int_{\Omega} \vert u_{n} \vert ^{q} \psi_{\epsilon,j}\,dx \biggr\vert &\leq \int _{B(x_{j},\epsilon)} \vert u_{n} \vert ^{q}\,dx \\ &\leq\biggl( \int_{B(x_{j},\epsilon)} \vert u_{n} \vert ^{q\frac {p^{*}}{q}}\,dx \biggr)^{\frac{q}{p^{*}}} \biggl( \int_{B(x_{j},\epsilon)}\,dx\biggr)^{\frac{p^{*}-q}{p^{*}}} \\ &\leq S_{\mu,0}^{-\frac{q}{p}} \Vert u_{n} \Vert ^{q} \biggl( \int_{B(x_{j},\epsilon )}\,dx\biggr)^{\frac{p^{*}-q}{p^{*}}} \\ &\leq S_{\mu,0}^{-\frac{q}{p}}\biggl( \int_{0}^{\epsilon}r^{N-1}\,dr\biggr)^{\frac {p^{*}-q}{p^{*}}} \Vert u_{n} \Vert ^{q} \\ &=\biggl(\frac{1}{N}\biggr)^{\frac{p^{*}-q}{p^{*}}}S_{\mu,0}^{-\frac{q}{p}} \epsilon ^{\frac{N(p^{*}-q)}{p^{*}}} \Vert u_{n} \Vert ^{q} \end{aligned}

and

\begin{aligned} \biggl\vert \int_{\Omega} \vert u_{n} \vert ^{p} \vert x \vert ^{\alpha-p} \psi_{\epsilon,j}(x)\,dx \biggr\vert &\leq \biggl( \int_{B(x_{j},\epsilon)} \vert u_{n} \vert ^{p\frac{p^{*}}{p}}\,dx \biggr)^{\frac{p}{p^{*}}}\biggl( \int_{B(x_{j},\epsilon)} \vert x \vert ^{\frac{p^{*}(\alpha-p)}{p^{*}-p}}\,dx \biggr)^{\frac{p^{*}-p}{p^{*}}} \\ &\leq\biggl( \int_{B(x_{j},\epsilon)} \vert u_{n} \vert ^{p\frac{p^{*}}{p}}\,dx \biggr)^{\frac{p}{p^{*}}}\biggl( \int_{B(x_{j},\epsilon)} \vert x-x_{j} \vert ^{\frac{p^{*}(\alpha-p)}{p^{*}-p}}\,dx\biggr)^{\frac{p^{*}-p}{p^{*}}} \\ &\leq S_{\mu,0}^{-1} \Vert u_{n} \Vert ^{p}\biggl( \int_{0}^{\epsilon} r^{N-1}r^{\frac{p^{*}(\alpha-p)}{p^{*}-p}}\,dr \biggr)^{\frac{p^{*}-p}{p^{*}}} \\ &=S_{\mu,0}^{-1} \Vert u_{n} \Vert ^{p}\biggl(\frac{p}{N\alpha} \epsilon^{\frac{N\alpha}{p}} \biggr)^{\frac{p^{*}-p}{p^{*}}}. \end{aligned}

Since $$\{u_{n}\}$$ is bounded in $$W_{0}^{1,p}(\Omega)$$, and $$u_{n}\rightharpoonup u$$ weakly in $$L^{p^{*}}(\Omega)$$, we conclude that

$$\lim_{\epsilon\rightarrow0}\lim_{n\rightarrow\infty} \int_{\Omega } \vert u_{n} \vert ^{q} \psi_{\epsilon,j}(x)\,dx=0$$

and

$$\lim_{\epsilon\rightarrow0}\lim_{n\rightarrow\infty} \int_{\Omega } \vert u_{n} \vert ^{p} \vert x \vert ^{\alpha-p}\psi_{\epsilon,j}(x)\,dx=0.$$

By (2.11), we have

\begin{aligned} 0=\lim_{\epsilon\rightarrow0}\lim_{n\rightarrow \infty}\bigl\langle I_{\lambda}^{\prime}(u_{n}),u_{n} \psi_{\epsilon,j}(x)\bigr\rangle \geq\mu_{j}-\nu_{j}. \end{aligned}

Since $$S_{0,0}\nu_{j}^{\frac{p}{p^{*}}}\leq\mu_{j}$$, we have $$\mu_{j}=\nu _{j}=0$$ or $$\mu_{j}\geq(S_{0,0})^{\frac{N}{p}}$$.

On the other hand, let $$\epsilon>0$$ be sufficiently small satisfying $$x_{j}\notin B(0, \epsilon)$$, $$\forall j\in\mathcal{J}$$. Let $$\psi _{\epsilon,0}(x)$$ a smooth cut-off function centered at the origin such that $$0\leq\psi_{\epsilon,0}(x)\leq1$$, $$\psi_{\epsilon,0}(x)=1$$ for $$\vert x \vert \leq\frac{\epsilon}{2}$$, $$\psi_{\epsilon,0}(x)=0$$ for $$\vert x \vert \geq \epsilon$$ and $$\vert \nabla\psi_{\epsilon,0}(x) \vert \leq\frac{4}{\epsilon}$$. Hence, we have

\begin{aligned} &\lim_{n\rightarrow\infty} \int_{\Omega} \vert \nabla u_{n} \vert ^{p}\psi _{\epsilon,0}(x)\,dx= \int_{\Omega} \psi_{\epsilon,0}(x)\,d\mu\geq \int_{\Omega} \vert \nabla u \vert ^{p} \psi_{\epsilon ,0}(x)\,dx+\mu_{0}, \\ &\lim_{n\rightarrow\infty} \int_{\Omega } \vert u_{n} \vert ^{p^{*}} \psi_{\epsilon,0}(x)\,dx = \int_{\Omega}\psi_{\epsilon,0}(x)\,d\nu= \int_{\Omega} \vert u \vert ^{p^{*}} \psi_{\epsilon,0}(x)\,dx+\nu_{0}, \\ &\lim_{n\rightarrow\infty} \int_{\Omega} \frac{ \vert u_{n} \vert ^{p}}{ \vert x \vert ^{p}}\psi_{\epsilon,0}(x)\,dx= \int_{\Omega}\psi_{\epsilon,0}(x)\,d\widetilde{\chi}= \int_{\Omega} \frac{ \vert u \vert ^{p}}{ \vert x \vert ^{p}}\psi_{\epsilon,0}(x)\,dx+ \widetilde{\chi_{0}}, \\ &\lim_{\epsilon\rightarrow0}\lim_{n\rightarrow\infty} \biggl\vert \int_{\Omega }u_{n} \vert \nabla u_{n} \vert ^{p-2}\nabla u_{n}\cdot\nabla\psi_{\epsilon,0}(x)\,dx \biggr\vert =0, \\ &\lim_{\epsilon\rightarrow0}\lim_{n\rightarrow\infty} \int_{\Omega } \vert u_{n} \vert ^{q} \psi_{\epsilon,0}(x)\,dx=0 \end{aligned}

and

$$\lim_{\epsilon\rightarrow0}\lim_{n\rightarrow\infty} \int_{\Omega } \vert u_{n} \vert ^{p} \vert x \vert ^{\alpha-p}\psi_{\epsilon,0}(x)\,dx=0.$$

Therefore

$$0=\lim_{\epsilon\rightarrow0}\lim_{n\rightarrow \infty}\bigl\langle I_{\lambda}^{\prime}(u_{n}),u_{n} \psi_{\epsilon,0}(x)\bigr\rangle \geq\mu_{0}-\mu\widetilde{ \chi_{0}}-\nu_{0}.$$

Combining the definition of $$S_{\mu,0}$$, we get that $$S_{\mu,0}\nu _{0}^{\frac{p}{p^{*}}}\leq\mu_{0}-\mu\widetilde{\chi_{0}}\leq\nu_{0}$$, which implies that $$\nu_{0}=0$$ or $$\nu_{0}\geq(S_{\mu,0})^{\frac {N}{p}}$$. Now, we prove that $$\mu_{j}\geq(S_{0,0})^{\frac{N}{p}}$$ and $$\nu_{0}\geq(S_{\mu,0})^{\frac{N}{p}}$$ are not true. If not, we have

\begin{aligned} \kappa_{\lambda}^{-}&=\lim_{n\rightarrow\infty } \biggl[I_{\lambda}(u_{n})-\frac{1}{p^{*}}\bigl\langle I_{\lambda }^{\prime}(u_{n}),u_{n}\bigr\rangle \biggr] \\ &\geq\lim_{n\rightarrow\infty}\biggl[ \biggl(\frac{1}{p}- \frac{1}{p^{*}}\biggr) \Vert u_{n} \Vert ^{p} + \biggl(\frac{1}{p^{*}}-\frac{1}{q}\biggr)\lambda \vert \Omega \vert ^{\frac {p^{*}-q}{p^{*}}}S_{\mu,0}^{-\frac{q}{p}} \Vert u_{n} \Vert ^{q}\biggr] \\ &=\lim_{n\rightarrow\infty}\biggl[\frac{1}{N} \Vert u_{n} \Vert ^{p} +\biggl(\frac{1}{p^{*}}- \frac{1}{q}\biggr)\lambda \vert \Omega \vert ^{\frac {p^{*}-q}{p^{*}}}S_{\mu,0}^{-\frac{q}{p}} \Vert u_{n} \Vert ^{q}\biggr] \\ &\geq\frac{1}{N}\biggl( \Vert u \Vert ^{p}+\sum _{j\in\mathcal{J}}\mu_{j} +\mu_{0}-\mu \widetilde{ \chi_{0}}\biggr) +\biggl(\frac{1}{p^{*}}-\frac{1}{q}\biggr) \lambda \vert \Omega \vert ^{\frac {p^{*}-q}{p^{*}}}S_{\mu,0}^{-\frac{q}{p}} \Vert u \Vert ^{q} \\ &\geq\frac{1}{N}S_{\mu,0}^{\frac{N}{p}}+\frac{1}{N} \Vert u \Vert ^{p} +\biggl(\frac{1}{p^{*}}-\frac{1}{q} \biggr)\lambda \vert \Omega \vert ^{\frac {p^{*}-q}{p^{*}}}S_{\mu,0}^{-\frac{q}{p}} \Vert u \Vert ^{q} \\ &= \frac{1}{N}S_{\mu,0}^{\frac{N}{p}}+\frac{1}{N} \Vert u \Vert ^{p} -\frac{p^{*} -q}{p^{*} q}\lambda \vert \Omega \vert ^{\frac{p^{*}-q}{p^{*}}}S_{\mu ,0}^{-\frac{q}{p}} \Vert u \Vert ^{q} \\ &\geq\frac{1}{N}S_{\mu,0}^{\frac{N}{p}}-D\lambda^{\frac{p}{p-q}}, \end{aligned}

where D is defined in (2.18). Hence, we conclude that $$\Lambda-D\lambda^{\frac{p}{p-q}}\leq\kappa _{\lambda}^{-}<\Lambda-D\lambda^{\frac{p}{p-q}}$$, which is a contradiction. It follows that $$\nu_{j}=0$$ for $$j\in\{0\}\cup\mathcal{J}$$, which means that $$\int_{\Omega} \vert u_{n} \vert ^{p^{*}}\,dx\rightarrow\int_{\Omega } \vert u \vert ^{p^{*}}\,dx$$ as $$n\rightarrow\infty$$. The proof is completed.â€ƒâ–¡

In the following, we need some estimates for the extremal function $$V_{\epsilon}$$ defined in LemmaÂ 2.7. Given $$R>0$$, let $$\varphi(x)\in W_{0}^{1,p}(\Omega)$$, $$0\leq\varphi(x)\leq1$$, $$\varphi(x)=1$$ for $$\vert x \vert \leq R$$, $$\varphi(x)=0$$ for $$\vert x \vert \geq2R$$. Set $$v_{\epsilon }(x)=\varphi(x)V_{\epsilon}(x)$$. For $$1< p< N$$ and $$1< q< p^{*}$$, we have the following estimates (see [4, 6]):

\begin{aligned} & \Vert v_{\epsilon} \Vert ^{p}=(S_{\mu,0})^{\frac{N}{p}}+O \bigl(\epsilon^{b(\mu)p+p-N}\bigr), \end{aligned}
(2.21)
\begin{aligned} &\int_{\Omega} \vert v_{\epsilon} \vert ^{p^{*}}\,dx=(S_{\mu,0})^{\frac {N}{p}}+O\bigl(\epsilon^{b(\mu)p^{*}-N} \bigr), \end{aligned}
(2.22)

then

$$\int_{\Omega} \vert v_{\epsilon} \vert ^{q}\,dx=\textstyle\begin{cases} C\epsilon^{N+q(1-\frac{N}{p})} &\frac{N}{b(\mu)}< q< p, \\ C\epsilon ^{N+q(1-\frac{N}{p})} \vert \ln\epsilon \vert & q=\frac{N}{b(\mu)}, \\ C\epsilon ^{q(b(\mu)+1-\frac{N}{p})} &1< q< \frac{N}{b(\mu)}, \end{cases}$$
(2.23)

where $$b(\mu)$$ is the zero of the function

$$f(\xi)=(p-1)\xi^{p}-(N-p)\xi^{p-1}+\mu,\quad \xi\geq0, 0\leq\mu< \overline{\mu},$$

satisfying $$0<\frac{N-p}{p}<b(\mu)<\frac{N-p}{p-1}$$.

Lemma 2.9

There exists $$\lambda_{0}>0$$ such that

$$\sup_{s\geq0}I_{\lambda}(sv_{\epsilon})< \Lambda-D \lambda^{\frac {p}{p-q}},\quad \textit{for }\lambda\in(0,\lambda_{0}),$$

where Î› and D are defined in LemmaÂ  2.8.

Proof

For two positive constants $$s_{0}$$ and $$s_{1}$$ (independent of Ïµ, Î»), we show that there exists $$s_{\epsilon}>0$$ with $$0< s_{0}\leq s_{\epsilon}\leq s_{1}<\infty$$ such that $$\sup_{s\geq0}I_{\lambda}(sv_{\epsilon})=I_{\lambda }(s_{\epsilon}v_{\epsilon})$$. In fact, since $$\lim_{s\rightarrow+\infty}I_{\lambda}(sv_{\epsilon })=-\infty$$, we can deduce that

$$s_{\epsilon}^{p-1} \Vert v_{\epsilon} \Vert ^{p}-\beta s_{\epsilon}^{p-1} \int _{\Omega} \vert v_{\epsilon} \vert ^{p} \vert x \vert ^{\alpha-p}\,dx- s_{\epsilon}^{p^{*}-1} \int_{\Omega} \vert v_{\epsilon} \vert ^{p^{*}}\,dx- \lambda s_{\epsilon}^{q-1} \int_{\Omega} \vert v_{\epsilon} \vert ^{q}\,dx=0$$
(2.24)

and

\begin{aligned} &(p-1)s_{\epsilon}^{p-2} \Vert v_{\epsilon} \Vert ^{p}-(p-1)\beta s_{\epsilon}^{p-2} \int_{\Omega} \vert v_{\epsilon } \vert ^{p} \vert x \vert ^{\alpha-p}\,dx \\ &\quad{}- \bigl(p^{*}-1\bigr)s_{\epsilon}^{p^{*}-2} \int_{\Omega} \vert v_{\epsilon} \vert ^{p^{*}}\,dx -(q-1)\lambda s_{\epsilon}^{q-2} \int_{\Omega} \vert v_{\epsilon} \vert ^{q}\,dx< 0. \end{aligned}
(2.25)

Equations (2.24) and (2.25) imply that

\begin{aligned} &(p-1)s_{\epsilon}^{p-2} \Vert v_{\epsilon} \Vert ^{p}-(p-1)\beta s_{\epsilon}^{p-2} \int_{\Omega} \vert v_{\epsilon } \vert ^{p} \vert x \vert ^{\alpha-p}\,dx- \bigl(p^{*}-1\bigr)s_{\epsilon}^{p^{*}-2} \int_{\Omega} \vert u_{\epsilon} \vert ^{p^{*}}\,dx \\ &\quad < (q-1)s_{\epsilon}^{p-2} \Vert v_{\epsilon} \Vert ^{p}- (q-1)\beta s_{\epsilon}^{p-2} \int_{\Omega} \vert v_{\epsilon} \vert ^{p} \vert x \vert ^{\alpha-p}\,dx -(q-1)s_{\epsilon}^{p^{*}-2} \int_{\Omega} \vert v_{\epsilon} \vert ^{p^{*}}\,dx. \end{aligned}

That is,

$$(p-q)s_{\epsilon}^{p-2} \Vert v_{\epsilon} \Vert ^{p}-(p-q)\beta s_{\epsilon }^{p-2} \int_{\Omega} \vert v_{\epsilon} \vert ^{p} \vert x \vert ^{\alpha-p}\,dx< \bigl(p^{*}-q\bigr)s_{\epsilon}^{p^{*}-2} \int_{\Omega} \vert v_{\epsilon} \vert ^{p^{*}}\,dx.$$
(2.26)

Hence, we can obtain from (2.26) that $$s_{\epsilon}$$ is bounded below. Moreover, it is clear to see from (2.24) that $$s_{\epsilon}$$ is bounded above for all $$\epsilon>0$$ small enough. Therefore, our claim holds.

Set

$$h(s_{\epsilon}v_{\epsilon})=\frac{s_{\epsilon}^{p}}{p} \Vert v_{\epsilon} \Vert ^{p}- \frac{s_{\epsilon}^{p^{*}}}{p^{*}} \int_{\Omega} \vert v_{\epsilon} \vert ^{p^{*}}\,dx.$$

In the following, we prove that

$$h(s_{\epsilon}v_{\epsilon})\leq\Lambda+O\bigl( \epsilon^{p(b(\mu)-\frac{N}{p}+1)}\bigr).$$
(2.27)

Let

$$\widetilde{h}(s)=\frac{s^{p}}{p} \Vert v_{\epsilon} \Vert ^{p}- \frac{s^{p^{*}}}{p^{*}} \int_{\Omega} \vert v_{\epsilon} \vert ^{p^{*}}\,dx.$$

Direct computations give us that $$\lim_{s\rightarrow\infty}\widetilde {h}(s)=-\infty$$ and $$\widetilde{h}(0)=0$$. Thus $$\sup_{s\geq0}\widetilde{h}(s)$$ is obtained at some $$S_{\epsilon }>0$$, and

$$S_{\epsilon}=\biggl(\frac{ \Vert v_{\epsilon} \Vert ^{p}}{ \int_{\Omega} \vert v_{\epsilon} \vert ^{p^{*}}\,dx}\biggr)^{\frac{1}{p^{*}-p}}.$$

Since $$\widetilde{h}^{\prime}(s) \vert _{S_{\epsilon}}=0$$, that is,

$$S_{\epsilon}^{p-1} \Vert v_{\epsilon} \Vert ^{p} -S_{\epsilon}^{p^{*}-1} \int_{\Omega} \vert v_{\epsilon} \vert ^{p^{*}}\,dx=0.$$

It is easy to check that $$h(s)$$ is increasing in $$[0,S_{\epsilon})$$, according to (2.21) and (2.22), we have

\begin{aligned} h(s_{\epsilon}v_{\epsilon})&\leq\widetilde {h}(S_{\epsilon}) \\ & =\biggl(\frac{1}{p}-\frac{1}{p^{*}}\biggr)\frac{( \Vert v_{\epsilon} \Vert ^{p})^{\frac {p^{*}}{p^{*}-p}}}{ (\int_{\Omega} \vert u_{\epsilon} \vert ^{p^{*}}\,dx)^{\frac{p}{p^{*}-p}}} \\ & =\biggl(\frac{1}{p}-\frac{1}{p^{*}}\biggr)\frac{((S_{\mu,0})^{\frac{N}{p}} +O(\epsilon^{b(\mu)p+p-N}))^{\frac{p^{*}}{p^{*}-p}}}{((S_{\mu,0})^{\frac{N}{p}} +O(\epsilon^{b(\mu)p^{*}-N}))^{\frac{p}{p^{*}-p}}} \\ &\leq\biggl(\frac{1}{p}-\frac{1}{p^{*}}\biggr) \frac{(S_{\mu,0})^{\frac{N}{p}\frac{p^{*}}{p^{*}-p}}}{(S_{\mu,0}) ^{\frac{N}{p}\frac{p}{p^{*}-p}}}+O \bigl(\epsilon^{b(\mu)p+p-N}\bigr) \\ &=\biggl(\frac{1}{p}-\frac{1}{p^{*}}\biggr) (S_{\mu,0})^{\frac{N}{p}}+O \bigl(\epsilon^{p(b(\mu)-\frac{N}{p}+1)}\bigr) \\ &=\Lambda+O\bigl(\epsilon^{p(b(\mu)-\frac{N}{p}+1)}\bigr). \end{aligned}
(2.28)

Therefore, by (2.27), we have

\begin{aligned} I_{\lambda}(s_{\epsilon}v_{\epsilon})&=h(s_{\epsilon} v_{\epsilon})-\frac{\beta s_{\epsilon}^{p}}{p} \int_{\Omega} \vert v_{\epsilon} \vert ^{p} \vert x \vert ^{\alpha-p}\,dx-\frac{\lambda s_{\epsilon}^{q}}{q} \int_{\Omega} \vert v_{\epsilon} \vert ^{q}\,dx \\ &\leq\Lambda+C\epsilon^{p(b(\mu)-\frac{N}{p}+1)}-\frac{\beta}{p}s_{0}^{p} \int_{\Omega} \vert v_{\epsilon} \vert ^{p} \vert x \vert ^{\alpha-p}\,dx -\frac{\lambda s_{0}^{q}}{q} \int_{\Omega} \vert v_{\epsilon} \vert ^{q}\,dx. \end{aligned}
(2.29)

Now, we consider the following cases:

(i) $$\frac{N}{b(\mu)}< q< p$$. Choose $$\epsilon=\lambda^{\frac{1}{(p-q)(b(\mu)-\frac{N}{p}+1)}}$$, for $$\lambda<\lambda_{1}:=(\frac{C_{1}+D}{C_{2}})^{\frac{(p-q)(b(\mu)-\frac {N}{p}+1)}{N-qb(\mu)}}$$, we have

\begin{aligned} C_{1} \epsilon^{p(b(\mu)-\frac{N}{p}+1)}-\lambda C_{2} \epsilon^{N+q(1-\frac{N}{p})} & =C_{1} \lambda^{\frac{p}{p-q}}-\lambda C_{2} \lambda^{\frac{N+q(1-\frac {N}{p})}{(p-q)(b(\mu)-\frac{N}{p}+1)}} \\ & =C_{1} \lambda^{\frac{p}{p-q}}-C_{2} \lambda^{\frac{N+q(1-\frac {N}{p})}{(p-q)(b(\mu)-\frac{N}{p}+1)}+1} \\ & =\lambda^{\frac{p}{p-q}}\bigl(C_{1}-C_{2} \lambda^{\frac{N-qb(\mu)}{(p-q)(b(\mu )-\frac{N}{p}+1)}}\bigr) \\ & < -D \lambda^{\frac{p}{p-q}}. \end{aligned}

(ii) $$q=\frac{N}{b(\mu)}$$. We still choose $$\epsilon=\lambda^{\frac{1}{(p-q)(b(\mu)-\frac {N}{p}+1)}}$$, for $$\lambda<\lambda_{2}:= e^{-(\frac{C_{1} +D}{C_{3}})}$$, we have

\begin{aligned} C_{1} \epsilon^{p(b(\mu)-\frac{N}{p}+1)}-\lambda C_{2} \epsilon^{N+q(1-\frac{N}{p})} \vert \ln\epsilon \vert & =C_{1} \lambda^{\frac{p}{p-q}}-\lambda C_{3} \lambda^{\frac{N+q(1-\frac {N}{p})}{(p-q)(b(\mu)-\frac{N}{p}+1)}} \vert \ln\lambda \vert \\ & =C_{1} \lambda^{\frac{p}{p-q}}-C_{3} \lambda^{\frac{N+q(1-\frac {N}{p})}{(p-q)(b(\mu)-\frac{N}{p}+1)}+1} \vert \ln\lambda \vert \\ & < \lambda^{\frac{p}{p-q}}\bigl(C_{1} -C_{3} \vert \ln \lambda \vert \bigr) \\ & < -D\lambda^{\frac{p}{p-q}}, \end{aligned}

where $$C_{3} =\frac{C_{2}}{(p-q)(b(\mu)-\frac{N}{p}+1)}$$.

(iii) $$1< q<\frac{N}{b(\mu)}$$. Put $$\epsilon^{p(b(\mu)-\frac {N}{p}+1)}\leq\lambda^{\frac{p}{p-q}}$$, for $$\lambda<\lambda_{3}:=(\frac{C_{2} -D}{C_{1}})^{\frac{p-q}{pq-p}}$$ with $$C_{2} >D$$, we have

\begin{aligned} C_{1} \epsilon^{p(b(\mu)-\frac{N}{p}+1)}-\lambda C_{2} \epsilon^{q(b(\mu)+1-\frac{N}{p})} & :=C_{1} \lambda^{\frac{pq}{p-q}}-\lambda C_{2} \lambda^{\frac{q}{p-q}} \\ & =\lambda^{\frac{p}{p-q}}\bigl(C_{1} \lambda^{\frac{pq-p}{p-q}}-C_{2} \bigr) \\ & < -D\lambda^{\frac{p}{p-q}}. \end{aligned}

Consequently, for $$\lambda<\lambda_{0}:=\min\{\lambda_{1}, \lambda_{2}, \lambda_{3}\}$$, we deduce that

$$I_{\lambda}(s_{\epsilon}v_{\epsilon})< \Lambda-D \lambda^{\frac{p}{p-q}}.$$

â€ƒâ–¡

3 Proof of main result

We can find a constant $$\delta>0$$ such that $$\Lambda-D\lambda^{\frac {p}{p-q}}>0$$ for $$\lambda<\delta$$. Let $$\lambda_{*}=\min\{T_{1}, \delta, \lambda_{0}\}$$. For $$\lambda\in(0,\lambda_{*})$$, Lemmas 2.1-2.4, 2.6 and 2.8 hold.

Let $$\{u_{n}\}\subset\mathcal{N}_{\lambda}$$ be a minimizing sequence of $$I_{\lambda}$$. It is easy to see that $$\{u_{n}\}$$ is bounded in $$W_{0}^{1,p}(\Omega)$$ and there exist a subsequence of $$\{u_{n}\}$$ (still denoted by $$\{u_{n}\}$$) and $$u_{\lambda}\in W_{0}^{1,p}(\Omega)$$ such that

$$\textstyle\begin{cases} u_{n}\rightharpoonup u_{\lambda} \quad\mbox{weakly in }W_{0}^{1,p}(\Omega), \\ u_{n}\rightarrow u_{\lambda} \quad\mbox{strongly in }L^{s}(\Omega)\ (1\leq s< p^{*}), \\ u_{n}(x)\rightarrow u_{\lambda}(x) \quad\mbox{a.e. in }\Omega, \end{cases}$$
(3.1)

as $$n\rightarrow\infty$$.

Firstly, by LemmaÂ 2.4, we can know that $$f_{n}^{\prime}(0)$$ is bounded with respect to $$n\in\mathbb{N}$$. Letting $$n\rightarrow\infty$$ in (2.11), we deduce that

\begin{aligned} &\int_{\Omega} \vert \nabla u_{*} \vert ^{p-2}\nabla u_{*}\cdot\nabla\phi-\mu \int _{\Omega}\frac{ \vert u_{*} \vert ^{p-2}u_{*}}{ \vert x \vert ^{p}}\phi \\ &\quad= \int_{\Omega } \vert u_{*} \vert ^{p^{*}-1}\phi +\lambda \int_{\Omega} \vert u_{*} \vert ^{q-1} \phi+\beta \int_{\Omega } \vert u_{*} \vert ^{p-1} \vert x \vert ^{\alpha-p} \phi \end{aligned}
(3.2)

for all $$\phi\in W_{0}^{1,p}(\Omega)$$. Equation (3.2) implies that $$u_{\lambda}$$ is a solution of (1.1). We claim that $$u_{\lambda}\not\equiv0$$. If not, $$u_{\lambda}=0$$, since $$u_{n}\in\mathcal{N}_{\lambda}$$, we have

$$\Vert u_{n} \Vert ^{p} - \int_{\Omega} \vert u_{n} \vert ^{p^{*}}- \beta \int_{\Omega} \vert u_{n} \vert ^{p} \vert x \vert ^{\alpha -p}-\lambda \int_{\Omega} \vert u_{n} \vert ^{q} =0.$$

Note that

$$\lim_{n\rightarrow\infty} \int_{\Omega} \vert u_{n} \vert ^{p} \vert x \vert ^{\alpha-p}\,dx=0,\qquad \lim_{n\rightarrow\infty} \int_{\Omega} \vert u_{n} \vert ^{q} \,dx=0.$$

Put $$\lim_{n\rightarrow\infty} \Vert u_{n} \Vert =m$$, we conclude that $$m\geq S_{\mu,0}^{\frac{p^{*}}{p(p^{*} -p)}}$$. By LemmaÂ 2.8, we obtain

\begin{aligned} \kappa_{\lambda}&=\lim_{n\rightarrow\infty}I_{\lambda}(u_{n}) \\ &= \lim_{n\rightarrow\infty}\biggl[\frac{1}{p} \Vert u_{n} \Vert ^{p} -\frac{\beta}{p} \int _{\Omega} \vert u_{n} \vert ^{p} \vert x \vert ^{\alpha-p} -\frac{1}{p^{*}} \int_{\Omega} \vert u_{n} \vert ^{p^{*}}\,dx- \frac{\lambda}{q} \int_{\Omega} \vert u_{n} \vert ^{q} \,dx \biggr] \\ &\geq \lim_{n\rightarrow\infty}\biggl(\frac{1}{p}-\frac{1}{p^{*}} \biggr) \Vert u_{n} \Vert ^{p} \\ &\geq\frac{p^{*} -p}{p p^{*}}S_{\mu,0}^{\frac{p^{*}}{p^{*} -p}} \\ & =\frac{1}{N}S_{\mu,0}^{\frac{N}{p}}, \end{aligned}

which contradicts with $$\kappa_{\lambda}<\Lambda-D\lambda^{\frac{p}{p-q}}$$ (from LemmaÂ 2.9).

Secondly, we prove that $$u_{\lambda}\in\mathcal{N}_{\lambda}^{+}$$. Suppose that this is not true, $$i.e$$., $$u_{\lambda}\in\mathcal {N}_{\lambda}^{-}$$. From LemmaÂ 2.1, we can find positive numbers $$s^{+}$$ and $$s^{-}$$ with $$s^{+}< s_{\max}< s^{-}=1$$ such that $$s^{+}u_{\lambda}\in \mathcal{N}_{\lambda}^{+}$$, $$s^{-}u_{\lambda}\in\mathcal{N}_{\lambda }^{-}$$ and

$$\kappa_{\lambda}< I_{\lambda}\bigl(s^{+}u_{\lambda} \bigr)< I_{\lambda }\bigl(s^{-}u_{\lambda} \bigr)=I_{\lambda}(u_{\lambda})=\kappa_{\lambda},$$

which is a contradiction. Hence $$u_{\lambda}\in\mathcal{N}_{\lambda}^{+}$$. Furthermore, combining with LemmaÂ 2.3, we can obtain

$$I_{\lambda}(u_{\lambda})=\kappa_{\lambda}^{+}= \kappa_{\lambda}< 0.$$

Therefore, we see that $$u_{\lambda}$$ is a non-negative ground state solution of problem (1.1).

In the following, we prove that problem (1.1) has a second solution $$v_{\lambda}$$ with $$v_{\lambda}\in\mathcal{N}_{\lambda}^{-}$$. Since $$I_{\lambda}$$ is coercive on $$\mathcal{N}_{\lambda}^{-}$$, according to the Ekeland variational principle and LemmaÂ 2.9, there exists a minimizing sequence $$\{v_{n}\}\subset\mathcal {N}_{\lambda}^{-}$$ of $$I_{\lambda}$$ such that

1. (i)

$$I_{\lambda}(v_{n})<\kappa_{\lambda}^{-}+\frac{1}{n}$$;

2. (ii)

$$I_{\lambda}(u)\geq I_{\lambda}(v_{n})-\frac{1}{n} \Vert u-v_{n} \Vert$$ for all $$u\in\mathcal{N}_{\lambda}^{-}$$.

Note that $$\{v_{n}\}$$ is bounded in $$W_{0}^{1,p}(\Omega)$$, there exist a subsequence (still denoted by $$\{v_{n}\}$$) and $$v_{\lambda}\in W_{0}^{1,p}(\Omega)$$ such that

$$\textstyle\begin{cases} v_{n}\rightharpoonup v_{\lambda} \quad\mbox{weakly in }W_{0}^{1,p}(\Omega), \\ v_{n}\rightarrow v_{\lambda}\quad \mbox{strongly in }L^{s}(\Omega)\ (1\leq s< p^{*}), \\ v_{n}(x)\rightarrow v_{\lambda}(x) \quad\mbox{a.e. in }\Omega, \end{cases}$$
(3.3)

as $$n\rightarrow\infty$$.

Similar to the above discussion, we can deduce that $$v_{n}\rightarrow v_{\lambda}$$ in $$W_{0}^{1,p}(\Omega)$$ and $$v_{\lambda}$$ is a non-negative solution of (1.1). Thirdly, we show that $$v_{\lambda}\neq0$$ in Î©. According to $$v_{n}\in\mathcal{N}_{\lambda}^{-}$$, we obtain

\begin{aligned} (p-q) \Vert v_{n} \Vert ^{p}&=\bigl(p^{*}-q \bigr) \int_{\Omega} \vert v_{n} \vert ^{p^{*}}\,dx+(p-q) \beta \int_{\Omega} \vert v_{n} \vert ^{p} \vert x \vert ^{\alpha-p}\,dx \\ & < \bigl(p^{*}-q\bigr)S_{\mu,0}^{-\frac{p^{*}}{p}} \Vert v_{n} \Vert ^{p^{*}} +(p-q)\frac{\beta}{\beta_{1}} \Vert v_{n} \Vert ^{p}, \end{aligned}

hence

$$\Vert v_{n} \Vert >\biggl[\frac{(p-q)(1-\frac{\beta}{\beta_{1}})S_{\mu,0}^{\frac{p^{*}}{p}}}{ p^{*}-q} \biggr]^{\frac{1}{p^{*}-p}}, \quad\forall v_{n}\in\mathcal{N}_{\lambda}^{-},$$
(3.4)

together with $$v_{n}\rightarrow v_{\lambda}$$ in $$W_{0}^{1,p}(\Omega)$$ means that $$v_{\lambda}\not\equiv0$$.

Lastly, we show that $$v_{\lambda}\in\mathcal{N}_{\lambda}^{-}$$. We only need to prove that $$\mathcal{N}_{\lambda}^{-}$$ is closed. In fact, for $$\{v_{n}\}\subset\mathcal{N}_{\lambda}^{-}$$, it follows from Lemmas 2.8 and 2.9 that

$$\lim_{n\rightarrow\infty} \int_{\Omega} \vert v_{n} \vert ^{p^{*}}\,dx= \int_{\Omega} \vert v_{\lambda} \vert ^{p^{*}}\,dx.$$

$$(p-q) \Vert v_{n} \Vert ^{p}-\bigl(p^{*}-q \bigr) \int_{\Omega} \vert v_{n} \vert ^{p^{*}}\,dx-(p-q)\beta \int_{\Omega} \vert v_{n} \vert ^{p} \vert x \vert ^{\alpha-p}\,dx< 0.$$

Thus

$$(p-q) \Vert v_{\lambda} \Vert ^{p}-\bigl(p^{*}-q \bigr) \int_{\Omega} \vert v_{\lambda} \vert ^{p^{*}}\,dx-(p-q)\beta \int_{\Omega} \vert v_{\lambda} \vert ^{p} \vert x \vert ^{\alpha-p}\,dx\leq0,$$

which means that $$v_{\lambda}\in\mathcal{N}_{\lambda}^{0}\cup\mathcal {N}_{\lambda}^{-}$$. Combining with LemmaÂ 2.1 and $$v_{\lambda}\not\equiv0$$, we see that $$\mathcal{N}_{\lambda}^{-}$$ is closed. Note that $$\mathcal{N}_{\lambda}^{+}\cap\mathcal{N}_{\lambda }^{-}=\emptyset$$, we know that $$u_{\lambda}$$ and $$v_{\lambda}$$ are different.

4 Conclusions

In this paper, we study the existence and multiplicity of positive solutions for the quasi-linear elliptic problem which consists of critical Sobolev exponent and a Hardy term.

The main conclusions of this work:

1. (1)

Adding a linear perturbation in the nonlinear term of elliptic equation.

2. (2)

The main challenge of this study is the lack of compactness of the embedding $$W_{0}^{1,p}\hookrightarrow L^{p^{*}}$$. We overcome it by the concentration compactness principle.

3. (3)

We apply the Ekeland variational principle to obtain a minimizing sequence with good properties.

5 Discussion

In the future, a natural question is whether the multiplicity of positive solutions for (1.1) can be established with negative exponent $$\frac{1}{u^{\gamma}}\ (0<\gamma<1)$$.

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Acknowledgements

This project is supported by the Natural Science Foundation of Shanxi Province (2016011003), Science Foundation of North University of China (110246), NSFC (11401583) and the Fundamental Research Funds for the Central Universities (16CX02051A).

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Correspondence to Yanbin Sang.

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Sang, Y., Guo, S. Solutions for the quasi-linear elliptic problems involving the critical Sobolev exponent. J Inequal Appl 2017, 217 (2017). https://doi.org/10.1186/s13660-017-1492-y