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The existence of ground state solution to elliptic equation with exponential growth on complete noncompact Riemannian manifold
Journal of Inequalities and Applications volume 2020, Article number: 74 (2020)
Abstract
In this paper, we consider the following elliptic problem:
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.
1 Introduction and main results
Suppose \(\varOmega\subset\mathbb{R}^{N}\) is a bounded smooth domain, and we consider the following problem:
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:
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
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
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:
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
equipped with the norm
then the assumption \(V(x)\geq V_{0}>0\) implies that E is a reflexive Banach space. For any \(p\ge N\), we define
and
The continuous embedding of \(E\hookrightarrow W^{1, N}(M) \hookrightarrow L^{p}(M)\) (\(p \geq N\)) 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.
- \((f_{4})\):
\(\limsup_{t\rightarrow0^{+}}\frac{NF(x, t)}{|t|^{N}}<\lambda_{a}\) 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)
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)
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.
2 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
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
Proof
Since
passing to a subsequence, there exists \(n_{0}\in\mathbb{N}\) such that
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 \(\frac{m}{\|u_{n}\|_{E}^{\frac{N}{N-1}}} \ge1\),
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
Hence for any \(\epsilon>0\), there exists some positive integral \(n_{0}\), such that when \(n>n_{0}\),
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
thus
From (2.1) and (2.2), when \(n>n_{0}\), we obtain
By Hölder’s inequality and the continuous embedding \(E \hookrightarrow L^{q}(M)\) (\(q\geq N\)), we have
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
Proof
On the one hand, using the Hölder inequality, we have
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
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
This completes the proof. □
3 Functionals and compactness analysis
We say that \(u\in E\) is a weak solution of problem (\(P_{a}\)) if, for all \(\phi\in E\),
Define the functional \(I: E\rightarrow\mathbb{R}\) by
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
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
Proposition B
Let\((E, \|\cdot\|_{E})\)be a real Banach space, \(I\in C^{1}(E, \mathbb{R})\), \(I(0)=0\)and satisfies:
- (i)
There exist positive constantsδandrsuch that
$$ I(u)\geq\delta\quad \textit{for }\|u\|_{E}=r $$and
- (ii)
there exists\(e\in E\)with\(\|e\|_{E}>r\)such that
$$ I(e)\leq0 . $$
Definecby
where
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
Proof
From \((f_{4})\), there exist \(\sigma, \epsilon>0\), such that if \(|u|\leq\epsilon\),
for all \(x\in M\). On the other hand, using \((f_{1})\) for each \(q> N\), we have
for \(|u|\geq\epsilon\) and \(x\in M\). Combining the above estimates, we obtain
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
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
and we derive that
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
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 \((x, s) \in\varOmega\times\mathbb{R}^{+}\),
Then
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.,
where
and
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
we can choose \(u_{n}\) such that
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
From \((a_{2})\), we have
This implies
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
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 \(\gamma:[0, 1]\rightarrow E\), \(\gamma(t)=tt_{0}u\) with \(e=t_{0}u\) as claimed above. We have \(\gamma\in\varGamma\) and therefore
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
Furthermore, uis a weak solution of (\(P_{a}\)).
Proof
We shall prove that \(\{u_{n}\}\) is bounded in E. Indeed, suppose by contradiction that
and set
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
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
From \((f_{2})\) we have
and
Thus
Since \(\{u_{n}\}\subset E\) is an arbitrary Cerami sequence of I, we have
then
so
This is a contradiction. Hence \(v\leq0\) a.e. and \(v_{n}^{+}\rightharpoonup0\) in E.
Let \(t_{n}\in[0, 1]\) be such that
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
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
In fact, from condition \((f_{1})\), we have
By using the Young inequality, for \(\frac{1}{p}+\frac{1}{q}=1\), \(p,q>1\), we have
So we have
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)
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
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
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 \(\delta>0\),
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
It follows that, for large n,
Thanks to the Trudinger–Moser inequality in Lemma 2.1, for sufficiently small \(\delta> 0\), there exists \(q > 1\) such that
For any \(L> 0\), we denote
It can be estimated that
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 \(\beta>0\),
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 \(\beta> 0\) is arbitrary,
On the other hand, we have by using the Hölder inequality, (3.6), and (3.9),
where \(\frac{1}{q}+\frac{1}{q'}=1\), \(\frac{1}{s}+\frac{1}{s'}=1\). Combining (3.12) and (3.13), we get
A covering argument implies that, for any compact set \(K\subset\subset M\setminus \varSigma_{\delta}\),
Next, we will prove for any compact set \(K\subset\subset M\setminus \varSigma_{\delta}\)
It suffices to prove for any \(x^{*}\in M\setminus \varSigma_{\delta}\) and r given by (3.9), we have
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
Since \(\nabla u_{n}\rightharpoonup\nabla u\) in \((L^{N}(M))^{N}\), we have
The Hölder inequality and (3.13) implies that
Note that
Hence,
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 \(n\rightarrow\infty\)
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,
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,
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
we obtain by combining (3.6) and (3.22),
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. □
4 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
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 \((f_{1})\),
thus
here we used Lemma 2.2 in the last step. From \(I^{\prime}(u_{n})u_{n}\rightarrow0\), we have
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
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
From \(I'(u)u=0\), we get
so
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,
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
Since \(u\in\varLambda\) is arbitrary, we have \(c\leq m\), thus \(c=m\). This completes the proof of Theorem 1.1. □
5 The nontrivial solution of perturbation problem
In this section, we consider the following perturbation problem:
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 \(\phi\in E\),
Define the functional \(I_{\epsilon}: E\rightarrow\mathbb{R}\) by
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
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.,
where
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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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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DOI: https://doi.org/10.1186/s13660-020-02338-4