Concentration-compactness principle for Trudinger-Moser inequalities on Heisenberg Groups and existence of ground state solutions

TL;DR

This paper extends the concentration-compactness principle to the Heisenberg group setting, improving Trudinger-Moser inequalities and proving the existence of ground state solutions for certain subelliptic equations with exponential nonlinearities.

Contribution

It develops a new approach to concentration-compactness on the Heisenberg group without symmetrization, extending inequalities and establishing ground state solutions.

Findings

Extended concentration-compactness principle to Heisenberg groups

Improved Trudinger-Moser inequalities on Heisenberg groups

Proved existence of ground state solutions for Q-Laplacian equations

Abstract

Let $\mathbb{H}^{n}=\mathbb{C}^{n}\times\mathbb{R}$ be the $n$-dimensional Heisenberg group, $Q=2n+2$ be the homogeneous dimension of $\mathbb{H}^{n}$. We extend the well-known concentration-compactness principle on finite domains in the Euclidean spaces of \ P. L. Lions to the setting of the Heisenberg group $\mathbb{H}^{n}$. Furthermore, we also obtain the corresponding concentration-compactness principle for the Sobolev space $HW^{1,Q}\left( \mathbb{H}^{n}\right) $ on the entire Heisenberg group $\mathbb{H}^{n}$. Our results improve the sharp Trudinger-Moser inequality on domains of finite measure in $\mathbb{H}^{n}$ by Cohn and the second author [8] and the corresponding one on the whole space $\mathbb{H}^n$ by Lam and the second author [21]. All the proofs of the concentration-compactness principles in the literature even in the Euclidean spaces use the rearrangement argument and the Poly\'a-Szeg\"{o} inequality. Due to the absence of the Poly\'a-Szeg\"{o} inequality on the Heisenberg group, we will develop a different argument. Our approach is surprisingly simple and general and can be easily applied to other settings where symmetrization argument does not work. As an application of the concentration-compactness principle, we establish the existence of ground state solutions for a class of $Q$- Laplacian subelliptic equations on $\mathbb{H}^{n}$ with nonlinear terms $f$ of maximal exponential growth $\exp\left( \alpha t^{\frac{Q}{Q-1}}\right) $ as $t\rightarrow+\infty$.

Equations

$$-\mathrm{div}\left({\left|{\nabla }_{\mathbb{H}}u\right|}^{Q-2}{\nabla }_{\mathbb{H}}u\right)+V\left(\xi \right){\left|u\right|}^{Q-2}u=\frac{f\left(u\right)}{\rho {\left(\xi \right)}^{\beta }}$$

$${\Vert u\Vert }_{W^{1,q}\left(\Omega \right)}={\left({\int }_{\Omega }\left({\left|u\right|}^q+{\left|\nabla u\right|}^q\right)dx\right)}^{\frac{1}{q}}.$$

$$W_0^{1,n}\left(\Omega \right)\hookrightarrow L^s\left(\Omega \right)\text{ for any }n\le s<+\infty \text{, }$$

$$\underset{{\Vert \nabla u\Vert }_{L^n\left(\Omega \right)}\le 1}{\underset{u\in W_0^{1,n}\left(\Omega \right)}{\sup }}{\int }_{\Omega }{e}^{\alpha {|u|}^{\frac{n}{n-1}}}dx<\infty \text{ iff }\alpha \le {\alpha }_n=n{\omega }_{n-1}^{\frac{1}{n-1}},$$

$$\underset{k}{\sup }{\int }_{\Omega }{e}^{{\alpha }_{n}p{|u_k|}^{\frac{n}{n-1}}}dx<\infty .$$

$${\Vert {\nabla }_{{\mathbb{H}}^n}{u}^{\ast }\Vert }_{L^p}\le {\Vert {\nabla }_{{\mathbb{H}}^n}u\Vert }_{L^p}$$

$$\underset{k}{\sup }{\int }_{{\mathbb{R}}^n}{e}^{{\alpha }_{n}p{|u_k|}^{\frac{n}{n-1}}}dx<\infty .$$

$$-\mathrm{div}\left({\left|{\nabla }_{\mathbb{H}}u\right|}^{Q-2}{\nabla }_{\mathbb{H}}u\right)+V\left(\xi \right){\left|u\right|}^{Q-2}u=\frac{f\left(u\right)}{\rho {\left(\xi \right)}^{\beta }}$$

$$\left(x,t\right)\circ \left(x^{\prime },t^{\prime }\right)=\left(x+x^{\prime },t+t^{\prime }+2\mathrm{im}\left(x\cdot {\bar{x}}^{\prime }\right)\right).$$

$$X_i=\frac{\partial }{\partial x_i}+2y_i\frac{\partial }{\partial t},Y_i=\frac{\partial }{\partial y_i}-2x_i\frac{\partial }{\partial t},T=\frac{\partial }{\partial t},$$

$$B_r=\left\{\xi :\left|\xi \right|<r\right\}$$

$$\left|B_r\right|={\omega }_Q{r}^Q,$$

$$\left|{\nabla }_{\mathbb{H}}u\right|=\sqrt{\sum {\left(X_{i}u\right)}^2+{\left(Y_{i}u\right)}^2}.$$

$$H{W}^{1,p}\left(\Omega \right)=\left\{u\in L^p\left(\Omega \right):{\Vert u\Vert }_{H{W}^{1,p}\left(\Omega \right)}<\infty \right\}$$

$${\Vert u\Vert }_{H{W}^{1,p}\left(\Omega \right)}={\left({\int }_{\Omega }\left({\left|{\nabla }_{\mathbb{H}}u\left(z,t\right)\right|}^p+{\left|u\left(z,t\right)\right|}^p\right)dxdt\right)}^{1/p}.$$

$$\left|{\nabla }_{\mathbb{H}}g\left(\xi \right)\right|=\frac{g^{\prime }\left(\rho \right)}{\rho }\left|x\right|.$$

$$\underset{{\Vert {\nabla }_{\mathbb{H}}u\Vert }_{L^Q}\le 1}{\underset{u\in H{W}_0^{1,Q}\left(\Omega \right)}{\sup }}{\int }_{\Omega }\frac{\exp \left(\alpha u{\left(\xi \right)}^{\frac{Q}{Q-1}}\right)}{\rho {\left(\xi \right)}^{\beta }}d\xi <c.$$

$$\underset{{\Vert f\Vert }_{H{W}^{1,Q}\left({\mathbb{H}}^n\right)}\le 1}{\underset{f\in H{W}^{1,Q}\left({\mathbb{H}}^n\right)}{\sup }}{\int }_{{\mathbb{H}}^n}\frac{\Phi \left(\alpha f{\left(\xi \right)}^{\frac{Q}{Q-1}}\right)}{\rho {\left(\xi \right)}^{\beta }}d\xi <c.$$

$$0<p<M_{Q,u}:=\frac{1}{{\left(1-{\Vert {\nabla }_{\mathbb{H}}u\Vert }_Q^Q\right)}^{1/\left(Q-1\right)}}$$

$$\underset{k}{\sup }{\int }_{\Omega }\frac{e^{{\alpha }_{Q,\beta }p{u}_k^{\frac{Q}{Q-1}}}}{\rho {\left(\xi \right)}^{\beta }}d\xi <\infty .$$

$$0<p<{\tilde{M}}_{Q,u}:=\frac{1}{{\left(1-{\Vert u\Vert }_{H{W}^{1,Q}\left({\mathbb{H}}^n\right)}^Q\right)}^{1/\left(Q-1\right)}},$$

$$\underset{k}{\sup }{\int }_{{\mathbb{H}}^n}\frac{\Phi \left({\alpha }_{Q,\beta }p{u}_k^{\frac{Q}{Q-1}}\right)}{\rho {\left(\xi \right)}^{\beta }}d\xi <\infty ,$$

$$J\left(u\right)=\frac{1}{Q}{\int }_{{\mathbb{H}}^n}\left({\left|{\nabla }_{\mathbb{H}}u\right|}^Q+V\left(\xi \right){\left|u\right|}^Q\right)d\xi -{\int }_{{\mathbb{H}}^n}\frac{F\left(u\right)}{\rho {\left(\xi \right)}^{\beta }}d\xi$$

$$\underset{k}{\sup }{\int }_{\Omega }\frac{\exp \left({\alpha }_{Q,\beta }{p}_1{u}_k^{\frac{Q}{Q-1}}\right)}{\rho {\left(\xi \right)}^{\beta }}d\xi =+\infty .$$

$${\Omega }_L^k=\left\{\xi \in \Omega :u_k\left(\xi \right)\ge L\right\},$$

$$u_k^{\frac{Q}{Q-1}}\le \left(1+\epsilon \right)v_k^{\frac{Q}{Q-1}}+C\left(\epsilon ,Q\right)L^{\frac{Q}{Q-1}}.$$

$${\int }_{\Omega }\frac{\exp \left({\alpha }_{Q,\beta }{p}_1{u}_k^{\frac{Q}{Q-1}}\right)}{\rho {\left(\xi \right)}^{\beta }}d\xi$$

$$\le {\int }_{{\Omega }_L^k}\frac{\exp \left({\alpha }_{Q,\beta }{p}_1{u}_k^{\frac{Q}{Q-1}}\right)}{\rho {\left(\xi \right)}^{\beta }}d\xi +c\exp \left({\alpha }_{Q,\beta }{p}_1{L}^{\frac{Q}{Q-1}}\right){\int }_{\Omega }\frac{1}{\rho {\left(\xi \right)}^{\beta }}d\xi$$

$$\le {\int }_{{\Omega }_L^k}\frac{\exp \left({\alpha }_{Q,\beta }{p}_1{u}_k^{\frac{Q}{Q-1}}\right)}{\rho {\left(\xi \right)}^{\beta }}d\xi +c\left(L,Q,|\Omega |,\beta \right),$$

$$\underset{k}{\sup }{\int }_{{\Omega }_L^k}\frac{\exp \left({\alpha }_{Q,\beta }{p}_1{u}_k^{\frac{Q}{Q-1}}\right)}{\rho {\left(\xi \right)}^{\beta }}d\xi =\infty .$$

$${\int }_{{\Omega }_L^k}\frac{\exp \left({\alpha }_{Q,\beta }{p}_1{u}_k^{\frac{Q}{Q-1}}\right)}{\rho {\left(\xi \right)}^{\beta }}d\xi \le \exp \left({\alpha }_{Q,\beta }{p}_{1}C\left(\epsilon ,Q\right)L^{\frac{Q}{Q-1}}\right)\cdot {\int }_{{\Omega }_L^k}\frac{\exp \left(\left(1+\epsilon \right){\alpha }_{Q,\beta }{p}_1{v}_k^{\frac{Q}{Q-1}}\right)}{\rho {\left(\xi \right)}^{\beta }}d\xi .$$

Full text

Our results improve the sharp Trudinger-Moser inequality on domains of finite measure in $\mathbb{H}^{n}$ by Cohn and the second author [8] and the corresponding one on the whole space $\mathbb{H}^{n}$ by Lam and the second author [21]. All the proofs of the concentration-compactness principles in the literature even in the Euclidean spaces use the rearrangement argument and the Polyá-Szegö inequality. Due to the absence of the Polyá-Szegö inequality on the Heisenberg group, we will develop a different argument. Our approach is surprisingly simple and general and can be easily applied to other settings where symmetrization argument does not work. As an application of the concentration-compactness principle, we establish the existence of ground state solutions for a class of $Q$- Laplacian subelliptic equations on $\mathbb{H}^{n}:$

[TABLE]

with nonlinear terms $f$ of maximal exponential growth $\exp\left(\alpha t^{\frac{Q}{Q-1}}\right)$ as $t\rightarrow+\infty$.

The research of the second author was partly supported by a US NSF grant and a Simons Fellowship from the Simons Foundation and the research of the third author was partly supported by Natural Science Foundation of China (11601190), Natural Science Foundation of Jiangsu Province (BK20160483) and Jiangsu University Foundation Grant (16JDG043).

Corresponding author: Guozhen Lu, Email: guozhen.lu@uconn.edu

1. Introduction

Let $\Omega\subseteq$ $\mathbb{R}^{n}$ and $W_{0}^{1,q}\left(\Omega\right)$ be the usual Sobolev space, that it, the completion of $C_{0}^{\infty}\left(\Omega\right)$ with the norm

[TABLE]

If $1\leq q<n$, the classical Sobolev embedding says that $W_{0}^{1,q}\left(\Omega\right)\hookrightarrow L^{s}\left(\Omega\right)$ for $1\leq s\leq q^{\ast}$, where $q^{\ast}:=\frac{nq}{n-q}$. When $q=n$, it is known that

[TABLE]

but $\ W_{0}^{1,n}\left(\Omega\right)\varsubsetneq L^{\infty}\left(\Omega\right)$. When $\Omega$ is of finite measure, the analogue of the Sobolev embedding is the well-known Trudinger’s inequality, which was established independently by Yudovič [42], Pohožaev [36], and Trudinger [40]. In 1971, Moser sharpened in [35] Trudinger’s inequality, and proved the following inequality:

[TABLE]

where $\omega_{n-1}$ is the $n-1$ dimensional surface measure of the unit ball in $\mathbb{R}^{n}$ and $|\Omega|<\infty$. Inequality (1.1) is known as the Trudinger-Moser inequality. In 1985, Lions [30] established the Concentration-Compactness Principle associated with (1.1), which tells us that, if $\left\{u_{k}\right\}$ is a sequence of functions in $W_{0}^{1,n}\left(\Omega\right)$ with $\left\|\nabla u_{k}\right\|_{n}=1$ such that $u_{k}\rightharpoonup u$ weakly in $W^{1,n}\left(\Omega\right)$, then for any $0<p<M_{n,u}:=\left(1-\left\|\nabla u\right\|_{n}^{n}\right)^{-1/\left(n-1\right)}$, one has

[TABLE]

This conclusion gives more precise information and is stronger than (1.1) when $u_{k}\rightharpoonup u\neq 0$ weakly in $W_{0}^{1,n}\left(\Omega\right)$.

When $\left|\Omega\right|=+\infty$, the inequality (1.1) is meaningless. In this case, the first related inequalities have been considered by D.M. Cao [5] in the case $N=2$ and for any dimension by do Ó [13] and Adachi-Tanaka [1]. For two-weighted subcritical Trudinger-Moser inequalities, see [18, 16]. Note that, unlike (1.1), all these results have been proved in the subcritical growth case, that is $\alpha<\alpha_{n}$. In [38], Ruf showed that in the case $N=2$, the exponent $\alpha_{2}=4\pi$ becomes admissible if the Dirichlet norm $\int_{\Omega}\left|\nabla u\right|^{2}dx$ is replaced by $W^{1,2}$ norm $\int_{\Omega}\left(\left|u\right|^{2}+\left|\nabla u\right|^{2}\right)dx$. Later, Y.X. Li and Ruf [28] established the same critical inequality as in [38] in arbitrary dimensions. These critical and subcritical inequalities have been proved to be equivalent in [26].

While there has been much progress for Trudinger-Moser type inequalities and the concentration-compactness phenomenon on the Euclidean spaces, much less is known on the Heisenberg group. We recall that most of the proofs for Trudinger-Moser inequalities in the Euclidean space are based on the rearrangement argument. When one considers the Trudinger-Moser inequalities in the subelliptic setting, one often attempts to use the radial non-increasing rearrangement $u^{\ast}$ of functions $u$. Unfortunately, it is not true that the $L^{p}$ norm of the subelliptic gradient of the rearrangement of a function is dominated by the $L^{p}$ norm of the subelliptic gradient of the function. In other words, the Pólya-Szegö type inequality in the subelliptic setting like

[TABLE]

is not available. Actually, from the work of D. Jerison and J. Lee [19] on sharp $L^{2}$ to $L^{\frac{2Q}{Q-2}}$ inequality on the Heisenberg group with applications to the solution to the CR Yamabe problem, we know that this inequality fails to hold for the case $p=2$ in Heisenberg groups.

The sharp Trudinger-Moser inequality on Heisenberg groups was due to Cohn and the second author [8] and has been extended to the Heisenberg type groups and Carnot groups in [9] and [4] and with singular weights in [22]. Furthermore, Lam and the second author developed in [21, 23] a rearrangement-free argument by considering the level sets of the functions under consideration, this argument enables them to deduce the global Trudinger-Moser inequalities on the entire space from the local ones on the level sets (see also [31] for adaptation of such an argument). Therefore, both sharp critical and subcritical Trudinger-Moser inequalities are established on the entire Heisenberg group in [21, 24].

More recently, Černý et al. in [7] discover a new approach to obtain and sharpen Lions’s concentration compactness principles (1.2) as well as fill in a gap in [30]. This approach was further extended to study the Concentration-compactness principle for the whole space $\mathbb{R}^{n}$ by do Ó et al. in [14]. Their results can be stated as follows: let $\left\{u_{k}\right\}$ be a sequence of functions in $W_{0}^{1,n}\left(\mathbb{R}^{n}\right)$ with $\left\|u_{k}\right\|_{W^{1,n}\left(\mathbb{R}^{n}\right)}=1$ such that $u_{k}\rightharpoonup u$ weakly in $W^{1,n}\left(\mathbb{R}^{n}\right)$, then for any $0<p<\tilde{M}_{n,u}:=\left(1-\left\|u\right\|_{W^{1,n}\left(\mathbb{R}^{n}\right)}\right)^{-1/\left(n-1\right)}$,

[TABLE]

Furthermore, $\tilde{M}_{n,u}$ is sharp in the sense that there exists a sequence $\left\{u_{k}\right\}$ satisfying $\left\|u_{k}\right\|_{W^{1,n}\left(\mathbb{R}^{n}\right)}=1$ and $u_{k}\rightharpoonup u$ weakly in $W^{1,n}\left(\mathbb{R}^{n}\right)$ such that the supremum (1.4) is infinite for $p\geq\tilde{M}_{n,u}$111The sequence $\left\{u_{k}\right\}$ constructed in [14] cannot show that the supremum (1.4) is infinite for $p=\tilde{M}_{n,u}$ (see Remark 1). We also note a recent work on sharp Trudinger-Moser type inequalities in the spirit of Lions’ work on the whole spaces [25].

Nevertheless, we mention that arguments of [7] and [14] still rely on the Polyá-Szegö inequality in the Euclidean spaces and such an inequality is not available in the subelliptic setting.

Now, it is fairly natural to ask whether the Concentration-compactness principles (1.2) and (1.4) still holds for the subelliptic setting in spite of its absence of the Polyá-Szegö inequality in such a setting. In this paper, we will give an affirmative answer to this question. More precisely, we first prove a concentration-compactness principle for domains with finite measure on Heisenberg groups (Theorem 2.1), and then prove the concentration-compactness principle for the Horizontal Sobolev space $HW^{1,Q}\left(\mathbb{H}^{n}\right)$–Theorem 2.2 (for definition of $HW^{1,Q}\left(\mathbb{H}^{n}\right)$ see Section 2). Theorem 2.1 sharpens the Trudinger-Moser inequality by Cohn and the second author [8] and recent one of Lam et al. [22], Theorem 2.2 improves the sharp Trudinger-Moser inequality by Lam and the second author [21].

In the proof of the Concentration-compactness principles on Heisenberg groups, we carry out a different argument from [7] and [14]. It is worthwhile to note that our approach can be easily applied to the other subelliptic setting such as Carnot groups with virtually no modifications.

As an application of Concentration-compactness principles on Heisenberg groups, we study the existence of positive ground state solution to a class of partial differential equations with exponential growth on $\mathbb{H}^{n}$ of the form:

[TABLE]

for any $0\leq\beta<Q$, where $V:\mathbb{H}^{n}\rightarrow\mathbb{R}$ is a continuous potential, and $f:\mathbb{R}\rightarrow\mathbb{R}$ behaves like $\exp\left(\alpha t^{\frac{Q}{Q-1}}\right)$ when $t\rightarrow\infty$ (for the meaning of $\nabla_{\mathbb{H}}$ and $\rho\left(\xi\right)$ see Section 2).

We remark that the Trudinger-Moser type inequalities play an important role in the study of the existence of solutions to nonlinear partial differential equations of exponential growth in Euclidean spaces. A good deal of works have been done and we just quote some of them on this subject, which are a good starting point for further bibliographic references: [2, 3, 6, 11, 12, 13, 15, 14, 17, 29, 20, 27, 32, 33, 39, 41, 43], etc.

Existence and multiplicity of nontrivial nonnegative solutions to the equations (1.5) on the Heisenberg groups have been proved in a series of papers [10, 21, 22, 24]. In their argument, they apply the Trudinger-Moser inequality in the whole space $\mathbb{H}^{n}$ (Lemma 2.3 in Section 2) combined with mountain-pass theorem, minimization and Ekelands variational principle. Nevertheless, the existence of ground state solutions to the sub-elliptic equation (1.5) on the Heisenberg groups has not been established yet so far. The concentration-compactness principles on Heisenberg groups proved in this paper makes it possible to establish such an existence result.

This paper is organized as follows: in Section 2 we recall some basic facts about Heisenberg Groups and state precisely our main results; in Section 3 we first prove the concentration compactness principles for Trudinger-Moser inequalities on domains with finite measure – Theorem 2.1, and then we give the proof for the concentration compactness principles for Horizontal Sobolev space $HW^{1,Q}\left(\mathbb{H}^{n}\right)$ –Theorem 2.2. As an application, in Section 4, we consider the equations (1.5) and establish the existence of the ground state solutions and  prove Theorem 2.3 by using the minimax argument and Theorem 2.2.

2.

Preliminaries and statement of the results

2.1. Background on Heisenberg groups

Let $\mathbb{H}^{n}=\mathbb{C}^{n}\times\mathbb{R}$ be the $n$-dimensional Heisenberg group, whose group structure is given by

[TABLE]

The Lie algebra of $\mathbb{H}^{n}$ is generated by the left invariant vector fields

[TABLE]

for $i=1,\ldots,n$. These generators satisfy the non-commutative relationship $\left[X_{i},Y_{i}\right]=4\delta_{ij}T$. Moreover, all the commutators of length greater than two vanish, and thus this is a nilpotent, graded, and stratified group of step two.

For each real number $r\in\mathbb{R}$, there is a dilation naturally associated with the Heisenberg group structure which is usually denoted as $\delta_{r}\left(x,t\right)=\left(rx,r^{2}t\right)$. The Jacobian determinant of $\delta_{r}$ is $r^{Q}$, where $Q=2n+2$ is the homogeneous dimension of $\mathbb{H}^{n}$.

We will use $\xi=(x,t)$ to denote any point $(x,t)\in\mathbb{H}^{n}$, then the anisotropic dilation structure on $\mathbb{H}^{n}$ introduces a homogeneous norm $\left|\xi\right|=\left(\left|x\right|^{4}+t^{2}\right)^{1/4}$. Let

[TABLE]

be the metric ball of center [math] and radius $r$ in $\mathbb{H}^{n}$. Since the Lebesgue measure in $\mathbb{R}^{2n+1}$ is the Haar measure on $\mathbb{H}^{n}$, one has (writing $\left|A\right|$ for the measure of $A$)

[TABLE]

where $\omega_{Q}$ is a positive constant only depending on $Q$ (see [8]).

We write $\left|\nabla_{\mathbb{H}}u\right|$ to express the norm of the subelliptic gradient of the function $u:\mathbb{H}^{n}\rightarrow\mathbb{R}:$

[TABLE]

Let $\Omega$ be an open set in $\mathbb{H}^{n}$ and $p>1$. We define the Horizontal Sobolev Spaces

[TABLE]

with the norm

[TABLE]

Also, we define the space $HW_{0}^{1,p}\left(\Omega\right)$ as the closure of $C_{0}^{\infty}\left(\Omega\right)$ in the norm of $HW^{1,p}\left(\Omega\right)$.

2.2. Some useful known results on Heisenberg groups

In this subsection, we collect some known results which will be used in the following.

Lemma 2.1** ([8]).**

Let $\rho=\left|\xi\right|$ be the homogeneous norm of the element $\xi=\left(x,t\right)\in\mathbb{H}^{n}$, and $g\left(\xi\right)=g\left(\rho\right)$ be a $C^{1}$ radial function on $\mathbb{H}^{n}$. Then

[TABLE]

Lemma 2.2** ([22]).**

Let $\alpha_{Q}=Q\left(2\pi^{n}\Gamma\left(\frac{1}{2}\right)\Gamma\left(\frac{Q-1}{2}\right)\Gamma\left(\frac{Q}{2}\right)^{-1}\Gamma\left(n\right)^{-1}\right)^{\frac{1}{Q-1}},$ $0\leq\beta<Q$. There exists a uniform constant $c$ depending only on $Q,\beta$ such that for all $\Omega\subset\mathbb{H}^{n}$ with $\left|\Omega\right|<\infty$ and $\alpha\leq\alpha_{Q,\beta}=\alpha_{Q}\left(1-\frac{\beta}{Q}\right)$, one has

[TABLE]

The constant $\alpha_{Q,\beta}$ is the best possible in the sense that if $\alpha>$ $\alpha_{Q,\beta}$, then the supremum above is infinite.

Lemma 2.3** ([21]).**

Let $0\leq\beta<Q$. There exists a uniform constant $c$ depending only on $Q,\beta$ such that for all $\alpha\leq\alpha_{Q,\beta}$, one has

[TABLE]

where $\Phi\left(t\right)=e^{t}-\underset{j=0}{\overset{Q-2}{\sum}}\frac{t^{j}}{j!}$. The constant $\alpha_{Q,\beta}$ is the best possible in the sense that if $\alpha>\alpha_{Q,\beta}$, then the supremum in the inequality (2.1) is infinite.

2.3.

Statement of the main results

Now, we are ready to state precisely the main results of this paper.

Theorem 2.1** (Concentration compactness for domains with finite measure).**

Let $0\leq\beta<Q$. Assume that $\left\{u_{k}\right\}$ is a sequence in $HW_{0}^{1,Q}\left(\Omega\right)$ with $\left|\Omega\right|<\infty$, such that $\left\|\nabla_{\mathbb{H}}u_{k}\right\|_{Q}=1$ and $u_{k}\rightharpoonup u\neq 0$ in $HW_{0}^{1,Q}\left(\Omega\right)$. If

[TABLE]

then

[TABLE]

Moreover, $M_{Q,u}$ is sharp in the sense that there exists a sequence $\left\{u_{k}\right\}$ satisfying $\left\|\nabla_{\mathbb{H}}u\right\|_{Q}^{Q}=1$ and $u_{k}\rightharpoonup u\neq 0$ in $HW_{0}^{1,Q}\left(\Omega\right)$ such that the supremum is infinite for $p\geq M_{Q,u}$.

Theorem 2.2** (Concentration compactness for $HW^{1,Q}\left(\mathbb{H}^{n}\right)$).**

Let $0\leq\beta<Q$.

Assume that $\left\{u_{k}\right\}$ is a sequence in $HW^{1,Q}\left(\mathbb{H}^{n}\right)$ such that $\left\|u_{k}\right\|_{HW^{1,Q}\left(\mathbb{H}^{n}\right)}^{Q}=1$ and $u_{k}\rightharpoonup u\neq 0$ in $HW^{1,Q}\left(\mathbb{H}^{n}\right)$. If

[TABLE]

then

[TABLE]

where $\Phi\left(t\right)=e^{t}-\underset{j=0}{\overset{Q-2}{\sum}}\frac{t^{j}}{j!}$. Furthermore, $\tilde{M}_{Q,u}$ is sharp in the sense that there exists a sequence $\left\{u_{k}\right\}$ satisfying $\left\|u_{k}\right\|_{HW^{1,Q}\left(\mathbb{H}^{n}\right)}^{Q}=1$ and $u_{k}\rightharpoonup u\neq 0$ in $HW^{1,Q}\left(\mathbb{H}^{n}\right)$ such that the supremum is infinite for $p>\tilde{M}_{Q,u}.$

The following natural question still remains open at this time.

Problem 1**.**

Does (2.3) still hold when $p=\tilde{M}_{Q,u}$?

Now, Let us give the definition of the ground state solution of (1.5):

Definition 1** (Ground state solution).**

A function $u$ is said to be the ground state solution of (1.5), if $u$ is positive and minimizes the energy functional associated to the equation (4.1) defined by

[TABLE]

within the set of nontrivial solutions of (1.5).

For the equation (4.1), we obtain the following

Theorem 2.3**.**

Under the hypotheses of (H1) and (H2) in Section 4, the  $Q-$sub-Laplacian equations (1.5) has a positive ground state solution.

Throughout this paper, denote by the letter $c$ some positive constant which may vary from line to line.

3. Concentration-Compactness principles on Heisenberg

groups

3.1. Concentration-Compactness principle for domains with finite

measure

In this subsection, we give the

Proof of Theorem 2.1.

Since $\left\|\nabla_{\mathbb{H}}u\right\|_{Q}\leq\underset{k}{\lim}\left\|\nabla_{\mathbb{H}}u_{k}\right\|_{Q}=1$, we split the proof into two cases.

Case 1:  $\left\|\nabla_{\mathbb{H}}u\right\|_{Q}<1.$ We assume by contradiction for some $p_{1}<M_{Q,u}$, we have

[TABLE]

Set

[TABLE]

where $L$ is some constant.  Let $v_{k}=u_{k}-L$. Then for any $\varepsilon>0$, one has

[TABLE]

Since $0\leq\beta<Q$, we have

[TABLE]

and then

[TABLE]

By (3.1) we have

[TABLE]

Thus

[TABLE]

where $\bar{p}_{1}=\left(1+\varepsilon\right)p_{1}<M_{Q,u}.$

Now, we define

[TABLE]

and choose $L$ such that

[TABLE]

We claim that

[TABLE]

If not, then up to a subsequence, one has

[TABLE]

Thus,

[TABLE]

For $L>0$ fixed, $T^{L}u_{k}$ is also bounded in $HW^{1,Q}\left(\Omega\right)$. Hence, up to a subsequence, $T^{L}u_{k}\rightharpoonup T^{L}u$ in $HW^{1,Q}\left(\Omega\right)$ and $T^{L}u_{k}\rightarrow T^{L}u$ almost everywhere in $\Omega$. By the lower semicontinuity of the norm in $HW^{1,Q}\left(\Omega\right)$ and the above inequality, we have

[TABLE]

combining with (3.2), we derive

[TABLE]

which is a contradiction.  Therefore

[TABLE]

By the Trudinger-Moser inequality (2.1), we derive

[TABLE]

which is also a contradiction. The proof is finished in this case.

Case 2: $\left\|\nabla_{\mathbb{H}}u\right\|_{Q}=1$. We can iterate the procedure as in Case $1$ and get

[TABLE]

where $\bar{p}_{1}=\left(1+\varepsilon\right)p_{1}$.  Then we have

[TABLE]

thus,

[TABLE]

On the other hand, since $\left\|\nabla_{\mathbb{H}}u\right\|_{Q}=1$, we can take $L$ large enough such that

[TABLE]

which is contradiction, and the proof is finished in this case.

Now, we prove the sharpness of $M_{Q,u}$. For some $r>0$, we defined $\omega_{k}\left(\xi\right)$ by

[TABLE]

where $c_{Q}=\int_{\Sigma}\left|x^{\ast}\right|^{Q}d\xi$, $x^{\ast}=\frac{x}{\left|\xi\right|}$ and $\Sigma$ is the unit sphere on $\mathbb{H}^{n}$.

We can verify that $\omega_{k}\left(\xi\right)\in HW_{0}^{1,Q}\left(\Omega\right)$. Actually, from Lemma 2.1 we have

[TABLE]

and $\omega_{k}\left(\xi\right)\rightharpoonup 0$ in $HW_{0}^{1,Q}\left(\Omega\right)$.

Now, for $R=3r$, we define

[TABLE]

where $A>0$ is chosen in such a way that $\left\|\nabla_{\mathbb{H}}u\right\|_{L^{Q}\left(\Omega\right)}=$ $\delta<1$. Defining

[TABLE]

Observing that $\nabla_{\mathbb{H}}u$ and $\nabla_{\mathbb{H}}\omega_{k}$ have disjoints supports, we have

[TABLE]

moreover, $u_{k}\rightharpoonup u$ in $HW_{0}^{1,Q}\left(\Omega\right)$.

Consequently,

[TABLE]

for some positive constant $C,C^{\prime},C^{\prime\prime}$, and the theorem is finished. ∎

3.2.

Concentration Compactness Principle for the whole space $\mathbb{H}^{n}$

In order to prove Theorem 2.2, we need the following

Lemma 3.1**.**

Let $\left\{u_{k}\right\}$ be a sequence in $HW^{1,Q}\left(\mathbb{H}^{n}\right)$ strongly convergent. Then there exist a subsequence $\left\{u_{k_{i}}\right\}$ of $\left\{u_{k}\right\}$ and $\omega\left(\xi\right)\in HW^{1,Q}\left(\mathbb{H}^{n}\right)$ such that $\left|u_{k_{i}}\right|\leq$ $\omega\left(\xi\right)$ almost everywhere on $\mathbb{H}^{n}$.

Proof.

The proof is similar as [14, Proposition 1], and we omit it. ∎

Now, we give the

Proof of Theorem 2.2.

As in the proof of Theorem 2.1, we split the proof into two cases.

Case 1:  $\left\|u\right\|_{HW^{1,Q}\left(\mathbb{H}^{n}\right)}<1.$ We assume by contradiction for some $p_{1}<\tilde{M}_{Q,u}$, we have

[TABLE]

Set

[TABLE]

and

[TABLE]

where $L$ is some constant which will be determined later.  We can easily verify that

[TABLE]

Now, we write

[TABLE]

Similar to the proof of [21, Theorem 1.6], we can show that

[TABLE]

Therefore, we have

[TABLE]

Let $v_{k}=u_{k}-L$. Then for any $\varepsilon>0$, one has

[TABLE]

By (3.7), we have

[TABLE]

Thus

[TABLE]

where $\bar{p}_{1}=\left(1+\varepsilon\right)p_{1}<\tilde{M}_{Q,u}$.

Since $\left|\Omega\left(u_{k}\right)\right|\leq 2^{\frac{1}{Q-1}}$, we have

[TABLE]

Now, we define

[TABLE]

and choose $L$ such that

[TABLE]

We claim that

[TABLE]

If not, up to a subsequence, one has

[TABLE]

Thus,

[TABLE]

For $L>0$ fixed, $T^{L}u_{k}$ is also bounded in $HW^{1,Q}\left(\mathbb{H}^{n}\right)$. Hence, up to a subsequence, $T^{L}u_{k}\rightharpoonup T^{L}u$ in $HW^{1,Q}\left(\mathbb{H}^{n}\right)$ and $T^{L}u_{k}\rightarrow T^{L}u$ almost everywhere on $\mathbb{H}^{n}$. By the lower semicontinuity of the norm in $HW^{1,Q}\left(\mathbb{H}^{n}\right)$ and the above inequality, we have

[TABLE]

combining with (3.8), we have

[TABLE]

which is a contradiction.  Therefore

[TABLE]

By the Trudinger-Moser inequality (2.1), we have

[TABLE]

which is also a contradiction. The proof is finished in this case.

Case 2: $\left\|u\right\|_{HW^{1,Q}\left(\mathbb{H}^{n}\right)}=1$. Since $HW^{1,Q}\left(\mathbb{H}^{n}\right)$ is uniformly convex Banach space and $u_{k}\rightharpoonup u$ weakly in $HW^{1,Q}\left(\mathbb{H}^{n}\right)$, by Radon’s Theorem, we have $u_{k}\rightarrow u$ strongly in $HW^{1,Q}\left(\mathbb{H}^{n}\right)$. Using Lemma 3.1, there exists some $\omega\left(\xi\right)\in HW^{1,Q}\left(\mathbb{H}^{n}\right)$, such that up to a subsequence, \left|u_{k}\right|\leq\omega\left(\xi\right)\a.e. in $\mathbb{H}^{n}$. Therefore

[TABLE]

Now, we show

[TABLE]

Set $\Omega\left(\omega\right)=\left\{\xi\in\mathbb{H}^{n}:\omega>1\right\}$, we have

[TABLE]

Similar as [21], we can derive

[TABLE]

Now, we only need to show

[TABLE]

Let $\omega^{\ast}\left(\xi\right)$ be the non-increasing rearrangement of $\omega\left(\xi\right)$ in $\Omega\left(\omega\right)$. Then

[TABLE]

where $\left|B_{R}\right|=$ $\left|\Omega\left(\omega\right)\right|$. We introduce the variable $t$ by $\ \rho\left(\xi\right)^{Q}=R^{Q}e^{-t}$, and set

[TABLE]

Then by Lemma 2.1 and the result of Manfredi and Vera De Serio [34] that there exists a constant $c\geq 1$ depending only on $Q$ such that,

[TABLE]

Moreover, we have

[TABLE]

This follows from the Hardy-Littlewood inequality implies by noticing that the rearrangement of $\rho\left(\xi\right)^{-\beta}$ is just itself.

Since $\int_{0}^{\infty}\left|\varphi^{\prime}\left(t\right)\right|^{Q}dt<\infty$, then for all $\varepsilon>0$, there exists $T=T\left(\varepsilon\right)$ such that

[TABLE]

Hence, by Hölder’s inequality

[TABLE]

There exists $\tilde{T}$ such that

[TABLE]

Therefore $\int_{\Omega\left(\omega\right)}\frac{\exp\left(\alpha_{Q,\beta}p_{1}\omega\left(\xi\right)^{\frac{Q}{Q-1}}\right)}{\rho\left(\xi\right)^{\beta}}d\xi<\infty$, and the proof is finished in this case.

Now, we prove the sharpness of $\tilde{M}_{Q,u}$. For some $r>0$ and $R=3r$, we define $\ \omega_{k}\left(\xi\right),u\in HW^{1,Q}\left(\mathbb{H}^{n}\right)$ as (3.4),(3.5), respectively. The constant $A$ is chosen in such a way that $\left\|u\right\|_{HW^{1,Q}\left(\mathbb{H}^{n}\right)}=\delta<1$. Defining

[TABLE]

We can easily verify that

[TABLE]

[TABLE]

and

[TABLE]

Moreover, from (3.11) we have

[TABLE]

where $\xi_{k}\rightarrow O\left(\left(\frac{1}{k}\right)^{1/Q}\right)$, and then we have $\left\|u_{k}\right\|_{HW^{1,Q}\left(\mathbb{H}^{n}\right)}^{Q}=1+\xi_{k}$. Set $v_{k}=\frac{u_{k}}{\left(1+\xi_{k}\right)^{1/Q}}$, we have

[TABLE]

Consequently, for any $\varepsilon_{0}>0$ and $p=\left(1+\varepsilon_{0}\right)\tilde{M}_{Q,u}$, one has

[TABLE]

(using the fact that $\xi_{k}\rightarrow 0$)

[TABLE]

for some positive constant $\varepsilon_{0}^{\prime},C,C^{\prime},C^{\prime\prime}$, and the theorem is finished. ∎

Remark 1**.**

The sequence $\left\{v_{k}\right\}$ is not enough to show that the supremum (2.3) is infinite when  $p=\tilde{M}_{Q,u}$. Actually, we have

[TABLE]

for some positive constant $\ C^{\prime}$,$C^{\prime\prime}$,$C^{\prime\prime\prime}$ and $C^{\prime\prime\prime\prime}$. We remark that this argument is also suitable for the sequence constructed in [15] for the sharpness of $\tilde{M}_{n,u}=\left(1-\left\|u\right\|_{W^{1,n}\left(\mathbb{R}^{n}\right)}\right)^{-1/\left(n-1\right)}$.

4. $Q-$sub-Laplacian equations of exponential growth on $\mathbb{H}^{n}$

.

In this section, let’s consider the following nonlinear partial differential equations on $\mathbb{H}^{n}:$

[TABLE]

where $0\leq\beta<Q$.

The main features of this class of equations (4.1) are that it is defined in the whole space $\mathbb{H}^{n}$ and involves critical growth and the nonlinear operator is $Q$-sub-Laplacian. In spite of a possible failure of the Palais–Smale (PS) compactness condition, we apply the minimax argument based on the Concentration-Compactness Principle for $HW^{1,Q}\left(\mathbb{H}^{n}\right)$ – Theorem 2.2 as in [15].

The basic assumptions about $f$ and V\are collected in the following:

(H1) Assumptions for potential $V$

The potential $V:\mathbb{H}^{n}\rightarrow\mathbb{R}$ is a continuous potential, and satisfies:

(V1) $V$ is a continuous function such that V(\xi)\geq 1\ for all $\xi\in\mathbb{H}^{n}$, and one of the following two conditions:

(V2) $V(\xi)\rightarrow\infty$ as $\rho\left(\xi\right)$ $\rightarrow\infty$; or more generally, for every $M>0$, $\left|\left\{\xi\in\mathbb{H}^{n}:V(\xi)>M\right\}\right|<\infty$;

(V3) the function $V(\xi)^{-1}$ belongs to $L^{1}(\mathbb{H}^{n})$.

(H2) Assumptions for $f$

The function $f\left(t\right):\mathbb{R}\rightarrow\mathbb{R}$ behaves like $\exp\left(\alpha t^{\frac{Q}{Q-1}}\right)$ when $\left|t\right|\rightarrow\infty$. Precisely, we assume that $f$ satisfies the following conditions:

(f1) there exist constants $\alpha_{0},b_{0},b_{1}>0$ such that for all $t\in\mathbb{R}$,

[TABLE]

(f2) there exists $\lambda>Q$ such that for all $\xi\in\mathbb{H}^{n}$ and $t>0,$

[TABLE]

(f3) there exist constant $R_{0},M_{0}>0$ such that for all $\xi\in\mathbb{H}^{n}$ and $t>R_{0},$

[TABLE]

(f4) there exist constant $\mu>Q$ and $C_{\mu}$ such that for all $t\geq 0,$

[TABLE]

with $C_{\mu}$ satisfying

[TABLE]

where

[TABLE]

(f5) $\underset{t\rightarrow 0^{+}}{\lim\sup}\frac{F\left(t\right)}{k_{0}\left|s\right|^{Q}}<\lambda_{Q}:=\underset{u\in\mathcal{S}\backslash 0}{\inf}\frac{\left\|u\right\|^{Q}}{\int_{\mathbb{H}^{n}}\frac{\left|u\right|^{Q}}{\rho\left(\xi\right)^{\beta}}d\xi}$.

Since we are interested in nonnegative weak solutions, we also suppose the following

(f6) $F\left(t\right)=0$ if $t\leq 0$.

From condition (f1), we conclude that for all $t\in\mathbb{R}\,$,

[TABLE]

for some constant $b_{2},\alpha_{1}>0$. From (3.10), we have $\frac{F\left(u\right)}{\rho\left(\xi\right)^{\beta}}\in L^{1}\left(\mathbb{H}^{n}\right)$ for all $u\in\mathcal{S}$. Therefore, the associated functional to the equation (4.1) defined by

[TABLE]

is well-defined. Moreover, $J$ is a $C^{1}$ functional on $\mathcal{S}$ with

[TABLE]

for all $v\in\mathcal{S}$. Thus, $DJ\left(u\right)=0$ if and only if $u$ $\in\mathcal{S}$ is a weak solution to equation (4.1).

We define the following space associated with the potential $V$:

[TABLE]

with the norm $\left\|u\right\|:=\left(\int_{\mathbb{H}^{n}}\left(\left|\nabla_{\mathbb{H}}u\right|^{Q}+V\left(\xi\right)\left|u\right|^{Q}\right)d\xi\right)^{1/Q}$.

From the hypothesis (H1), we have the following compactness result:

Lemma 4.1**.**

If $V\left(\xi\right)$ satisfy the hypothesis (H1), then for all $Q\leq q<\infty$, the embedding

[TABLE]

is compact.

Proof.

The proof is analogous to the proof for the Euclidean case in [Costa, 37], for the completeness, we give the details here.

Let $\left\{u_{k}\right\}$ be a sequence such that $\left\|u_{k}\right\|^{Q}<C$. In order to prove this result, we only need to show that $u_{k}\rightarrow 0$ strongly in $L^{q}\left(\mathbb{H}^{n}\right)$ for any $Q\leq q<\infty$, whenever $u_{k}\rightharpoonup 0$ weakly in $\mathcal{S}$, as $k\rightarrow\infty$.

For any $\varepsilon>0$, from (V2), we can choose some $R>0$ such that

[TABLE]

for all $\xi$ satisfying $\rho\left(\xi\right)\geq R$. Since the embedding $HW^{1,Q}\left(B_{R}\right)\hookrightarrow L^{q}\left(B_{R}\right)$ is compact, we know $u_{k}\rightarrow 0$ strongly in $L^{q}\left(B_{R}\right)$, and then there exists a integer $N>0$ such that when $k>N$,

[TABLE]

On the other hand, from (4.4) we have

[TABLE]

that is

[TABLE]

Combine (4.5) and (4.6) we obtain

[TABLE]

For any $q<p<\infty$, we define

[TABLE]

Then $\lambda>0$ and $\mu>0$.   By Hölder’s inequality, the Trudinger-Moser inequality (2.2) and (4.7), we have

[TABLE]

The proof is finished. ∎

4.1. Palais–Smale compactness condition

In this subsection, we analyze the compactness of Palais–Smale sequences of the functional $J$. This is the crucial step in the study of existence results for equation (4.1).

First, we recall the definition of Palais-Smale Condition:

Definition 2** (Palais–Smale Condition).**

A sequence\ \left\{u_{k}\right\}\ in $\mathcal{S}$ is called a local Palais–Smale sequence at level $d$ for the

functional $J$ ($\left(PS\right)_{d}$ sequence), if

[TABLE]

the functional $J$ is said to satisfy the Palais–Smale condition at level $d$ ($\left(PS\right)_{d}$ condition), if any $\left(PS\right)_{d}$ sequence has a convergent subsequence.

Lemma 4.2**.**

Under the hypotheses of (H1) and (H2). The functional $J$ satisfies the Palais–Smale condition at level $d$ for any $d<\frac{1}{Q}\left(\frac{\alpha_{Q,\beta}}{\alpha_{0}}\right)^{Q-1}$.

Proof.

The proof is analogous to the proof of [15, Proposition 4.1]. For the completeness, we give the details here.

Let $\left\{u_{k}\right\}$ be a $\left(PS\right)_{d}$ sequence for $J$, that is,

[TABLE]

and $\left|DJ\left(u_{k}\right)v\right|\rightarrow 0$ for all $v\in\mathcal{S}$, as   $k\rightarrow\infty$. Then

[TABLE]

for all $v\in\mathcal{S}$, where $\varepsilon_{k}\rightarrow 0$, as $k\rightarrow\infty$.

Choosing $v=u_{k}$ in (4.9), by (4.8) we get

[TABLE]

From (f2), we have

[TABLE]

hence, $u_{k}$ is bounded in $\mathcal{S}$. Since for any $q\geq Q$, the embedding $\mathcal{S}\hookrightarrow L^{q}\left(\mathbb{H}^{n}\right)$ is compact, we can assume that,

[TABLE]

From (4.10), we can verify that

[TABLE]

for any $s\in\left[Q,\infty\right)$ and $\beta\in\left[0,Q\right)$. Actually, for any $R>0$,

[TABLE]

by Hölder’s inequality, for any $p,\gamma\in\left(1,\infty\right)$, we have

[TABLE]

where $p^{\prime}=\frac{p}{p-1}$ and $\gamma^{\prime}=\frac{\gamma}{\gamma-1}$.

Choosing $p^{\prime}$ and $\gamma$ sufficiently closed to $1$ such that

[TABLE]

then by (4.10), we get $\int_{\mathbb{H}^{n}}\frac{\left|u_{k}-u\right|^{s}}{\rho\left(\xi\right)^{\beta}}d\xi\rightarrow 0$.

Thanks to [21, Lemma 5.5 and (6.7)], we have

[TABLE]

From this convergence and passing the limit in (4.9), we get

[TABLE]

for any $v\in C_{0}^{\infty}\left(\mathbb{H}^{n}\right)$. By density, taking $v=u$, we have

[TABLE]

from (f2), we get

[TABLE]

thus, $J\left(u\right)\geq 0$.

In the following, we prove the strong convergence of $\left\{u_{k}\right\}$. For this purpose, we split the proof into two cases:

Case 1: $d=0$. From (4.12) and (4.8), we have

[TABLE]

hence $J\left(u\right)\leq 0$. Therefore $J\left(u\right)=0$ and $\underset{k}{\lim}\left\|u_{k}\right\|^{Q}$ $=\left\|u\right\|^{Q}$. Since $\mathcal{S}$ is a uniformly convex Banach space, by Radon’s Theorem, $u_{k}\rightarrow$ $u$ strongly in $\mathcal{S}$.

Case 2: $d\neq 0$. We first claim that $u\neq 0$. We assume by contradiction for $u=0$. By (4.12), (f2) and (f1), we have $\int_{\mathbb{H}^{n}}\frac{F\left(u_{k}\right)}{\rho\left(\xi\right)^{\beta}}d\xi\rightarrow 0$, as $k\rightarrow\infty$. From (4.8) we obtain

[TABLE]

Since $d<\frac{1}{Q}\left(\frac{\alpha_{Q,\beta}}{\alpha_{0}}\right)^{Q-1}$, we can choose some $q>1$ close to $1$ sufficiently such that

[TABLE]

Then, by Lemma 2.3, we have

[TABLE]

By Hölder’s inequality, combining (f1), (4.14), (4.10) and (4.11) we get

[TABLE]

where $q^{\prime}=\frac{q}{q-1}$.

On the other hand, since $u_{k}$ is a $\left(PS\right)_{d}$ sequence, $DJ\left(u_{k}\right)u_{k}\rightarrow 0$, i.e.,

[TABLE]

thus, $\int_{\mathbb{H}^{n}}\left(\left|\nabla_{\mathbb{H}}u_{k}\right|^{Q}+V\left(\xi\right)\left|u_{k}\right|^{Q}\right)d\xi\rightarrow 0$, which is a contradiction, and the claim is proved.

Set $\tilde{u}_{k}=\frac{u_{k}}{\left\|u_{k}\right\|}$ and $\tilde{u}=\frac{u}{\underset{k}{\lim}\left\|u_{k}\right\|}$. Then $\left\|u_{k}\right\|=1$ and $\tilde{u}_{k}\rightharpoonup\tilde{u}$ weakly in $\mathcal{S}$. If $\left\|\tilde{u}\right\|=1$, we have $\underset{k}{\lim}\left\|u_{k}\right\|=\left\|u\right\|$, and then $u_{k}\rightarrow$ $u$ strongly in $\mathcal{S}$.

If $\left\|u\right\|<1$,  by (4.8) and (4.12) and the fact that $J\left(u\right)\geq 0$, one has

[TABLE]

thus,

[TABLE]

that is

[TABLE]

therefore when $k$ large, we can choose some $q>1$ close to $1$ sufficiently such that

[TABLE]

By Theorem 2.2, $\frac{\Phi\left(q\alpha_{0}u_{k}^{\frac{Q}{Q-1}}\right)}{\rho\left(\xi\right)^{q\beta}}$ is bounded in $L^{1}\left(\mathbb{H}^{n}\right)$.

By Hölder’s inequality, combining (f1), (4.15), (4.10), (4.11), we get

[TABLE]

Since $DJ\left(u_{k}\right)\left(u_{k}-u\right)\rightarrow 0$,  from (4.16) we derive

[TABLE]

On the other hand, since $u_{k}\rightharpoonup u$ in $\mathcal{S}$, we have

[TABLE]

Combining (4.17) and (4.18), we obtain there is a constant $c>0$ so that

[TABLE]

where we have used the inequality $\left(\left|a\right|^{Q-2}a-\left|b\right|^{Q-2}b\right)\left(a-b\right)\geq 2^{2-Q}\left|a-b\right|^{Q}$, for all $a,b\in\mathbb{R}^{2n}$. The proof is finished. ∎

From the proof of [21, Lemma 5.1 and Lemma 5.2.], we have the following geometric conditions of the mountain-pass theorem:

Lemma 4.3**.**

Suppose that the hypotheses of (H1) and (H2) hold. Then

(i)

there exists $r,\delta>0$ such that $J\left(u\right)\geq\delta$ if $\left\|u\right\|=r$;

(ii) there exists $e\in\mathcal{S}$ with $\left\|e\right\|>r$, such that $J\left(e\right)<0$.

Now, we define the minimax level by

[TABLE]

where $\Gamma=\left\{g\in C\left(\left[0,1\right],\mathcal{S}\right):g\left(0\right)=0\text{ and }g\left(1\right)<0\right\}$. From Lemma 4.3, we have $d_{\infty}>0$.

Lemma 4.4**.**

Under the hypotheses of (H1) and (H2). We have $d_{\infty}<\frac{1}{Q}\left(\frac{\alpha_{Q,\beta}}{\alpha_{0}}\right)^{Q-1}$.

Proof.

Let $\left\{v_{k}\right\}$ be in $\mathcal{S}$ with $\int_{\mathbb{H}^{n}}\frac{\left|v_{k}\right|^{\mu}}{\rho\left(\xi\right)^{\beta}}d\xi=1$ and $\left\|v_{k}\right\|^{Q}\rightarrow\lambda_{\mu}$. Then $\left\{v_{k}\right\}$ is bounded in $\mathcal{S}$. Using the compactness of embedding  $\mathcal{S}\hookrightarrow L^{q}\left(\mathbb{H}^{n}\right)$ for all $q\geq Q$, up to a subsequence, we have

[TABLE]

(4.11) implies that $\int_{\mathbb{H}^{n}}\frac{\left|v_{0}\right|^{\mu}}{\rho\left(\xi\right)^{\beta}}d\xi=\underset{k}{\lim}\int_{\mathbb{H}^{n}}\frac{\left|v_{k}\right|^{\mu}}{\rho\left(\xi\right)^{\beta}}d\xi=1$. By the semicontinuity of the norm $\left\|\cdot\right\|$, we infer that

[TABLE]

thus $\lambda_{\mu}$ is attained by $v_{0}$, we may assume that $v_{0}\geq 0$.

From (4.2), we know that $F\left(t\right)\geq\frac{C_{\mu}}{\mu}t^{\mu}$ for some $\mu>Q$. Hence,

[TABLE]

Setting $\tilde{v}_{0}\left(t\right)=tt_{0}v_{0}$ with $t_{0}$ sufficiently large, then $\tilde{v}_{0}\left(t\right)\in\Gamma$. By (f4), we have

[TABLE]

and this completes the proof. ∎

Finally, we come to the

Proof of Theorem 2.3.

Let $\left\{u_{k}\right\}$ be a sequence in $\mathcal{S}$ such that

[TABLE]

and $DJ\left(u_{k}\right)\rightarrow 0$. By Lemmas 4.2 and 4.4, the sequence $\left\{u_{k}\right\}$ converges weakly to a weak solution $u_{0}$ of (4.1).   Now, we show $u_{0}>0$ in $\mathbb{H}^{n}$.

Set $u_{0+}:=\max\left\{u_{0},0\right\}$ and $u_{0-}:=\max\left\{-u_{0},0\right\}$.  Since $u_{0}$ satisfies $DJ\left(u_{0}\right)=0$, we have $DJ\left(u_{0}\right)u_{0-}=0$, that is,

[TABLE]

On the other hand, from (f6) we have $\int_{\mathbb{H}^{n}}\frac{f\left(u_{0}\right)u_{0-}}{\rho\left(\xi\right)^{\beta}}d\xi=0$, and then $\left\|\ u_{0-}\right\|^{Q}=0$. Therefore, $u_{0}\geq 0$ on $\mathbb{H}^{n}$. From $J\left(u_{0}\right)=d_{\infty}>0$, we know u_{0}\is positive on $\mathbb{H}^{n}$.

Now, let

[TABLE]

where $\mathcal{P}:=\left\{u\in\mathcal{S}:DJ\left(u\right)=0\right\}$.

In order to show that $u_{0}$ is a ground state solution of (4.1), we only need to prove $d_{\infty}$ $\leq M_{\infty}$. For any $u\in$ $\mathcal{P}\backslash 0$, we define $m\left(t\right)$ by $m\left(t\right)=J\left(tu\right)$. Since $J\in C^{1}\left(\mathcal{S},\mathbb{R}\right)$, we have $m\left(t\right)$ is differentiable and

[TABLE]

for any $t>0$.

From $DJ\left(u\right)u=0$, we derive

[TABLE]

By (f2), we know $\frac{f\left(t\right)}{t^{Q-1}}$ is increasing for all $s>0$. From this and the fact $m^{\prime}\left(1\right)=0$, we know $m^{\prime}\left(t\right)>0$ if $t\in\left(0,1\right)$, and $m^{\prime}\left(t\right)<0$ if $t\in\left(1,\infty\right)$. Thus, $J\left(u\right)=\underset{t\geq 0}{\max}J\left(tu\right)$.

Setting $\tilde{u}\left(t\right)=tt_{0}u$ with $t_{0}$ sufficiently large, we get $\tilde{u}\left(t\right)\in\Gamma$, and then

[TABLE]

Therefore, $d_{\infty}$ $\leq M_{\infty}$. The proof is completed. ∎

Acknowledgement The results of this paper were presented by the third author at the Workshop in Fourier Analysis in Sanya Mathematical Forum in August, 2016 and by the second author at the AMS special session on Geometric Aspects of Harmonic Analysis in Maine in September, 2016.