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

In this paper, we consider the following elliptic problem: −divg(|∇gu|N−2∇gu)+V(x)|u|N−2u=f(x,u)a(x)in M,(Pa)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ -\mathtt{div}_{g}\bigl( \vert \nabla_{g} u \vert ^{N-2}\nabla_{g} u \bigr)+V(x) \vert u \vert ^{N-2}u = \frac{f(x, u)}{a(x)}\quad \mbox{in } M, \qquad (P_{a}) $$\end{document} where (M,g)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$(M, g)$\end{document} be a complete noncompact N-dimensional Riemannian manifold with negative curvature, N≥2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$N\geq2$\end{document}, V is a continuous function satisfying V(x)≥V0>0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$V(x) \geq V_{0 }> 0$\end{document}, a is a nonnegative function and f(x,t)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$f(x, t)$\end{document} 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 (Pa\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$P_{a}$\end{document}). Moreover, we also get a nontrivial weak solution to the perturbation problem.

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 blowup 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 ≥ 2, 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 Ndimensional Riemannian manifold with negative curvature. We state it below for further use.
Proposition A ( [34]) Let (M, g) be a complete, simply connected N -dimensional Riemannian manifold with negative curvature, N ≥ 2. Then for any τ > 0 there exists a positive constant C = C(τ , N, M) such that 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 Ndimensional Riemannian manifold with negative curvature, N ≥ 2. And on (M, g), we consider the following quasilinear equations: where > 0, h ∈ E * , h ≥ 0 and h ≡ 0, (M, g) is a complete noncompact N -dimensional Riemannian manifold with negative curvature, N ≥ 2, ∇ g denotes its covariant derivative (the Levi-Civita connection), and div g denotes its divergence operator, V is a continuous function satisfying V (x) ≥ 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) ∈ M × (-∞, 0]. Furthermore, let O be a fixed point of M and d g (·, ·) be the geodesic distance between two points of (M, g). We assume the function a satisfies: We assume the following conditions on V .
Define a function space E as then the assumption V (x) ≥ V 0 > 0 implies that E is a reflexive Banach space. For any p ≥ N , we define The continuous embedding of E → W 1,N (M) → L p (M) (p ≥ N ) and the Hölder inequality imply here s is stated in the condition (a 1 ). Thus we have S p > 0. We now introduce the following three conditions.
Our main results can be stated as follows. 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 Suppose a satisfies (a 1 ) and (a 2 ), V satisfies condition (V 1 ). Let α > 0 and {u n } be a sequence satisfying passing to a subsequence, there exists n 0 ∈ N such that Choose k > ts > 1 such that kαm < α N and t < s, here s is stated in the condition (a 1 ) and 1 s + 1 s = 1. Now for each n ≥ n 0 , combining with The property Φ(q) p ≤ Φ(pq) (see [37]) has been used above. Finally, the result is derived from Proposition A by taking τ = V 0 in condition (V 1 ).
Proof From the condition (V 1 ), we have E → W 1,N (M), together with the standard Sobolev embedding theorem W 1,N (M) → L q (M) for any q ≥ N , we immediately derive that E can be continuously embedded into L q (M). Next, for any q ≥ N , one needs to prove that the above continuous embedding E → L q (M) is compact. Let {u n } be a sequence of functions with u n N E ≤ 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 ≥ N , there exists u 0 ∈ E such that up to a subsequence Hence for any > 0, there exists some positive integral n 0 , such that when n > n 0 , On the other hand, for any > 0, from the condition (V 2 ), we can assume (2.2) From (2.1) and (2.2), when n > n 0 , we obtain By Hölder's inequality and the continuous embedding Combining with (2.3), we see that {u n } strongly converges in L q (M).

From (2.4) and (2.5), by Proposition A and the continuous embedding of
This completes the proof.

Functionals and compactness analysis
We say that u ∈ E is a weak solution of problem (P a ) if, for all φ ∈ E, Define the functional I : E → R by where F(x, t) = t 0 f (x, s) ds. I is well defined and I ∈ C 1 (E, R) thanks to the Trudinger-Moser inequality. A straightforward calculation shows that for all u, φ ∈ 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, · E ) be a real Banach space with its dual space (E * , · E * ). Suppose Then I possesses a (C) c sequence.
Next we will check the geometry of the functional I. Proof From (f 4 ), there exist σ , > 0, such that if |u| ≤ , On the other hand, using (f 1 ) for each q > N , we have for |u| ≥ and x ∈ M. Combining the above estimates, we obtain for all (x, u) ∈ M × R. Fix r > 0 small enough such that α 0 r N N-1 < α N , then Lemma 2.3 implies for u E ≤ r. Hence, I is bounded from below for u E ≤ r. Since σ > 0 and q > N , we may choose a small r that satisfies the inequality σ Nλ a r N -Cr q ≥ σ 2Nλ a r N , and we derive that This completes the proof.

Lemma 3.2 If the condition (f 2 ) is satisfied, then there exists e ∈ B c r (0) such that
where r is given in Lemma 3.1.
Then  Proof Firstly, we claim the best constant S p (p > N ) in (1.3) can be obtained. In fact, since , we can choose u n such that M |u n | p a(x) dv g = 1 and u n E → S p as n → ∞, so u n is bounded in E. From Lemma 2.2, there exists u ∈ E such that up to a subsequence u n u in E, u n → u in L p (M) and u n (x) → u(x) almost everywhere in M, so u n → u in L q (B ρ (O)) for all q > 1 and a ∈ L s (B ρ (O)), by the Hölder inequality From (a 2 ), we have We also have u E ≤ lim n→∞ 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 namely, we can take e = t 0 u as in Lemma 3.2. In fact, for t > 0, by condition (f 5 ), we have Since p > N , by choosing t 0 large enough we get the claim clearly. From the definition of c, take γ : [0, 1] → E, γ (t) = tt 0 u with e = t 0 u as claimed above. We have γ ∈ Γ and therefore c ≤ max 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.
Furthermore, u is a weak solution of (P a ).
Proof We shall prove that {u n } is bounded in E. Indeed, suppose by contradiction that u n E → +∞ and set v n = u n u n E , then v n = 1. From Lemma 2.2, we can assume that, for any q ≥ N , there exists v ∈ E such that up to a subsequence We will show that v + = 0 a.e. in M. In fact, if Λ + = {x ∈ M : v + (x) > 0} has a positive measure, then in Λ + , we have lim n→∞ u + n = lim n→∞ v + n u n = +∞.
From (f 2 ) we have Since {u n } ⊂ E is an arbitrary Cerami sequence of I, we have This is a contradiction. Hence v ≤ 0 a.e. and v + n 0 in E. Let t n ∈ [0, 1] be such that 1] I(tu n ).

For any given
, for the sake of simplicity, let In the following argument we will take A → ((1 -1 s ) α N α 0 ) N-1 N and so we have → 0. From condition (f 1 ), there exists C > 0 such that In fact, from condition (f 1 ), we have By using the Young inequality, for 1 p + 1 q = 1, p, q > 1, we have ab ≤ a p p + -q/p b q q .
So we have Now we take p = α 0 + α 0 and q = α 0 + > N . One can see that near infinity |t| q can be estimated from above by Φ((α 0 + )|t| N N-1 ), and near the origin |t| q can be estimated from above by |t| N .
We also have A u n ∈ [0, 1] with sufficient large n, so by using (3.4) Since v + n 0 in E and the embedding E → L q (M) (q ≥ N ) is compact, by using the Hölder Notice I(0) = 0 and I(u n ) → c, we can assume t n ∈ (0, 1), and so I (t n u n )t n u n = 0, it follows from (f 6 ) that which is a contradiction to (3.5). This proves that {u n } is bounded in E. It then follows from (3.3) that By (f 3 ), there exists C > 0 such that From Lemma 2.2 and the generalized Lebesgue dominated convergence theorem, discussed as Lemma 7.6 in [33], we can derive that and Now we prove the remaining part of the lemma. Up to a subsequence, we can define an energy concentration set for any fixed δ > 0, |∇ g u n | N + |u n | N dv g ≥ δ .
Since {u n } is bounded in E, Σ δ must be a finite set. For any x * ∈ M \ Σ δ , there exists r : 0 < r < d g (x * , Σ δ ) such that |∇ g u n | N + |u n | N dv g < δ.
It follows that, for large n, |∇ g u n | N + |u n | N dv g < δ. (3.8) Thanks to the Trudinger-Moser inequality in Lemma 2.1, for sufficiently small δ > 0, there exists q > 1 such that For any L > 0, we denote It can be estimated that Here we have used (3.9) in the last inequality. Since u ∈ L q t (B r (x * )), we have, for any β > 0, provided that L is chosen sufficiently large. It follows from (3.6) that Combining (3.10) and (3.11), we have and thanks to the fact that β > 0 is arbitrary, On the other hand, we have by using the Hölder inequality, (3.6), and (3.9), |f (x, u n )f (x, u)u| a(x) dv g = 0.
This equality holds for all ϕ ∈ L Since C ∞ 0 (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 ≡ 0, since F(x, 0) = 0 for all x ∈ R N , from Lemma 3.4, we have