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The existence of ground state solution to elliptic equation with exponential growth on complete noncompact Riemannian manifold

Abstract

In this paper, we consider the following elliptic problem:

$$ -\mathtt{div}_{g}\bigl( \vert \nabla_{g} u \vert ^{N-2}\nabla_{g} u \bigr)+V(x) \vert u \vert ^{N-2}u = \frac{f(x, u)}{a(x)}\quad \mbox{in } M, \qquad (P_{a}) $$

where \((M, g)\) be a complete noncompact N-dimensional Riemannian manifold with negative curvature, \(N\geq2\), V is a continuous function satisfying \(V(x) \geq V_{0 }> 0\), a is a nonnegative function and \(f(x, t)\) has exponential growth with t in view of the Trudinger–Moser inequality. By proving some estimates together with the variational techniques, we get a ground state solution of (\(P_{a}\)). Moreover, we also get a nontrivial weak solution to the perturbation problem.

Introduction and main results

Suppose \(\varOmega\subset\mathbb{R}^{N}\) is a bounded smooth domain, and we consider the following problem:

$$ -\mathtt{div}\bigl( \vert \nabla u \vert ^{p-2}\nabla u\bigr) =f(x, u),\quad x \in\varOmega, $$
(1.1)

when \(p=2\), \(|f(x, u)|\leq c(|u|+|u|^{q-1})\), \(1< q\leq2^{*}=\frac{2N}{N-2}\), \(N>2\), for corresponding results one may refer to Brézis [1], Brézis and Nirenberg [2], Bartsch and Willem [3], Capozzi, Fortunato and Palmieri [4]. Garcia and Alonso [5] generalized Brézis and Nirenberg’s existence and nonexistence results to the p-Laplace equation. When \(p = N\) and \(f(x, u)\) behaves like \(e^{\alpha|u|^{\frac{N}{N-1}}}\) as \(|u|\rightarrow\infty\), problem (1.1) was studied by Adimurthi [6], Adimurthi and Yadava [7], Ruf et al. [8, 9], do Ó [10], Panda [11] and the references therein. Specially, Lam and Lu [12] study the case which does not satisfy the Ambrosetti–Rabinowitz condition. All these results are based on the Trudinger–Moser inequality [13, 14] and critical point theory.

For the following semilinear Schrödinger equation:

$$ -\mathtt{div}\bigl( \vert \nabla u \vert ^{p-2}\nabla u\bigr)+V(x) \vert u \vert ^{p-2}u =f(x, u),\quad x \in\mathbb{R}^{N}, $$
(1.2)

when \(p=2\), \(|f(x, u)|\leq c(|u|+|u|^{q-1})\), \(1< q\leq2^{*}=\frac{2N}{N-2}\), \(N>2\), for the relevant literature one is referred to Kryszewski and Szulkin [15], Alama and Li [16], Ding and Ni [17] and Jeanjean [18]. When \(p = N\), \(f (x, s)\) is continuous in \(\mathbb{R}^{N}\times\mathbb{R}\) and behaves like \(e^{\alpha|s|^{\frac{N}{N-1}}}\) as \(|s|\rightarrow\infty\), it was studied by Cao [19] in the case \(N = 2\), by Panda [20], do Ó [21], Alves, Figueiredo [22] and do Ó et al. [23] in general dimensional cases. We note that the Sobolev embedding theorem can be applied to the \(p< N\) case, while the Trudinger–Moser type embedding theorem can be applied to the \(p=N\) case.

The Trudinger–Moser inequalities have been generalized to some Riemannian manifolds. In the case of compact Riemannian manifolds, the study of Trudinger–Moser inequalities can be traced back to Aubin [24], Cherrier [25, 26], and Fontana [27]. On a compact Riemannian surface, Ding et al. [28] established a Trudinger–Moser inequality and used it to deal with the prescribed Gaussian curvature problem by the method of blow-up analysis. Motivated by the ideas of Ding et al., the existence of extremal functions was proved by Li in [29, 30], Yang also got extremal functions for several Trudinger–Moser type inequalities in [31, 32].

When \((M, g)\) is any complete noncompact N-dimensional Riemannian manifold with no boundary, \(N\geq2\), if its Ricci curvature is bounded from below and injectivity radius is bounded from below by a positive number, Yang in [33] established some Trudinger–Moser inequalities and applied them to some quasilinear equations. Recently, Yang et al. [34] derived a sharp Trudinger–Moser inequality on a complete, simply connected N-dimensional Riemannian manifold with negative curvature. We state it below for further use.

Proposition A

([34])

Let\((M, g)\)be a complete, simply connectedN-dimensional Riemannian manifold with negative curvature, \(N\geq2\). Then for any\(\tau>0\)there exists a positive constant\(C=C(\tau,N,M)\)such that

$$ \sup_{u\in W^{1, N}(M), \Vert u \Vert _{1, \tau}\leq1} \int_{M} \bigl(\varPhi\bigl( \alpha_{N} \vert u_{n} \vert ^{\frac{N}{N-1}}\bigr) \bigr)\,dv_{g}\le C. $$

Furthermore, the constant\(\alpha_{N}=N\omega_{N-1}^{1/(N-1)}\)is sharp, where\(\omega_{N-1}\)is the surface measure of the unit sphere in\(\mathbb{R}^{N}\). \(\varPhi(x)=e^{x}-\sum_{k=0}^{N-2}\frac{x^{k}}{k!}\)and

$$ \Vert u \Vert _{1, \tau}= \biggl( \int_{M} \vert \nabla_{g}u \vert ^{N}\,dv_{g}+\tau \int _{M} \vert u \vert ^{N}\,dv_{g} \biggr)^{\frac{1}{N}}. $$

More recently, Kristály [35] studied some geometric features of Trudinger–Moser inequalities on complete noncompact Riemannian manifolds and proved the existence of a non-zero isometry-invariant solution for a class of N-Laplacian equation by combining variational techniques and the symmetrization-compactness principle.

In this paper, we always assume that \((M, g)\) is a complete, simply connected N-dimensional Riemannian manifold with negative curvature, \(N\geq2\). And on \((M,g)\), we consider the following quasilinear equations:

figurea

where \(\epsilon>0\), \(h\in E^{*}\), \(h\geq0\) and \(h\not\equiv0\), \((M, g)\) is a complete noncompact N-dimensional Riemannian manifold with negative curvature, \(N\geq2\), \(\nabla_{g}\) denotes its covariant derivative (the Levi–Civita connection), and \(\mathtt{div}_{g}\) denotes its divergence operator, V is a continuous function satisfying \(V(x) \geq V_{0 }> 0\), a is a nonnegative function and \(f(x, t)\) has exponential growth with t in view of the Trudinger–Moser inequality. Since we are concerned with nonnegative weak solutions, we require that \(f(x, t)=0\) for all \((x, t)\in M\times(-\infty, 0]\). Furthermore, let O be a fixed point of M and \(d_{g}(\cdot, \cdot)\) be the geodesic distance between two points of \((M, g)\). We assume the function a satisfies:

\((a_{1})\):

\(a^{-1}\in L_{\mathrm{loc}}^{s}(M)\) for some \(s>1\), i.e., \(a^{-1}\in L^{s}(B_{\rho}(O))\) for any \(\rho>0\).

\((a_{2})\):

\(a^{-1}\) is bounded in \(M \setminus B_{\rho}(O)\) for any \(\rho>0\).

We assume the following conditions on V.

\((V_{1})\):

There exists a constant \(V_{0}>0\) such that \(V(x)\geq V_{0}\) for all \(x\in M\).

\((V_{2})\):

\(V(x)\rightarrow+\infty\) as \(d_{g}(O, x)\rightarrow+\infty\).

The function f satisfies:

\((f_{1})\):

There exist constants \(\alpha_{0}, c_{1}, c_{2} > 0\) such that, for all \((x, t)\in M\times\mathbb{R}^{+}\),

$$ f(x, t)\leq c_{1} \vert t \vert ^{N-1}+c_{2} \varPhi\bigl(\alpha_{0} \vert u_{n} \vert ^{\frac{N}{N-1}}\bigr), $$

where

$$ \varPhi(x)=e^{x}-\sum_{k=0}^{N-2} \frac{x^{k}}{k!}. $$
\((f_{2})\):

\(\lim_{|t|\rightarrow+\infty}\frac{F(x, t)}{|t|^{N}}=\infty\) uniformly on \(x \in M\), where \(F(x,t)=\int_{0}^{t}f(x,s)\,ds\).

\((f_{3})\):

There exists \(C>0\) such that, for all \((x, t)\in M\times\mathbb{R}^{+}\), \(F(x,t)\leq C|t|^{N}+Cf(x, t)\).

Define a function space E as

$$ E= \biggl\{ u \in W^{1, N}(M): \int_{M} \bigl( \vert \nabla_{g} u \vert ^{N}+V(x) \vert u \vert ^{N} \bigr)\,dv_{g}< \infty \biggr\} $$

equipped with the norm

$$ \Vert u \Vert _{E}= \biggl( \int_{M} \bigl( \vert \nabla_{g} u \vert ^{N}+V(x) \vert u \vert ^{N} \bigr)\,dv_{g} \biggr)^{\frac{1}{N}}, $$

then the assumption \(V(x)\geq V_{0}>0\) implies that E is a reflexive Banach space. For any \(p\ge N\), we define

$$ S_{p}=\inf_{u\in E\setminus \{0\}} \frac{ \Vert u \Vert _{E}}{(\int_{M} \frac{ \vert u \vert ^{p}}{a(x)}\,dv_{g})^{\frac{1}{p}}} $$
(1.3)

and

$$ \lambda_{a}=\inf_{u\in E\setminus \{0\}} \frac{ \Vert u \Vert _{E}^{N}}{\int_{\mathbb{R}^{N}} \frac{ \vert u \vert ^{N}}{a(x)}\,dx}. $$

The continuous embedding of \(E\hookrightarrow W^{1, N}(M) \hookrightarrow L^{p}(M)\) (\(p \geq N\)) and the Hölder inequality imply

$$\begin{aligned} \int_{M}\frac{ \vert u \vert ^{p}}{a(x)}\,dv_{g} &= \int_{M \setminus B_{\rho}(O)} \frac{ \vert u \vert ^{p}}{a(x)}\,dv_{g}+ \int_{ B_{ \rho}(O)}\frac{ \vert u \vert ^{p}}{a(x)}\,dv_{g} \\ &\leq C \int_{M \setminus B_{\rho}(O)} \vert u \vert ^{p}\,dv_{g} + \biggl( \int_{ B_{\rho}(O)} \vert u \vert ^{ps'}\,dv_{g} \biggr)^{ \frac{1}{s'}} \biggl( \int_{ B_{\rho}(O)} \biggl(\frac{1}{a(x)} \biggr)^{s}\,dv_{g} \biggr)^{\frac{1}{s}} \\ &\leq C \Vert u \Vert _{E}^{p}, \end{aligned}$$

here s is stated in the condition \((a_{1})\). Thus we have \(S_{p} > 0\). We now introduce the following three conditions.

\((f_{4})\):

\(\limsup_{t\rightarrow0^{+}}\frac{NF(x, t)}{|t|^{N}}<\lambda_{a}\) uniformly in M.

\((f_{5})\):

There exist constants \(p > N\) and \(C_{p}\) such that

$$ f(x, t)\geq C_{p}t^{p-1}, $$

where

$$ C_{p}> \biggl(\frac{p-N}{p} \biggr)^{\frac{p-N}{N}} \biggl( \frac{s\alpha_{0}}{(s-1)\alpha_{N}} \biggr)^{\frac{(p-N)(N-1)}{N}}S_{p}^{p}. $$
\((f_{6})\):

\(H (x, t) = tf (x, t)-NF (x, t)\) is strictly increasing in \(t > 0\).

Our main results can be stated as follows.

Theorem 1.1

SupposeVsatisfies\((V_{1})\)and\((V_{2})\), asatisfies\((a_{1})\)and\((a_{2})\), fsatisfies\((f_{1})\)\((f_{6})\). Then the problem (\(P_{a}\)) has a positive ground state solution.

Remark 1.2

  1. (1)

    When \((M, g)\) is the standard Euclidean space \(\mathbb{R^{N}}\) and \(a(x)=|x|^{\eta}\) (\(0\leq\eta< N\)), the equation (\(P_{a}\)) and the perturbation problem (P) have been studied by many authors such as Adimurthi and Yang [36], Yang [37], Lam and Lu [38].

  2. (2)

    Comparing with the existing literature, we firstly study the ground state solution of problem (\(P_{a}\)) on complete noncompact Riemannian manifold without the Ambrosetti–Rabinowitz condition.

Theorem 1.3

SupposeVsatisfies\((V_{1})\)and\((V_{2})\), asatisfies\((a_{1})\)and\((a_{2})\), fsatisfies\((f_{1})\)\((f_{4})\)and\((f_{6})\). Then the perturbation problem (P) has a nontrivial weak solution of mountain-pass type.

This paper is organized as follows: In Sect. 2, some preliminary results are introduced. In Sect. 3, we study the functionals and related compact analysis. In Sect. 4, we give a proof of Theorem 1.1. Finally, in Sect. 5, we prove Theorem 1.3.

Preliminaries

In this section, we give some preliminaries for later use.

Lemma 2.1

Supposeasatisfies\((a_{1})\)and\((a_{2})\), Vsatisfies condition\((V_{1})\). Let\(\alpha> 0\)and\(\{u_{n}\}\)be a sequence satisfying

$$ \limsup_{n\rightarrow\infty} \Vert u_{n} \Vert _{E}^{\frac{N}{N-1}}< \frac{\alpha_{N}}{\alpha}, $$

where\(\alpha_{N}=N\omega_{N-1}^{1/(N-1)}\)is the sharp constant in Proposition A. Then there exist constants\(t > 1\), \(C > 0\), independent ofn, such that

$$ \int_{M}\frac{[\varPhi(\alpha \vert u_{n} \vert ^{\frac{N}{N-1}})]^{t}}{a(x)}\,dv_{g} \leq C. $$

Proof

Since

$$ \limsup_{n\rightarrow\infty} \Vert u_{n} \Vert _{E}< \biggl( \frac{\alpha_{N}}{\alpha} \biggr)^{\frac{N-1}{N}}, $$

passing to a subsequence, there exists \(n_{0}\in\mathbb{N}\) such that

$$ \Vert u_{n} \Vert _{E}^{\frac{N}{N-1}}\leq m< \frac{\alpha_{N}}{\alpha}, \quad \forall n\geq n_{0}. $$

Choose \(k>ts'>1\) such that \(k\alpha m<\alpha_{N}\) and \(t< s\), here s is stated in the condition \((a_{1})\) and \(\frac{1}{s}+\frac{1}{s'}=1\). Now for each \(n\geq n_{0}\), combining with \(\frac{m}{\|u_{n}\|_{E}^{\frac{N}{N-1}}} \ge1\),

$$\begin{aligned}& \int_{M}\frac{[\varPhi(\alpha \vert u_{n} \vert ^{\frac{N}{N-1}})]^{t}}{a(x)}\,dv_{g} \\& \quad = \int_{ B_{\rho}(O)} \frac{[\varPhi(\alpha \vert u_{n} \vert ^{\frac{N}{N-1}})]^{t}}{a(x)}\,dv_{g}+ \int _{M \setminus B_{\rho}(O)} \frac{[\varPhi(\alpha \vert u_{n} \vert ^{\frac{N}{N-1}})]^{t}}{a(x)}\,dv_{g} \\& \quad \leq \bigl\Vert a^{-1} \bigr\Vert _{L^{s}(B_{\rho}(O))} \biggl( \int_{ B_{\rho}(O)}\bigl( \varPhi\bigl(\alpha \vert u_{n} \vert ^{\frac{N}{N-1}}\bigr)\bigr)^{ts'}\,dv_{g} \biggr)^{ \frac{1}{s'}}+C \int_{M \setminus B_{\rho}(O)}\bigl(\varPhi\bigl(\alpha \vert u_{n} \vert ^{ \frac{N}{N-1}}\bigr)\bigr)^{t}\,dv_{g} \\& \quad \leq \bigl\Vert a^{-1} \bigr\Vert _{L^{s}(B_{\rho}(O))} \biggl( \int_{M}\varPhi\bigl(ts' \alpha \vert u_{n} \vert ^{\frac{N}{N-1}}\bigr)\,dv_{g} \biggr)^{\frac{1}{s'}}+C \int _{M}\varPhi\bigl(t\alpha \vert u_{n} \vert ^{\frac{N}{N-1}}\bigr)\,dv_{g} \\& \quad \leq C \int_{M}\varPhi\bigl(k\alpha \vert u_{n} \vert ^{\frac{N}{N-1}}\bigr)\,dv_{g} \leq C \int_{M}\varPhi\biggl(k\alpha m\biggl(\frac{ \vert u_{n} \vert }{ \Vert u_{n} \Vert _{E}} \biggr)^{ \frac{N}{N-1}}\biggr)\,dv_{g}. \end{aligned}$$

The property \(\varPhi(q)^{p}\le\varPhi(pq)\) (see [37]) has been used above. Finally, the result is derived from Proposition A by taking \(\tau=V_{0}\) in condition \((V_{1})\). □

Lemma 2.2

SupposeVsatisfies\((V_{1})\)and\((V_{2})\), asatisfies\((a_{1})\)and\((a_{2})\), we have\(E \hookrightarrow L^{q}(M)\), \(q \geq N\). Moreover, this continuous embedding is compact.

Proof

From the condition \((V_{1})\), we have \(E \hookrightarrow W^{1, N}(M)\), together with the standard Sobolev embedding theorem \(W^{1, N}(M) \hookrightarrow L^{q}(M)\) for any \(q\geq N\), we immediately derive that E can be continuously embedded into \(L^{q}(M)\).

Next, for any \(q\geq N\), one needs to prove that the above continuous embedding \(E \hookrightarrow L^{q}(M)\) is compact. Let \(\{u_{n}\}\) be a sequence of functions with \(\|u_{n}\|_{E}^{N}\leq C\) for some constant C, it suffices to prove that, up to a subsequence, \(\{u_{n}\}\) strongly converges in \(L^{q}(M)\). Obviously \(\{u_{n}\}\) is also bounded in \(W^{1, N}(M)\), thus we can assume that, for any \(q\geq N\), there exists \(u_{0}\in E\) such that up to a subsequence

$$ \textstyle\begin{cases} u_{n}\rightharpoonup u_{0} \quad \mbox{in } E, \\ u_{n}\rightarrow u_{0}\quad \mbox{in } L^{q}(B_{\rho}(O)), \\ u_{n}(x)\rightarrow u_{0}(x)\quad \mbox{a.e. in } M. \end{cases} $$

Hence for any \(\epsilon>0\), there exists some positive integral \(n_{0}\), such that when \(n>n_{0}\),

$$ \int_{ B_{\rho}(O)} \vert u_{n}-u_{0} \vert ^{N}\,dv_{g}< \frac{\epsilon}{2}. $$
(2.1)

On the other hand, for any \(\epsilon>0\), from the condition \((V_{2})\), we can assume \(V(x)>4C/\epsilon\) when \(x\in M \setminus B_{\rho}(O)\), so

$$ \frac{4C}{\epsilon} \int_{M \setminus B_{\rho}(O)} \vert u_{n}-u_{0} \vert ^{N}\,dv_{g}< \int_{M \setminus B_{\rho}(O)}V \vert u_{n}-u_{0} \vert ^{N}\,dv_{g}\leq2C, $$

thus

$$ \int_{M \setminus B_{\rho}(O)} \vert u_{n}-u_{0} \vert ^{N}\,dv_{g}< \frac{\epsilon}{2}. $$
(2.2)

From (2.1) and (2.2), when \(n>n_{0}\), we obtain

$$ \int_{M} \vert u_{n}-u_{0} \vert ^{N}\,dv_{g}< \epsilon. $$
(2.3)

By Hölder’s inequality and the continuous embedding \(E \hookrightarrow L^{q}(M)\) (\(q\geq N\)), we have

$$\begin{aligned} \int_{M} \vert u_{n}-u_{0} \vert ^{q}\,dv_{g}&= \int_{M} \vert u_{n}-u_{0} \vert \cdot \vert u_{n}-u_{0} \vert ^{q-1}\,dv_{g} \\ &\leq \biggl( \int_{M} \vert u_{n}-u_{0} \vert ^{N}\,dv_{g} \biggr)^{ \frac{1}{N}} \biggl( \int_{M} \vert u_{n}-u_{0} \vert ^{\frac{N(q-1)}{N-1}}\,dv_{g} \biggr)^{\frac{N-1}{N}} \\ &\leq C \biggl( \int_{M} \vert u_{n}-u_{0} \vert ^{N}\,dv_{g} \biggr)^{ \frac{1}{N}}. \end{aligned}$$

Combining with (2.3), we see that \(\{u_{n}\}\) strongly converges in \(L^{q}(M)\). □

Lemma 2.3

SupposeVsatisfies\((V_{1})\), asatisfies\((a_{1})\)and\((a_{2})\). Let\(\beta>0\)and\(\|u\|_{E}\leq M\)such that\(\beta M ^{\frac{N}{N-1}}<\alpha_{N}\)and\(q>N\), then

$$ \int_{M} \vert u \vert ^{q} \frac{\varPhi(\beta \vert u \vert ^{\frac{N}{N-1}})}{a(x)}\,dv_{g} \leq C(\beta, N) \Vert u \Vert _{E}^{q}. $$

Proof

On the one hand, using the Hölder inequality, we have

$$\begin{aligned}& \int_{B_{\rho}(O)} \vert u \vert ^{q} \frac{\varPhi(\beta \vert u \vert ^{\frac{N}{N-1}})}{a(x)}\,dv_{g} \\& \quad \leq \bigl\Vert a^{-1} \bigr\Vert _{L^{s}(B_{\rho}(O))} \biggl( \int_{B_{\rho}(O)} \vert u \vert ^{qk}\,dv_{g} \biggr)^{\frac{1}{k}} \biggl( \int_{ M} \bigl(\varPhi\bigl(\beta \vert u \vert ^{ \frac{N}{N-1}}\bigr) \bigr)^{t}\,dv_{g} \biggr)^{\frac{1}{t}}, \end{aligned}$$
(2.4)

where \(t>1\) is sufficiently close 1 such that \(\beta t M^{\frac{N}{N-1}}<\alpha_{N}\) and \(1/s+1/k+1/t=1\).

On the other hand, we get

$$\begin{aligned}& \int_{M \setminus B_{\rho}(O)} \vert u \vert ^{q} \frac{\varPhi(\beta \vert u \vert ^{\frac{N}{N-1}})}{a(x)}\,dv_{g} \\& \quad \leq C \biggl( \int_{M} \vert u \vert ^{q\gamma'}\,dv_{g} \biggr)^{ \frac{1}{\gamma'}} \biggl( \int_{ M} \bigl(\varPhi\bigl(\beta \vert u \vert ^{ \frac{N}{N-1}}\bigr) \bigr)^{\gamma}\,dv_{g} \biggr)^{\frac{1}{\gamma}}, \end{aligned}$$
(2.5)

where \(\gamma>1\) is sufficiently close 1 such that \(\beta\gamma M^{\frac{N}{N-1}}<\alpha_{N}\), \(1/\gamma'+1/\gamma=1\).

From (2.4) and (2.5), by Proposition A and the continuous embedding of \(E\hookrightarrow L^{q}(\mathbb{R}^{N})\) (\(q\geq N\)), we have

$$ \int_{M} \vert u \vert ^{q} \frac{\varPhi(\beta \vert u \vert ^{\frac{N}{N-1}})}{a(x)}\,dv_{g} \leq C(\beta, N) \Vert u \Vert _{E}^{q}. $$

This completes the proof. □

Functionals and compactness analysis

We say that \(u\in E\) is a weak solution of problem (\(P_{a}\)) if, for all \(\phi\in E\),

$$ \int_{M}\bigl( \vert \nabla_{g} u \vert ^{N-2}\nabla_{g} u \nabla_{g} \phi+V(x) \vert u \vert ^{N-2}u \phi\bigr)\,dv_{g}- \int_{M}\frac{f(x, u)}{a(x)}\phi \,dv_{g}=0. $$

Define the functional \(I: E\rightarrow\mathbb{R}\) by

$$ I(u)=\frac{1}{N} \Vert u \Vert _{E}^{N} - \int_{M}\frac{F(x, u)}{a(x)}\,dv_{g}, $$
(3.1)

where \(F(x, t)=\int_{0}^{t}f(x, s)\,ds\). I is well defined and \(I\in C^{1}(E, \mathbb{R})\) thanks to the Trudinger–Moser inequality. A straightforward calculation shows that

$$ \bigl\langle I'(u), \phi\bigr\rangle = \int_{M} \vert \nabla_{g} u \vert ^{N-2}\nabla_{g} u \nabla_{g} \phi \,dv_{g}+ \int_{M}V(x) \vert u \vert ^{N-2}u\phi \,dv_{g}- \int_{M} \frac{f(x, u)}{a(x)}\phi \,dv_{g}, $$
(3.2)

for all \(u, \phi\in E\). Hence, a critical point of I defined in (3.1) is a weak solution of (\(P_{a}\)).

In this paper, we will use the mountain-pass theorem for the existence of the Cerami sequence which was introduced in [39, 40].

Definition A

Let \((E, \|\cdot\|_{E})\) be a real Banach space with its dual space \((E^{*}, \|\cdot\|_{E^{*}})\). Suppose \(I\in C^{1}(E, \mathbb{R})\). For \(c\in\mathbb{R}\), we call \(\{u_{n}\}\subset E\) a \((C)_{c}\) sequence of the functional I, if

$$ I(u_{n})\rightarrow c \quad \mbox{and}\quad \bigl(1+ \Vert u_{n} \Vert _{E}\bigr) \bigl\Vert I'(u_{n}) \bigr\Vert _{E^{*}}\rightarrow0 \quad \mbox{as } n \rightarrow \infty. $$

Proposition B

Let\((E, \|\cdot\|_{E})\)be a real Banach space, \(I\in C^{1}(E, \mathbb{R})\), \(I(0)=0\)and satisfies:

  1. (i)

    There exist positive constantsδandrsuch that

    $$ I(u)\geq\delta\quad \textit{for }\|u\|_{E}=r $$

    and

  2. (ii)

    there exists\(e\in E\)with\(\|e\|_{E}>r\)such that

    $$ I(e)\leq0 . $$

Definecby

$$ c=\inf_{\gamma\in\varGamma} \max_{t\in[0, 1]} I\bigl( \gamma(t)\bigr), $$

where

$$ \varGamma=:\bigl\{ \gamma\in C\bigl([0, 1]:E\bigr):\gamma(0)=0, \gamma(1)=e\bigr\} . $$

ThenIpossesses a\((C)_{c}\)sequence.

Next we will check the geometry of the functional I.

Lemma 3.1

Assume that\((V_{1})\), \((f_{1})\)and\((f_{4})\)hold, asatisfies\((a_{1})\)and\((a_{2})\). Then there exist positive constantsδandrsuch that

$$ I(u)\geq\delta \quad \textit{for } \|u\|_{E}=r . $$

Proof

From \((f_{4})\), there exist \(\sigma, \epsilon>0\), such that if \(|u|\leq\epsilon\),

$$ F(x,u)\leq\frac{\lambda_{a}-\sigma}{N} \vert u \vert ^{N} $$

for all \(x\in M\). On the other hand, using \((f_{1})\) for each \(q> N\), we have

$$ \begin{aligned}[b] F(x,u)&\leq\frac{c_{1}}{N} \vert u \vert ^{N}+c_{2} \vert u \vert \varPhi\bigl( \alpha_{0} \vert u_{n} \vert ^{ \frac{N}{N-1}}\bigr) \\ &\leq C \vert u \vert ^{q}\varPhi\bigl(\alpha_{0} \vert u_{n} \vert ^{\frac{N}{N-1}}\bigr) \end{aligned} $$

for \(|u|\geq\epsilon\) and \(x\in M\). Combining the above estimates, we obtain

$$ F(x,u)\leq\frac{\lambda_{a}-\sigma}{N} \vert u \vert ^{N}+C \vert u \vert ^{q}\varPhi\bigl( \alpha_{0} \vert u_{n} \vert ^{\frac{N}{N-1}}\bigr), $$

for all \((x, u)\in M\times\mathbb{R}\). Fix \(r>0\) small enough such that \(\alpha_{0} r^{\frac{N}{N-1}}<\alpha_{N}\), then Lemma 2.3 implies

$$\begin{aligned} I(u)&=\frac{1}{N} \Vert u \Vert _{E}^{N} - \int_{M}\frac{F(x, u)}{a(x)}\,dv_{g} \\ &\geq\frac{1}{N} \Vert u \Vert _{E}^{N}- \frac{\lambda_{a}-\sigma}{N} \int_{M} \frac{ \vert u \vert ^{N}}{a(x)}\,dv_{g}-C \int_{M} \frac{ \vert u \vert ^{q}\varPhi(\alpha_{0} \vert u_{n} \vert ^{\frac{N}{N-1}})}{a(x)}\,dv_{g} \\ &\geq\frac{1}{N} \Vert u \Vert _{E}^{N}- \frac{\lambda_{a}-\sigma}{N} \int_{M} \frac{ \vert u \vert ^{N}}{a(x)}\,dv_{g}-C \Vert u \Vert _{E}^{q} \\ &\geq\frac{1}{N}\biggl(1-\frac{\lambda_{a}-\sigma}{\lambda_{a}}\biggr) \Vert u \Vert _{E}^{N}-C \Vert u \Vert _{E}^{q}{= \frac{\sigma}{N\lambda_{a}} \Vert u \Vert _{E}^{N}-C \Vert u \Vert _{E}^{q}}, \end{aligned}$$

for \(\|u\|_{E}\leq r\). Hence, I is bounded from below for \(\|u\|_{E}\leq r\). Since \(\sigma>0\) and \(q >N\), we may choose a small r that satisfies the inequality

$$ {\frac{\sigma}{N\lambda_{a}} r^{N}-Cr^{q}\geq \frac{\sigma}{2N\lambda_{a}}r^{N}}, $$

and we derive that

$$ I(u)\geq\frac{\sigma}{2N\lambda_{a}}r^{N}:=\delta>0 \quad \mbox{for } \|u \|_{E}=r . $$

This completes the proof. □

Lemma 3.2

If the condition\((f_{2})\)is satisfied, then there exists\(e\in B_{r}^{c}(0)\)such that

$$ I(e)< \inf_{\|u\|_{E}=r}I(u) , $$

whereris given in Lemma 3.1.

Proof

Let \(u\in E\setminus\{0\}\), \(u\geq0\) with compact support \(\varOmega=\operatorname{supp} (u)\), by \((f_{2})\), there exists d such that, for all \((x, s) \in\varOmega\times\mathbb{R}^{+}\),

$$ F(x, s)\geq M s^{N}-d. $$

Then

$$\begin{aligned} I(tu)&\leq \frac{t^{N}}{N} \Vert u \Vert _{E}^{N}-Mt^{N} \int_{\varOmega} \frac{ \vert u \vert ^{N}}{a(x)}\,dv_{g}+O(1) \\ &\leq t^{N} \biggl(\frac{ \Vert u \Vert _{E}^{N}}{N}-M \int_{\varOmega} \frac{ \vert u \vert ^{N}}{a(x)}\,dv_{g} \biggr)+O(1). \end{aligned}$$

Now, choose \(M> \frac{\|u\|_{E}^{N}}{\int_{\varOmega}\frac{|u|^{N}}{a(x)}\,dv_{g}}\), which implies that \(I(tu)\rightarrow-\infty\) as \(t\rightarrow\infty\). Setting \(e = tu\) with t sufficiently large, the proof of the lemma is completed. □

From Lemmas 3.1 and 3.2, we get a \((C)_{c}\) sequence \(\{u_{n}\}\subset E\) by using Proposition B, i.e.,

$$ I(u_{n})\rightarrow c>0\quad \mbox{and} \quad \bigl(1+ \Vert u_{n} \Vert _{E}\bigr) \bigl\Vert I'(u_{n}) \bigr\Vert _{E^{*}}\rightarrow0\quad \mbox{as } n \rightarrow \infty $$
(3.3)

where

$$ c=\inf_{\gamma\in\varGamma} \max_{t\in[0, 1]} I\bigl( \gamma(t)\bigr) $$

and

$$ \varGamma=:\bigl\{ \gamma\in C\bigl([0, 1]:E\bigr):\gamma(0)=0, \gamma(1)=e\bigr\} . $$

Lemma 3.3

SupposeVsatisfies\((V_{1})\)and\((V_{2})\), asatisfies\((a_{1})\)and\((a_{2})\), fsatisfies\((f_{5})\), then the level\(c\in (0, \frac{1}{N} ((1-\frac{1}{s}) \frac{\alpha_{N}}{\alpha_{0}} )^{N-1} )\).

Proof

Firstly, we claim the best constant \(S_{p}\) (\(p> N\)) in (1.3) can be obtained. In fact, since

$$ S_{p}=\inf_{u\in E\setminus \{0\}} \frac{ \Vert u \Vert _{E}}{(\int_{M} \frac{ \vert u \vert ^{p}}{a(x)}\,dv_{g})^{\frac{1}{p}}}, $$

we can choose \(u_{n}\) such that

$$ \int_{M}\frac{ \vert u_{n} \vert ^{p}}{a(x)}\,dv_{g}=1 \quad \mbox{and}\quad \Vert u_{n} \Vert _{E}\rightarrow S_{p} \quad \mbox{as } n \rightarrow\infty, $$

so \(u_{n}\) is bounded in E. From Lemma 2.2, there exists \(u\in E\) such that up to a subsequence \(u_{n}\rightharpoonup u\) in E, \(u_{n}\rightarrow u\) in \(L^{p}(M)\) and \(u_{n}(x)\rightarrow u(x)\) almost everywhere in M, so \(u_{n}\rightarrow u\) in \(L^{q}(B_{\rho}(O))\) for all \(q>1\) and \(a\in L^{s}(B_{\rho}(O))\), by the Hölder inequality

$$ \int_{ B_{\rho}(O)}\frac{ \vert u_{n} \vert ^{p}}{a(x)}\,dv_{g} \rightarrow \int_{ B_{\rho}(O)}\frac{ \vert u \vert ^{p}}{a(x)}\,dv_{g}. $$

From \((a_{2})\), we have

$$\begin{aligned} \int_{M\setminus B_{\rho}(O)} \biggl\vert \frac{ \vert u_{n} \vert ^{p}- \vert u \vert ^{p}}{a(x)} \biggr\vert \,dv_{g} &\leq p C \int _{M}\bigl( \vert u_{n} \vert ^{p-1}+ \vert u \vert ^{p-1}\bigr) \vert u_{n}-u \vert \,dv_{g} \\ &\leq C \biggl( \int_{M} \vert u_{n}-u \vert ^{p}\,dv_{g} \biggr)^{\frac{1}{p}} \rightarrow0. \end{aligned}$$

This implies

$$ \int_{M}\frac{ \vert u \vert ^{p}}{a(x)}\,dx=\lim_{n\rightarrow\infty} \int_{M} \frac{ \vert u_{n} \vert ^{p}}{a(x)}\,dx=1. $$

We also have \(\| u \|_{E}\leq\lim_{n\rightarrow\infty}\| u_{n} \|_{E}=S_{p}\), thus \(\| u \|_{E}=S_{p}\). We finished the proof of the claim.

We claim that there exists a number \(t_{0}>r\) such that for the above u we have

$$ I(t_{0}u)< 0, $$

namely, we can take \(e=t_{0}u\) as in Lemma 3.2. In fact, for \(t>0\), by condition \((f_{5})\), we have

$$ I(tu)\le\frac{t^{N}}{N}S_{p}^{N}- \frac{t^{p}}{p}C_{p}. $$

Since \(p>N\), by choosing \(t_{0}\) large enough we get the claim clearly.

From the definition of c, take \(\gamma:[0, 1]\rightarrow E\), \(\gamma(t)=tt_{0}u\) with \(e=t_{0}u\) as claimed above. We have \(\gamma\in\varGamma\) and therefore

$$\begin{aligned} c&\leq\max_{t\in[0, 1]}I \bigl(\gamma(t)\bigr)\leq\max_{t\geq0}I(tu) \\ &\leq \max_{t\geq0} \biggl(\frac{t^{N}}{N}S_{p}^{N}- \frac{t^{p}}{p}C_{p} \biggr) \\ &=\frac{p-N}{Np}\frac{S_{p}^{\frac{Np}{p-N}}}{C_{p}^{\frac{N}{p-N}}} \\ &< \frac{1}{N} \biggl(\biggl(1-\frac{1}{s}\biggr) \frac{\alpha_{N}}{\alpha_{0}} \biggr)^{N-1}. \end{aligned}$$

In the last estimation we have used the inequality in condition \((f_{5})\). The proof of Lemma 3.3 is completed. □

It is well known that the absence of the Cerami compactness condition brings about difficulties in studying this class of elliptic problems involving critical growth and unbounded domains. In the next lemma, we will analyze the compactness of Cerami sequences for I.

Lemma 3.4

Suppose that the conditions\((V_{1})\)\((V_{2})\), \((f_{1})\)\((f_{4})\)and\((f_{6})\)are satisfied, asatisfies\((a_{1})\)and\((a_{2})\), let\(\{u_{n}\}\subset E\)be an arbitrary Cerami sequence ofI, then there exists a subsequence of\(\{u_{n}\}\) (still denoted by\(\{u_{n}\}\)) and\(u\in E\)such that

$$ \textstyle\begin{cases} \frac{f(x, u_{n})}{a(x)}\rightarrow\frac{f(x, u)}{a(x)}\quad \textit{strongly in } L^{1}_{\mathrm{loc}}(M), \\ \frac{F(x, u_{n})}{a(x)}\rightarrow\frac{F(x, u)}{a(x)}\quad \textit{strongly in }L^{1}(M), \\ \nabla_{g} u_{n}\rightarrow\nabla_{g} u \quad \textit{almost everywhere in }M, \\ \vert \nabla_{g} u_{n} \vert ^{N-2}\nabla_{g} u_{n}\rightharpoonup \vert \nabla_{g} u \vert ^{N-2}\nabla_{g} u\quad \textit{weakly in } (L^{N/(N-1)}(M) )^{N}. \end{cases} $$

Furthermore, uis a weak solution of (\(P_{a}\)).

Proof

We shall prove that \(\{u_{n}\}\) is bounded in E. Indeed, suppose by contradiction that

$$ \Vert u_{n} \Vert _{E}\rightarrow+\infty $$

and set

$$ v_{n}=\frac{u_{n}}{ \Vert u_{n} \Vert _{E}}, $$

then \(\|v_{n}\|=1\). From Lemma 2.2, we can assume that, for any \(q\geq N\), there exists \(v\in E\) such that up to a subsequence

$$ \textstyle\begin{cases} v_{n}^{+}\rightharpoonup v^{+} \quad \mbox{in } E, \\ v_{n}^{+}\rightarrow v^{+} \quad \mbox{in } L^{q}(M), \\ v_{n}^{+}\rightarrow v^{+}\quad \mbox{a.e. in } M. \end{cases} $$

We will show that \(v^{+}=0\) a.e. in M. In fact, if \(\varLambda^{+}=\{x\in M: v^{+}(x)>0\}\) has a positive measure, then in \(\varLambda^{+}\), we have

$$ \lim_{n\rightarrow\infty}u_{n}^{+}=\lim _{n\rightarrow\infty}v_{n}^{+} \Vert u_{n} \Vert =+\infty. $$

From \((f_{2})\) we have

$$ \lim_{n\rightarrow\infty} \frac{F(x, u_{n}^{+}(x))}{a(x) \vert u_{n}^{+}(x) \vert ^{N}}=+\infty \quad \mbox{a.e. in } \varLambda^{+} $$

and

$$ \lim_{n\rightarrow\infty} \frac{F(x, u_{n}^{+}(x))}{a(x) \vert u_{n}^{+}(x) \vert ^{N}} \bigl\vert v_{n}^{+}(x) \bigr\vert ^{N}=+ \infty\quad \mbox{a.e. in } \varLambda^{+}. $$

Thus

$$ \int_{M}\liminf_{n\rightarrow\infty} \frac{F(x, u_{n}^{+}(x))}{a(x) \vert u_{n}^{+}(x) \vert ^{N}} \bigl\vert v_{n}^{+}(x) \bigr\vert ^{N}\,dv_{g}=+ \infty. $$

Since \(\{u_{n}\}\subset E\) is an arbitrary Cerami sequence of I, we have

$$ \Vert u_{n} \Vert _{E}^{N}=Nc+N \int_{M} \frac{F(x, u_{n}^{+}(x))}{a(x)}\,dv_{g}+o_{n}(1) $$

then

$$ \int_{M}\frac{F(x, u_{n}^{+}(x))}{a(x)}\,dv_{g}\rightarrow+ \infty, $$

so

$$\begin{aligned}& \liminf_{n\rightarrow\infty} \int_{M} \frac{F(x, u_{n}^{+}(x))}{a(x) \vert u_{n}^{+}(x) \vert ^{N}} \bigl\vert v_{n}^{+}(x) \bigr\vert ^{N}\,dv_{g} \\& \quad =\liminf_{n\rightarrow\infty} \int_{M} \frac{F(x, u_{n}^{+}(x))}{a(x) \Vert u_{n} \Vert ^{N}}\,dv_{g} \\& \quad =\liminf_{n\rightarrow\infty} \frac{\int_{M}\frac{F(x, u_{n}^{+}(x))}{a(x)}\,dv_{g}}{Nc+N\int_{M}\frac {F(x, u_{n}^{+}(x))}{a(x)}\,dv_{g}+o_{n}(1)} \\& \quad =\frac{1}{N}. \end{aligned}$$

This is a contradiction. Hence \(v\leq0\) a.e. and \(v_{n}^{+}\rightharpoonup0\) in E.

Let \(t_{n}\in[0, 1]\) be such that

$$ I(t_{n}u_{n})=\max_{t\in[0, 1]}I(tu_{n}). $$

For any given \(A\in (0, ((1-\frac{1}{s})\frac{\alpha_{N}}{\alpha_{0}} )^{\frac{N-1}{N}} )\), for the sake of simplicity, let

$$ \epsilon=\frac{(1-\frac{1}{s})\alpha_{N}}{A^{\frac{N}{N-1}}}-\alpha_{0}>0. $$

In the following argument we will take \(A\to ((1-\frac{1}{s})\frac{\alpha_{N}}{\alpha_{0}} )^{\frac{N-1}{N}}\) and so we have \(\epsilon\to0\). From condition \((f_{1})\), there exists \(C > 0\) such that

$$ F(x, t)\leq C \vert t \vert ^{N}+\epsilon\varPhi \bigl(( \alpha_{0}+\epsilon) \vert t \vert ^{ \frac{N}{N-1}} \bigr),\quad \forall(x, t)\in M\times\mathbb{R}^{+}. $$
(3.4)

In fact, from condition \((f_{1})\), we have

$$ F(x,t)\le\frac{C}{N} \vert t \vert ^{N}+ \vert t \vert \varPhi \bigl(\alpha_{0} \vert t \vert ^{\frac{N}{N-1}} \bigr). $$

By using the Young inequality, for \(\frac{1}{p}+\frac{1}{q}=1\), \(p,q>1\), we have

$$ ab \le\epsilon\frac{a^{p}}{p}+\epsilon^{-q/p} \frac{b^{q}}{q}. $$

So we have

$$ F(x,t)\le\frac{C}{N} \vert t \vert ^{N}+ \frac{\epsilon\varPhi (p\alpha_{0} \vert t \vert ^{\frac{N}{N-1}} )}{p} + \epsilon^{-q/p} \frac{ \vert t \vert ^{q}}{q}. $$

Now we take \(p=\frac{\alpha_{0}+\epsilon}{\alpha_{0}}\) and \(q=\frac{\alpha_{0}+\epsilon}{\epsilon}>N\). One can see that near infinity \(|t|^{q}\) can be estimated from above by \(\varPhi ((\alpha_{0}+\epsilon)|t|^{\frac{N}{N-1}} )\), and near the origin \(|t|^{q}\) can be estimated from above by \(|t|^{N}\).

We also have \(\frac{A}{\|u_{n}\|}\in[0,1]\) with sufficient large n, so by using (3.4)

$$\begin{aligned} I(t_{n}u_{n}) &\geq I\biggl(\frac{A}{ \Vert u_{n} \Vert }u_{n}\biggr)=I(Av_{n}) \\ &=\frac{A^{N}}{N}- \int_{M}\frac{F(x, Av_{n})}{a(x)}\,dv_{g} \\ &=\frac{A^{N}}{N}- \int_{M}\frac{F(x, Av_{n}^{+})}{a(x)}\,dv_{g} \\ &\geq\frac{A^{N}}{N}-\frac{C}{N}A^{N} \int_{M} \frac{ \vert v_{n}^{+} \vert ^{N}}{a(x)}\,dv_{g} -\epsilon \int_{M} \frac{\varPhi ((\alpha_{0}+\epsilon) \vert Av_{n}^{+} \vert ^{\frac {N}{N-1}} )}{a(x)}\,dv_{g} \\ &\geq\frac{A^{N}}{N}-\frac{C}{N}A^{N} \int_{M} \frac{ \vert v_{n}^{+} \vert ^{N}}{a(x)}\,dv_{g} -\epsilon \int_{M} \frac{\varPhi ((1-\frac{1}{s})\alpha_{N} \vert v_{n}^{+} \vert ^{\frac {N}{N-1}} )}{a(x)}\,dv_{g}. \end{aligned}$$

Since \(v_{n}^{+}\rightharpoonup0\) in E and the embedding \(E\hookrightarrow L^{q}(M)\) (\(q\geq N\)) is compact, by using the Hölder inequality, we have \(\int_{M}\frac{|v_{n}^{+}|^{N}}{a(x)}\,dv_{g} \rightarrow0\). By Proposition A, \(\int_{M} \frac{\varPhi ((1-\frac{1}{s})\alpha_{N}| v_{n}^{+}|^{\frac {N}{N-1}} )}{a(x)}\,dv_{g}\) is bounded. Note that when \(A\rightarrow (\frac{(s-1)\alpha_{N}}{s\alpha_{0}} )^{\frac{N-1}{N}}\), \(\epsilon\rightarrow0\), we can show

$$ \liminf_{n\rightarrow\infty}I(t_{n}u_{n})\geq \frac{1}{N} \biggl(\biggl(1- \frac{1}{s}\biggr) \frac{\alpha_{N}}{\alpha_{0}} \biggr)^{N-1} >c. $$
(3.5)

Notice \(I(0)=0\) and \(I(u_{n})\rightarrow c\), we can assume \(t_{n}\in(0, 1)\), and so \(I'(t_{n}u_{n})t_{n}u_{n}=0\), it follows from \((f_{6})\) that

$$\begin{aligned} NI(t_{n}u_{n})&=NI(t_{n}u_{n})-I'(t_{n}u_{n})t_{n}u_{n} \\ &= \Vert t_{n}u_{n} \Vert ^{N}-N \int_{M}\frac{F(x, t_{n}u_{n})}{a(x)}\,dv_{g}- \Vert t_{n}u_{n} \Vert ^{N}+ \int_{M}\frac{f(x, t_{n}u_{n})t_{n}u_{n}}{a(x)}\,dv_{g} \\ &= \int_{M}\frac{H(x, t_{n}u_{n})}{a(x)}\,dv_{g} \leq \int_{M} \frac{H(x, u_{n})}{a(x)}\,dv_{g} \\ &=NI(u_{n})-I'(u_{n})u_{n} \\ &=NI(u_{n})+o_{n}(1) =Nc+o_{n}(1), \end{aligned}$$

which is a contradiction to (3.5). This proves that \(\{u_{n} \}\) is bounded in E. It then follows from (3.3) that

$$ \int_{M}\frac{f(x, u_{n})u_{n}}{a(x)}\,dv_{g}\leq C,\qquad \int_{M} \frac{F(x, u_{n})}{a(x)}\,dv_{g}\leq C. $$

By \((f_{3})\), there exists \(C>0\) such that

$$ F(x, u_{n})\leq C \vert u_{n} \vert ^{N}+Cf(x, u_{n}). $$

From Lemma 2.2 and the generalized Lebesgue dominated convergence theorem, discussed as Lemma 7.6 in [33], we can derive that

$$ \frac{f(x, u_{n})}{a(x)}\rightarrow\frac{f(x, u)}{a(x)} \quad \mbox{strongly in } L^{1}_{\mathrm{loc}}(M) $$
(3.6)

and

$$ \frac{F(x, u_{n})}{a(x)}\rightarrow\frac{F(x, u)}{a(x)}\quad \mbox{strongly in } L^{1}(M). $$
(3.7)

Now we prove the remaining part of the lemma. Up to a subsequence, we can define an energy concentration set for any fixed \(\delta>0\),

$$ \varSigma_{\delta}=\biggl\{ x\in M:\lim_{r\rightarrow0}\lim _{n\rightarrow \infty} \int_{\mathbb{B}_{r}(x)}\bigl( \vert \nabla_{g} u_{n} \vert ^{N}+ \vert u_{n} \vert ^{N}\bigr)\,dv_{g} \geq\delta\biggr\} . $$

Since \(\{u_{n}\}\) is bounded in E, \(\varSigma_{\delta}\) must be a finite set. For any \(x^{*}\in M\setminus \varSigma_{\delta}\), there exists \(r: 0< r< d_{g}(x^{*}, \varSigma_{\delta})\) such that

$$ \lim_{n\rightarrow\infty} \int_{\mathbb{B}_{r}( x^{*})}\bigl( \vert \nabla_{g} u_{n} \vert ^{N}+ \vert u_{n} \vert ^{N}\bigr)\,dv_{g}< \delta. $$

It follows that, for large n,

$$ \int_{\mathbb{B}_{r}(x^{{*}})}\bigl( \vert \nabla_{g} u_{n} \vert ^{N}+ \vert u_{n} \vert ^{N}\bigr)\,dv_{g}< \delta. $$
(3.8)

Thanks to the Trudinger–Moser inequality in Lemma 2.1, for sufficiently small \(\delta> 0\), there exists \(q > 1\) such that

$$ \int_{\mathbb{B}_{r}(x^{*})}\frac{ \vert f(x, u_{n}) \vert ^{{q}}}{a(x)}\,dv_{g} \leq C. $$
(3.9)

For any \(L> 0\), we denote

$$ A_{L} = \bigl\{ x\in\mathbb{B}_{r} \bigl(x^{*}\bigr) : \bigl\vert u(x) \bigr\vert \geq L\bigr\} . $$

It can be estimated that

$$\begin{aligned}& \int_{A_{L}}\frac{ \vert f(x, u_{n})-f(x, u) \vert \vert u \vert }{a(x)}\,dv_{g} \\& \quad \leq \biggl( \int_{A_{L}}\frac{ \vert f(x, u_{n})-f(x, u) \vert ^{q}}{a(x)}\,dv_{g} \biggr)^{\frac{1}{q}} \biggl( \int_{A_{L}}\frac{ \vert u \vert ^{q'}}{a(x)}\,dv_{g} \biggr)^{\frac{1}{q'}} \\& \quad \leq \biggl[ \biggl( \int_{\mathbb{B}_{r}(x^{*})} \frac{ \vert f(x, u_{n}) \vert ^{q}}{a(x)}\,dv_{g} \biggr)^{\frac{1}{q}}+ \biggl( \int_{\mathbb{B}_{r}(x^{*})}\frac{ \vert f(x, u) \vert ^{q}}{a(x)}\,dv_{g} \biggr)^{ \frac{1}{q}} \biggr] \\& \qquad {} \times \biggl( \int_{\mathbb{B}_{r}(x^{*})} \frac{1}{a(x)^{s}}\,dv_{g} \biggr)^{\frac{1}{q's}} \biggl( \int_{A_{M}} \vert u \vert ^{q't'}\,dx \biggr)^{ \frac{1}{q't'}} \\& \quad \leq C \biggl( \int_{A_{L}} \vert u \vert ^{q't'}\,dv_{g} \biggr)^{\frac{1}{q't'}}, \end{aligned}$$

where \(\frac{1}{q}+\frac{1}{q'}=1\), \(\frac{1}{s}+\frac{1}{s'}=1\). Here we have used (3.9) in the last inequality. Since \(u\in L^{q't'}(\mathbb{B}_{r}(x^{*}))\), we have, for any \(\beta>0\),

$$ \int_{A_{L}}\frac{ \vert f(x, u_{n})-f(x, u) \vert \vert u \vert }{a(x)}\,dv_{g}< \beta, $$
(3.10)

provided that L is chosen sufficiently large. It follows from (3.6) that

$$ \lim_{n\rightarrow\infty} \int_{\mathbb{B}_{r}(x^{*})\setminus A_{L}} \frac{ \vert f(x, u_{n})-f(x, u) \vert \vert u \vert }{a(x)}\,dv_{g}=0. $$
(3.11)

Combining (3.10) and (3.11), we have

$$ \lim_{n\rightarrow\infty} \int_{\mathbb{B}_{r}(x^{*})} \frac{ \vert f(x, u_{n})-f(x, u) \vert \vert u \vert }{a(x)}\,dv_{g}\leq\beta, $$

and thanks to the fact that \(\beta> 0\) is arbitrary,

$$ \lim_{n\rightarrow\infty} \int_{\mathbb{B}_{r}(x^{*})} \frac{ \vert f(x, u_{n})-f(x, u) \vert \vert u \vert }{a(x)}\,dv_{g}=0. $$
(3.12)

On the other hand, we have by using the Hölder inequality, (3.6), and (3.9),

$$ \begin{aligned}[b] &\int_{\mathbb{B}_{r}(x^{*})}\frac{ \vert f(x, u_{n}) \vert \vert u_{n}-u \vert }{a(x)}\,dv_{g} \\ &\quad \leq \biggl( \int_{\mathbb{B}_{r}(x^{*})} \frac{ \vert f(x, u_{n}) \vert ^{q}}{a(x)}\,dv_{g} \biggr)^{\frac{1}{q}} \biggl( \int_{\mathbb{B}_{r}(x^{*})}\frac{1}{a(x)^{s}}\,dv_{g} \biggr)^{ \frac{1}{q's}} \Vert u_{n}-u \Vert _{L^{q's'}(\mathbb{B}_{r}(x^{*}))} \\ &\quad \leq C \Vert u_{n}-u \Vert _{L^{q's'}(\mathbb{B}_{r}(x^{*}))}\rightarrow0, \end{aligned} $$
(3.13)

where \(\frac{1}{q}+\frac{1}{q'}=1\), \(\frac{1}{s}+\frac{1}{s'}=1\). Combining (3.12) and (3.13), we get

$$ \lim_{n\rightarrow\infty} \int_{\mathbb{B}_{r}(x^{*})} \frac{ \vert f(x, u_{n})-f(x, u)u \vert }{a(x)}\,dv_{g}=0. $$

A covering argument implies that, for any compact set \(K\subset\subset M\setminus \varSigma_{\delta}\),

$$ \lim_{n\rightarrow\infty} \int_{K} \frac{ \vert f(x, u_{n})-f(x, u)u \vert }{a(x)}\,dv_{g}=0. $$
(3.14)

Next, we will prove for any compact set \(K\subset\subset M\setminus \varSigma_{\delta}\)

$$ \lim_{n\rightarrow\infty} \int_{K} \vert \nabla_{g} u_{n}-\nabla_{g} u \vert ^{N}\,dv_{g}=0. $$
(3.15)

It suffices to prove for any \(x^{*}\in M\setminus \varSigma_{\delta}\) and r given by (3.9), we have

$$ \lim_{n\rightarrow\infty} \int_{\mathbb{B}_{\frac{r}{2}}(x^{*})} \vert \nabla_{g} u_{n}-\nabla_{g} u \vert ^{N}\,dv_{g}=0. $$
(3.16)

For this purpose, we take \(\phi\in C_{0}^{\infty}(\mathbb{B}_{r}(x^{*}))\) with \(0\leq\phi\leq1\) and \(\phi\equiv1\) in \(\mathbb{B}_{\frac{r}{2}}(x^{*})\). By the Hölder inequality and the compact embedding of \(E\hookrightarrow L^{N}(M)\), we have

$$ \lim_{n\rightarrow\infty} \int_{\mathbb{B}_{r}(x^{*})} \vert \nabla_{g} u_{n} \vert ^{N-2} \nabla_{g} u_{n} \nabla_{g} \phi(u-u_{n})\,dv_{g}=0. $$
(3.17)

Since \(\nabla u_{n}\rightharpoonup\nabla u\) in \((L^{N}(M))^{N}\), we have

$$ \lim_{n\rightarrow\infty} \int_{\mathbb{B}_{r}(x^{*})}\phi \vert \nabla_{g} u_{n} \vert ^{N-2}\nabla_{g} u( \nabla_{g} u-\nabla_{g} u_{n})\,dv_{g}=0. $$
(3.18)

The Hölder inequality and (3.13) implies that

$$ \lim_{n\rightarrow\infty} \int_{\mathbb{B}_{r}(x^{*})}\phi(u_{n}-u) \frac{f(x, u_{n})}{a(x)}\,dv_{g} \rightarrow0. $$
(3.19)

Note that

$$ \bigl\langle \vert x \vert ^{N-2}x- \vert y \vert ^{N-2}y, x-y \bigr\rangle = \frac{ \vert x \vert ^{N-2}+ \vert y \vert ^{N-2}}{2} \vert x-y \vert ^{2}+ \frac{ ( \vert x \vert ^{N-2}- \vert y \vert ^{N-2} )( \vert x \vert ^{2}- \vert y \vert ^{2})}{2}. $$

Hence,

$$ \bigl\langle \vert x \vert ^{N-2}x- \vert y \vert ^{N-2}y, x-y \bigr\rangle \geq \frac{ \vert x \vert ^{N-2}+ \vert y \vert ^{N-2}}{2} \vert x-y \vert ^{2}\geq2^{2-N} \vert x-y \vert ^{N}. $$
(3.20)

Obviously \(\phi u_{n}\) is a bounded sequence in E. Inserting \(\varphi= \phi u_{n}\) and \(\varphi= \phi u\) into (3.3), respectively, we have from (3.17)–(3.20) and \(\tau_{n}\rightarrow0\) as \(n\rightarrow\infty\)

$$\begin{aligned} 0&\leq2^{2-N} \int_{\mathbb{B}_{r}(x^{*})} \vert \nabla_{g} u_{n}-\nabla_{g} u \vert ^{N}\,dv_{g} \\ &\leq \int_{\mathbb{B}_{r}(x^{*})}\phi \bigl\langle \vert \nabla_{g} u_{n} \vert ^{N-2} \nabla_{g} u_{n}- \vert \nabla_{g} u \vert ^{N-2} \nabla_{g} u, \nabla_{g} u_{n}- \nabla_{g} u \bigr\rangle \,dv_{g} \\ &\leq \int_{\mathbb{B}_{r}(x^{*})} \vert \nabla_{g} u_{n} \vert ^{N-2}\nabla_{g} u_{n} \nabla_{g} \phi(u-u_{n})\,dv_{g}+ \int_{B_{r}(x^{*})} \phi \vert \nabla_{g} u \vert ^{N-2}\nabla_{g} u(\nabla_{g} u- \nabla_{g} u_{n})\,dv_{g} \\ &\quad {}+ \int_{\mathbb{B}_{r}(x^{*})} \phi( u_{n}-u) \frac{f(x, u_{n})}{a(x)}\,dv_{g}+ \tau_{n} \Vert \phi u_{n} \Vert + \tau_{n} \Vert \phi u \Vert \rightarrow0. \end{aligned}$$
(3.21)

We derive (3.16) from (3.21). Hence, (3.15) holds thanks to a covering argument. Since \(\varSigma_{\delta}\) is a finite set, it follows that \(\nabla_{g} u_{n}\) converges to \(\nabla_{g} u\) almost everywhere in M. This immediately implies, up to a subsequence,

$$ \vert \nabla_{g} u_{n} \vert ^{N-2} \nabla_{g} u_{n}\rightharpoonup \vert \nabla_{g} u \vert ^{N-2}\nabla u \quad \mbox{in } \bigl(L^{\frac{N}{N-1}}\bigl(\mathbb{B}_{R}(0)\bigr) \bigr)^{N} $$

for all \(R>0\). For any fixed \(\varphi\in C_{0}^{\infty}(M)\), there exists some \(R_{0} > 0\) such that the support of φ is contained in the ball \(\mathbb{B}_{R_{0}}(O)\). Hence,

$$ \lim_{n\rightarrow\infty} \int_{M}\bigl( \vert \nabla_{g} u_{n} \vert ^{N-2} \nabla_{g} u_{n}- \vert \nabla_{g} u \vert ^{N-2} \nabla_{g} u\bigr)\varphi \,dv_{g} \rightarrow0. $$

This equality holds for all \(\varphi\in L^{\frac{N}{N-1}}(M)\), thanks to the density of \(C_{0}^{\infty}(M)\) in \(L^{\frac{N}{N-1}}(M)\). Hence, we obtain

$$ \vert \nabla_{g} u_{n} \vert ^{N-2} \nabla_{g} u_{n}\rightharpoonup \vert \nabla_{g} u \vert ^{N-2}\nabla_{g} u\quad \mbox{in } \bigl(L^{\frac{N}{N-1}}(M)\bigr)^{N}, $$
(3.22)

we obtain by combining (3.6) and (3.22),

$$ \bigl\langle I'(u), \varphi\bigr\rangle =0, \quad \forall\varphi\in C_{0}^{ \infty}(M). $$

Since \(C_{0}^{\infty}(M)\) is dense in E, the above equation implies that u is a weak solution of (\(P_{a}\)). This completes the proof of the lemma. □

The ground state solution

Proof of Theorem 1.1

By the process of proof in Lemma 3.4, we see that the Cerami sequence \(\{u_{n}\}\) is bounded in E and its weak limit u is a critical point of the functional I.

We will show that u is nonzero. If \(u\equiv0\), since \(F(x, 0)=0\) for all \(x\in\mathbb{R}^{N}\), from Lemma 3.4, we have

$$ \lim_{n\rightarrow\infty} \Vert u_{n} \Vert _{E}^{N}=Nc>0. $$
(4.1)

Lemma 3.3 implies \(0< c <\frac{1}{N} ((1-\frac{1}{s})\frac{\alpha_{N}}{\alpha_{0}} )^{N-1}\), so there exist some \(\epsilon_{0}>0\) and \(n_{*}>0\) such that \(\|u_{n}\|_{E}^{N}\leq ((1-\frac{1}{s}) \frac{\alpha_{N}}{\alpha_{0}}-\epsilon_{0} )^{N-1}\) for all \(n>n_{*}\). Choose \(q>1\) sufficiently close to 1 such that \(q\alpha_{0}\|u_{n}\|_{E}^{\frac{N}{N-1}}\leq(1-\frac{1}{s})\alpha_{N}- \epsilon_{0}\alpha_{0}/2\) for all \(n>n_{*}\). By \((f_{1})\),

$$ f(x, u_{n})\leq c_{1} \vert u_{n} \vert ^{N}+c_{2} \vert u_{n} \vert \varPhi\bigl(\alpha_{0} \vert u_{n} \vert ^{ \frac{N}{N-1}}\bigr), $$

thus

$$\begin{aligned}& \int_{M}\frac{ \vert f(x, u_{n})u_{n} \vert }{a(x)}\,dv_{g} \\& \quad \leq c_{1} \int_{M}\frac{ \vert u_{n} \vert ^{N}}{a(x)}\,dv_{g}+c_{2} \int_{M} \frac{ \vert u_{n} \vert \varPhi(\alpha_{0} \vert u_{n} \vert ^{\frac{N}{N-1}})}{a(x)}\,dv_{g} \\& \quad \leq c_{1} \int_{M}\frac{ \vert u_{n} \vert ^{N}}{a(x)}\,dv_{g}+c_{2} \biggl( \int_{M} \frac{\varPhi(q\alpha_{0} \vert u_{n} \vert ^{\frac{N}{N-1}})}{a(x)}\,dv_{g} \biggr)^{ \frac{1}{q}} \biggl( \int_{M}\frac{ \vert u_{n} \vert ^{q'}}{a(x)}\,dv_{g} \biggr)^{ \frac{1}{q'}} \\& \quad \leq c_{1} \int_{M}\frac{ \vert u_{n} \vert ^{N}}{a(x)}\,dv_{g}+C \biggl( \int_{M} \frac{ \vert u_{n} \vert ^{q'}}{a(x)}\,dv_{g} \biggr)^{\frac{1}{q'}}\rightarrow0, \end{aligned}$$

here we used Lemma 2.2 in the last step. From \(I^{\prime}(u_{n})u_{n}\rightarrow0\), we have

$$ \lim_{n\rightarrow\infty} \Vert u_{n} \Vert _{E}^{N}=0, $$
(4.2)

which is a contradiction with (4.1), so u is nonzero.

Now, we will show the existence of positive ground state solution for (\(P_{a}\)). Setting

$$ m=\inf_{u\in\varLambda}I(u), \quad \varLambda:=\bigl\{ u\in E\setminus \{0 \}:I'(u)u=0 \bigr\} . $$

Let c be the mountain-pass level, obviously \(m\leq c\).

On the other hand, let \(u\in\varLambda\), then \(u>0\). In fact, denote \(u^{-}:=\min\{u, 0\}\), from \(I'(u)u^{-}=0\), we have \(\|u^{-}\|=0\), thus \(u\geq0\) on M, from \(I(u) = c > 0\), we know u is positive on M. Define \(h:(0, +\infty)\rightarrow\mathbb{R}\) by \(h(t)=I(tu)\). We see that h is differentiable and

$$ h'(t)=I'(tu)u=t^{N-1} \Vert u \Vert ^{N}- \int_{M}\frac{f(x, tu)u}{a(x)}\,dv_{g},\quad \forall t>0. $$

From \(I'(u)u=0\), we get

$$ h'(t)=I'(tu)u-t^{N-1}I'(u)u, $$

so

$$ h'(t)=t^{N-1} \int_{M} \biggl(\frac{f(x, u)}{u^{N-1}}- \frac{f(x, tu)}{(tu)^{N-1}} \biggr)\frac{u^{N}}{a(x)}\,dv_{g}, \quad \forall t>0. $$

By \((f_{6})\) and \(u>0\), we conclude that \(h'(t)>0\) for \(0< t<1\) and \(h'(t)<0\) for \(t>1\), since \(h'(1)=0\), thus,

$$ I(u)=\max_{t\geq0}I(tu). $$

Now, define \(\gamma:[0, 1]\rightarrow E\), \(\gamma(t)=tt_{0}u\), where \(t_{0}\) is a real number which satisfies \(I(t_{0}u)<0\), we have \(\gamma \in\varGamma\), and therefore

$$ c\leq\max_{t\in[0, 1]}I\bigl(\gamma(t)\bigr)\leq\max _{t\geq0}I(tu)=I(u). $$

Since \(u\in\varLambda\) is arbitrary, we have \(c\leq m\), thus \(c=m\). This completes the proof of Theorem 1.1. □

The nontrivial solution of perturbation problem

In this section, we consider the following perturbation problem:

figureb

where \(\epsilon>0\), \(h\in E^{*}\), \(h\geq0\) and \(h\not\equiv0\).

We say that \(u\in E\) is a weak solution of problem (P) if for all \(\phi\in E\),

$$ \int_{M}\bigl( \vert \nabla_{g} u \vert ^{N-2}\nabla_{g} u \nabla\phi+V(x) \vert u \vert ^{N-2}u \phi\bigr)\,dv_{g}- \int_{M}\frac{f(x, u)}{a(x)}\phi \,dv_{g}- \epsilon \int_{ \mathbb{R}^{N}}h(x)\phi \,dv_{g}=0. $$

Define the functional \(I_{\epsilon}: E\rightarrow\mathbb{R}\) by

$$ I_{\epsilon}(u)=\frac{1}{N} \Vert u \Vert _{E}^{N} - \int_{\mathbb{R}^{N}} \frac{F(x, u)}{a(x)} \,dv_{g}-\epsilon \int_{M}h(x)u \,dv_{g}, $$
(5.1)

where \(F(x, t)=\int_{0}^{t}f(x, s)\,ds\). I is well defined and \(I \in C^{1}(E, \mathbb{R})\) thanks to the singular Trudinger–Moser inequality. A straightforward calculation shows that

$$\begin{aligned} \bigl\langle I'(u), \phi\bigr\rangle =& \int_{M}\bigl( \vert \nabla_{g} u \vert ^{N-2}\nabla_{g} u \nabla_{g} \phi+V(x) \vert u \vert ^{N-2}u\phi\bigr)\,dv_{g} \\ &{}- \int_{M} \frac{f(x, u)}{a(x)}\phi \,dv_{g}- \epsilon \int_{M}h(x)\phi \,dv_{g}, \end{aligned}$$
(5.2)

for all \(u, \phi\in E\), hence, a critical point of (5.2) is a weak solution of (P).

Next, we will prove Theorem 1.3, one solution is obtained by using the mountain-pass theorem without Cerami condition.

Proof of Theorem 1.3

As in previous sections, it is similar to the first part of Theorem 1.1, \(I_{\epsilon}\) satisfies the geometry condition of the mountain-pass theorem; i.e., \(I_{\epsilon}\in C^{1}(E, \mathbb{R})\), there exists \(\epsilon_{1}>0\) such that when \(0<\epsilon<\epsilon_{1}\), \(I_{\epsilon}(u)\geq\delta_{\epsilon}>0\) for \(\|u\|_{E}=r_{\epsilon}\); \(I_{\epsilon}(e)<0\) for some \(e\in E\) with \(\|e\|_{E}>r_{\epsilon}\), where \(r_{\epsilon}\) can be further chosen such that \(r_{\epsilon}\rightarrow0\) as \(\epsilon\rightarrow0\). Then, by the mountain-pass theorem without the Cerami condition, we get a \((C)_{c}\) sequence \(\{u_{n}\}\subset E\); i.e.,

$$ I_{\epsilon}(u_{n})\rightarrow c_{\epsilon}\quad \mbox{and}\quad \bigl(1+ \Vert u_{n} \Vert _{E}\bigr) \bigl\Vert I'_{\epsilon}(u_{n}) \bigr\Vert _{E^{*}}\rightarrow0\quad \mbox{as } n \rightarrow\infty $$
(5.3)

where

$$ c_{\epsilon}=\inf_{\gamma\in\varGamma} \max_{t\in[0, 1]} I_{\epsilon}\bigl(\gamma(t)\bigr)\geq\delta_{\epsilon} $$

is the min–max value of \(I_{\epsilon}\). By a similar argument to Lemma 3.4 to \(I_{\epsilon}\), there exists a subsequence of \(\{u_{n}\}\) that is bounded and its weak limit u is a critical point of the functional \(I_{ \epsilon}\). This completes the proof of Theorem 1.3. □

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Acknowledgements

The authors are very grateful to the editorial board and the reviewers, whose comments improved the quality of the paper.

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This work was supported by NNSF of China (11471170, 11790271), Guangdong Basic and Applied basic Research Foundation (2020A1515011019), Innovation and development project of Guangzhou University, and Nankai Zhide Foundation.

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Liu, C., Liu, Y. The existence of ground state solution to elliptic equation with exponential growth on complete noncompact Riemannian manifold. J Inequal Appl 2020, 74 (2020). https://doi.org/10.1186/s13660-020-02338-4

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MSC

  • 35J60
  • 35B33
  • 46E30

Keywords

  • Trudinger–Moser inequality
  • Riemannian manifold
  • Exponential growth
  • The ground state solution