Positive ground state solutions for asymptotically periodic generalized quasilinear Schrödinger equations

1.   Introduction

In this paper, we are concerned with the following generalized quasilinear Schrödinger equations

where $N\geq 3$, $g\in C^{1}(\mathbb{R}, \mathbb{R}^{+})$ is an even function with $g'(t)\geq 0$ for all $t\geq 0$ and $g(0) = 1$, $h\in C(\mathbb{R}^{N}\times \mathbb{R}, \mathbb{R})$ and $V(x)$ is 1-periodic or asymptotically periodic potential.

The equations are related to the solitary wave solutions for quasilinear Schrödinger equations

where $W: \mathbb{R}^{N} \rightarrow \mathbb{R}$ is a given potential, $z: \mathbb{R} \times \mathbb{R}^{N}\rightarrow \mathbb{C}$, $h$ and $l$ can be used to model physical phenomenon. The form of (1.2) has many applications in physics. For instance, the case $l(s) = s$ models the time evolution of the condensate wave function in superfluid film ^[13,21]. In the case of $l(s) = \sqrt{1+s}$, problem (1.2) models the self-channeling of a high-power ultrashort laser in matter ^[1,30]. For more physical background, we can refer to ^[2,5,20,29] and references therein.

Putting $z(t, x) = \exp(-iEt)u(x)$ in (1.2), where $E\in \mathbb{R}$ and $u$ is a real function, we obtain the following elliptic equation

When $l(s) = s$, Eq (1.3) is the superfluid film equation in plasma physics

Note that Eq (1.4) is a special case of (1.1) if we choose $g^{2}(u) = 1+ 2u^{2}$. Equation (1.4) has been extensively studied, to see ^{[3,23,27,33,34]}.

When $l(s) = \sqrt{1+s}$, (1.3) derives the following equation

which is used as a model of the self-channeling of a high-power ultrashort laser in matter. We obverse that Eq (1.5) is also a particular case of (1.1) if we take $g^{2}(u) = 1+ \frac{u^{2}}{2(1+u^{2})}$.

For (1.5), there were many papers studying the existence of solutions, to see ^[9,12,32,39] and references therein. In ^[19], when the nonlinear term is autonomous, the authors obtained a positive ground state solution by a perturbation approach. In ^[4], the authors proved that (1.5) has at least a positive solution by using a change of variable, monotonicity trick developed by Jeanjean and a priori estimate. In ^[16], the authors got the existence of infinitely many nontrivial solutions by using a revised Clark theorem and a priori estimate of the solution.

Furthermore, if we take $g^{2}(u) = 1+ \frac{[(l(u^{2}))']^{2}}{2}$, then (1.3) turns into (1.1).

In recent years, many scholars have studied (1.1), for example ^{[6,7,10,11,22,24,28,31,35]}. Particularly, in ^[31], the authors obtained a positive solution with the autonomous nonlinear term. In ^[6], the authors acquired the existence of ground state solutions with periodic potential. In ^[10,11], the authors established the existence of positive solutions with the critical exponents, where critical exponents are $2^{*}$ and $\alpha2^{*}$, respectively. With regard to generalized quasilinear Schrödinger-Maxwell systems and generalized quasilinear Schrödinger equation of Kirchhoff type, we can refer to ^[8,26,40] and references therein. To the best of our knowledge, there is no work concerning with the unified asymptotic process on $V$ and $h$ at infinity for general quasilinear Schrödinger equations.

Motivated by above papers, we establish the existence of a positive ground state solution for Eq (1.1) and the corresponding periodic equation. We point out that Eq (1.1) is more general than (1.4) and (1.5). There are several difficulties in dealing with Eq (1.1). The first one is that the energy functional associated with Eq (1.1) is not well defined in the whole space $H^{1}(\mathbb{R}^{N})$ due to the presence of second order nonhomogeneous term. Another difficulty is the lack of compactness owing to the unboundedness of the domain. The final difficulty is that the functional loses the translation invariance because of the asymptotically periodic potential. To overcome the above difficulties, we firstly introduce a change of variable and we reformulate quasilinear elliptic Eq (1.1) into semilinear elliptic equation, whose associated functional is well defined in our working space. Secondly, we employ the energy comparison method to overcome the loss of translation invariance. Finally, to find a positive ground state solution, we use the Concentration-Compactness Principle, Nehari manifold method and the strong maximun principle. Our results are a complement and generalization of some results obtained by ^[18,34,37].

Now, let us recall some basic facts. Set

with the norm

It is well known that in the study of the elliptic equations, the potential function $V$ plays an important role in dealing with compactness problem. Let us introduce the following working space

endowed with the norm

Then, the subsequent condition $(V)$ implies that the norm $||\cdot ||$ is equivalent to the norm $||\cdot ||_{H}$ (see ^[18]).

We define the energy functional associated with (1.1) by

where $H(x, u) = \int_{0}^{u}h(x, \tau)d\tau$. However, $\Psi$ is not well defined in the usual Sobolev space $H^{1}(\mathbb{R}^{N})$ because of the term $\int_{\mathbb{R}^{N}}g^{2}(u)|\nabla u|^{2}dx$. To overcome this difficulty, we make use of the change of variable introduced by ^[35],

After the change of variable, we can obtain the following functional

where $G^{-1}(v)$ is the inverse function of $G(u)$. Since $g$ is a nondecreasing positive function, we can get $|G^{-1}(v)|\leq \frac{1}{g(0)}|v|$. From this and our hypotheses, it is easy to verify that $\Phi$ is well defined on $X$ and $\Phi\in \mathcal{C}^{1}$.

In order to obtain a critical point of (1.1), it suffices to study the following semilinear equations

If $v$ is said to be a weak solution for Eq (1.8), then it should satisfy

for any $\varphi \in C_{0}^{\infty}(\mathbb{R}^{N})$. We note that if $v$ is a critical point of the functional $\Phi$, then $u = G^{-1}(v)$ is a critical point of the functional $\Psi$, i.e. $u = G^{-1}(v)$ is a solution of Eq (1.1).

Next, we give the following condition on potential $V$:

On nonlinearity term $h$, since we look for a positive ground state solution, we assume that $h(x, t) = 0$ for all $(x, t)\in \mathbb{R}^{N}\times \mathbb{R}^{-}$. Moreover, the function $h$ satisfies the following assumptions:

We employ $(H_{1})$, $(H_{2})$ and $(H_{4})$ conditions to ensure that the energy functional $\Phi$ has the mountain pass geometry structure. However, under these hypotheses, this functional does not satisfy the Cerami compactness condition, since the domain is all $\mathbb{R}^{N}$. We observe that the condition $(H_{3})$ and (3) of $(H_{4})$ are used in the proof of the boundedness of the Cerami sequence of the functional associated with (1.1). We also observe that the asymptotic process of $(V)$ and (1) of $(H_{4})$ is uniform at infinity due to ^[18], and then was used in ^[17,25,38].

Now, we present the first result of this paper.

Theorem 1.1. (Asymptotically periodic case). Assume that $(V)$ and $(H_{1})-(H_{4})$ hold. Then, Eq (1.1) has a positive ground state solution.

Remark 1.2. Compared with the known results in ^{[6,18,24,34,37]}, our results are new and different due to the following some facts:

(1) While ^[18,34,37], our model is more general and they are our special case. In our results, there is no need to assume $h(x, t)\in C^{1}(\mathbb{R}^{N}, \mathbb{R})$. The constrained manifold also need not be of class $C^{1}$.

(2) Contrasting with ^[34], we give a new asymptotic process of potential and nonlinearity term at infinity. To some extent, we extend and complement their results.

(3) In ^[6,24], they verified the mountain pass geometry structure under the following condition

Our condition $(H_{2})$ is somewhat weaker than $(H^{\prime}_{2})$.

(4) We choose condition $(H_{3})$ and (3) of $(H_{4})$ to be weaker than Ambrosetti-Rabinowitz type condition.

In the special case: $V = V_{p}, h = h_{p}$, Theorem 1.1 clearly gives us a solution for the periodic equation. Indeed, considering the following equation

under the hypothesis:

$(V_{1})$ The function $V_{p}$ is 1-periodic in $x_{i}, 1\leq i\leq N$ and there exists a constant $V_{0} > 0$ such that

Corollary 1.3. (Periodic case). Assume that $(V_{1})$ holds, $h_{p}$ satisfies $(H_{1}), (H_{2})$, (2) of $(H_{4})$ and (3) of $(H_{4})$. Then, Eq (1.10) has a positive ground state solution.

Remark 1.4. To the best of our knowledge, even for the periodic case, our method is new. In ^[26], under condition $(V_{1})$, $h_{p}$ satisfies $(H_{1}), (H_{2})$ and other conditions. They could get at least one nontrivial ground state solution by the Mountain Pass Theorem. While, we obtain the existence of a positive ground state solution with Eq (1.10) by Nehari manifold method.

The rest of this article is organized as follows: In Section 2, we present some preliminary lemmas. In Section 3, we give the proof for our result.

Notations

$\bullet$ $L^{s}(\mathbb{R}^{N})$ is the usual Lebesgue space endowed with the norm

$\bullet$ $||u||_{\infty} = \text{ess}\sup_{x\in \mathbb{R}^{N}}|u(x)|$ denotes the usual norm in $L^{\infty}(\mathbb{R}^{N})$.

$\bullet$ $(X^{*}, || \cdot ||_{*})$ is the dual space of $(X, || \cdot ||)$.

$\bullet$ $B_{R}(x_{0}): = \{x\in \mathbb{R}^{N}:|x-x_{0}| < R\}$.

$\bullet$ $C$ represent different positive constants.

2.   Some preliminary lemmas

Before we prove the existence of a positive ground state solution for Eq (1.1), we present some useful lemmas.

Lemma 2.1. ^[24,40] The functions $g, G, G^{-1}$ satisfy the following properties:

Lemma 2.2. Assume that $(H_{1})-(H_{4})$ hold, then for all $(x, s)\in \mathbb{R}^{N}\times \mathbb{R}$, we have

For any $\delta > 0$, there exist $r_{\delta} > 0, C_{\delta} > 0$ and $\alpha\in(2, 2^{*})$ such that

Proof. We give the proof the first inequality of (2.1). From $(H_{3})$, we have

that is

then, taking $t = G^{-1}(s)$, (2.1) is proved. According to $(H_{1})-(H_{4})$ conditions, it is easy to deduce the inequalities of (2.2)–(2.7).

Lemma 2.3. ^[18] Assume that condition $(V)$ holds. Then, there are two positive constants $d_{1}$ and $d_{2}$ such that $d_{1}||u||_{H}\leq ||u||\leq d_{2}||u||_{H}$ for all $u\in X$.

Define the Nehari manifold

and set $c: = \inf_{v\in \mathcal{N}}\Phi(v)$.

Lemma 2.4. Assume that $(V)$ and $(H_{1})-(H_{4})$ are satisfied, then for any $v\in X, v\neq 0$, there exists a unique $t_{v} > 0$ such that $t_{v}v\in \mathcal{N}$. Moreover, the maximun of $\Phi(tv)$ for $t\geq 0$ is arrived at $t_{v}$.

Proof. Let $v\in X\setminus\{0\}$ and define a function $f(t): = \Phi(tv)$ on $[0, \infty)$. It follows from Lemma 2.1-(10), $(H_{1})$ and the Lebesgue dominated convergence theorem that

when $t\rightarrow 0$, which shows that $f(t) > 0$ for $t > 0$ small enough. Set $\Omega = \{x\in \mathbb{R}^{N}: v(x) > 0\}$. Using the fact that $(1), (3)$ of $(H_{4})$ and the Fatou lemma, We have

Hence, from Lemma 2.1-(6) and (2.8), we can get

which deduces $f(t)\rightarrow -\infty$ as $t\rightarrow +\infty$. So there exists $t_{v} > 0$ such that $f(t_{v}) = \max_{t > 0}f(t)$ and $f'(t_{v}) = 0$, i.e., $\Phi(t_{v}v) = \max_{t > 0}\Phi(tv)$ and $t_{v}v\in \mathcal{N}$.

The condition $f'(t) = 0$ is equivalent to

By $(H_{3})$ and Lemma 2.1-(9), the right side of (2.9) is increasing for $t > 0$. So, there is a unique $t_{v} > 0$ such that $f'(t_{v}) = 0$.

From Lemma 2.4, we can obtain the following lemma easily.

Lemma 2.5. Assume that $(V)$ and $(H_{1})-(H_{4})$ are satisfied. Then, the functional $\Phi$ satisfies the following mountain pass geometry structure:

(i) there exist positive constants $\rho$ and $b$ such that $\Phi(v)\geq b$ for $||v|| = \rho$;

(ii) there exists a function $v_{0}\in X$ such that $||v_{0}|| > \rho$ and $\Phi(v_{0}) < 0$.

Lemma 2.6. Assume that $(V)$ and $(H_{1})-(H_{4})$ hold, then there exists a bounded Cerami sequence $\{v_{n}\}\subset X$ for $\Phi$.

Proof. From Lemma 2.5, we know that $\Phi$ satisfies the mountain pass geometry structure. Thus, the mountain pass theorem deduces that there exists a Cerami sequence $\{v_{n}\}\subset X$ such that

where $c: = \inf_{v\in \mathcal{N}}\Phi(v)$.

We claim that a Cerami sequence $\{v_{n}\}\subset X$ satisfies

for some $M > 0$.

From (2.10), we have

and

From (2.1), (2.12), (2.13) and Lemma 2.1-(8), we obtain

that is

Thus, (2.11) holds.

Next, we claim that $\int_{\mathbb{R}^{N}}v_{n}^{2}dx$ is bounded. In fact, by Lemma 2.1-(2) and (2.14), we have

Moreover, by the Sobolev inequality and (2.14), we deduce that

Obviously, there is a constant $C > 0$ such that

The claim is proved, then combining (2.14) with (2.17), $\{v_{n}\}\subset X$ is bounded.

3.   Proof of Theorem 1.1

In this section, before proving Theorem 1.1, we give three important lemmas to help us complete the proof of Theorem 1.1.

Lemma 3.1. ^[18] Assume that $(V)$ and $(H_{1})-(H_{4})$ are satisfied. If $v\in \mathcal{N}$ and $\Phi(v) = c$, then $v$ is a solution of problem (1.1).

Since $V(x)$ is asymptotically periodic. In this case, the functional $\Phi$ loses the translation invariance. The following two lemmas give careful estimates among $V$ and $V_{p}$, $h$ and $h_{p}$, $H$ and $H_{p}$, which is inspired by ^[18,37].

Lemma 3.2. Assume that $(V), (H_{1}), (H_{2})$ and (1) of $(H_{4})$ hold. Suppose that $\{v_{n}\}$ is bounded in $X$ and $v_{n}\rightarrow 0$ in $L_{loc}^{\gamma}(\mathbb{R}^{N})$, for any $\gamma\in [2, 2^{*})$, then up to a subsequence, one has

Proof. (i) The proof of (3.1). Firstly, we assert that for any given $k(x)\in \mathcal{F}_{0}$ and $\varepsilon > 0$, there exists $R_{\varepsilon} > 0$ such that

where $C$ is a constant and independent on $\varepsilon$. (3.3) has already been proved in ^[18]. For the convenience of readers, we give a brief proof as following. According to the definition of $\mathcal{F}_{0}$, for any $\varepsilon > 0$, there exists $R_{\varepsilon} > 0$ such that

Covering $\mathbb{R}^{N}$ by balls $B_{1}(y_{i}), i\in \mathbb{N}$, where each point of $\mathbb{R}^{N}$ is contained in at most $N+1$ balls. Without loss of generality, we assume that $|y_{i}| < R_{\varepsilon}$, $i = 1, 2, \cdot \cdot \cdot, n_{\varepsilon}$ and $|y_{i}|\geq R_{\varepsilon}$, $i = n_{\varepsilon}+1, n_{\varepsilon}+2, \cdot \cdot \cdot, +\infty$. By the H$\ddot{o}$lder and Sobolev inequalities, we have

where $\Omega_{i} = \{ x\in B_{1}(y_{i}): |k(x)|\geq \varepsilon\}$.

Set $k(x): = V(x) - V_{p}(x)\in \mathcal{F}_{0}$, by Lemma 2.1-(6), we have

Let $\varepsilon\rightarrow 0$, this completes the proof of (3.1).

(ii) The proof of (3.2). Let $\overline{h}(x, s): = h(x, s)- h_{p}(x, s) \in \mathcal{F}$. By the definition of $\mathcal{F}$, for any $\varepsilon > 0$, there exists $R_{\varepsilon} > 0$ such that

covering $\mathbb{R}^{N}$ by balls $B_{1}(y_{i}), i\in \mathbb{N}$, where each point of $\mathbb{R}^{N}$ is contained in at most $N+1$ balls. Without loss of generality, we assume that $|y_{i}| < R_{\varepsilon}$, $i = 1, 2, \cdot \cdot \cdot, n_{\varepsilon}$ and $|y_{i}|\geq R_{\varepsilon}$, $i = n_{\varepsilon}+1, n_{\varepsilon}+2, \cdot \cdot \cdot, +\infty$. Using the mean value theorem and Lemma 2.1-(4), there exists $t_{n}\in (0, 1)$ such that

Set

Then, we obtain

From (2.4), we have

Set

By (2.2), we obtain

From (2.4), H$\ddot{o}$lder and Sobolev inequalities, we get

Due to (2.5), one has

Now, let $\varepsilon\rightarrow 0$ and then $\delta\rightarrow 0$, we have $\Phi_{i}\rightarrow 0, i = 1, 2, 3, 4$, the proof of (3.2) is completed.

Lemma 3.3. Assume that $(V), (H_{1}), (H_{2})$ and (1) of $(H_{4})$ are satisfied. If $\{v_{n}\}$ is bounded in $X$ and $|y_{n}|\rightarrow +\infty$. Then, for any $\varphi\in C_{0}^{\infty}(\mathbb{R}^{N})$, one has

Proof. (i) The proof of (3.4). By Lemma 2.1-(3) and (6), we have

Due to $\varphi\in C_{0}^{\infty}(\mathbb{R}^{N})$, one has

Set $k(x): = V(x) - V_{p}(x)\in \mathcal{F}$, in view of (3.3), (3.6), (3.7) and the H$\ddot{o}$lder inequality, we have

Let $\varepsilon\rightarrow 0$, (3.4) is proved.

(ii) The proof of (3.5). Let $\overline{h}(x, s): = h(x, s)- h_{p}(x, s)\in \mathcal{F}$. As the proof of Lemma 3.2, covering $\mathbb{R}^{N}$ by balls $B_{1}(y_{i})$. Set

We obtain

From (2.4) and (3.7), we have

On account of Lemma 2.1-(3), we obtain

From (2.4), the H$\ddot{o}$lder, Yong and Sobolev inequalities, one has

By using (2.5) and H$\ddot{o}$lder inequality, we have

As before, let $\varepsilon\rightarrow 0$ and then $\delta\rightarrow 0$, we get $\Phi_{i}\rightarrow 0, i = 5, 6, 7, 8$, we complete the proof of (3.5).

Now we are ready to prove Theorem 1.1.

Proof of Theorem 1.1. We can follow the energy comparison method in ^[15] to give the proof of Theorem 1.1. The functional $\Phi$ is rewritten as

We also need to consider the corresponding periodic functional $\Phi_{p}: X\rightarrow \mathbb{R}$,

By Lemma 2.5, there exists a Cerami sequence $\{v_{n}\}\subset X$, such that

According to Lemma 2.6, the Cerami sequence $\{v_{n}\}$ is bounded in $X$. Thus, there exists $v\in X$ such that $v_{n}\rightharpoonup v$ in $X$ and then $\Phi'(v) = 0$, that is $v$ is a weak solution of problem (1.8). The proof of this result is standard, so we omit here.

(i) The case $v = 0$. Then $v_{n}\rightharpoonup 0$ in $X$, $v_{n}\rightarrow 0$ in $L_{loc}^{\gamma}(\mathbb{R}^{N}), 2\leq \gamma < 2^{*}$ and $v_{n}(x)\rightarrow 0$ a.e. on $\mathbb{R}^{N}$. From Lemma 3.2 and Lemma 3.3, we can deduce that

and taking $\varphi \in X$ with $||\varphi|| = 1$, we obtain that

From (3.8) and (3.9), we can get that $\{v_{n}\}$ is also a Cerami sequence for $\Phi_{p}$. Namely,

Define

if $\beta = 0$, the Lions lemma ^[14], we have $v_{n}\rightarrow 0$ in $L^{\alpha}(\mathbb{R}^{N})$ for all $\alpha\in (2, 2^{*})$.

From (2.5) and (2.7), we have

It is implied by Lemma 2.1-(8) that

Then, combining with (3.11) and (3.12), we obtain

This contradiction shows $\beta > 0$. Up to a subsequence, there exist a sequence $\{y_{n}\}\subset \mathbb{Z}^{N}$ and $r > 0$ such that $|y_{n}|\rightarrow \infty$,

where $w_{n}(x): = v_{n}(x + y_{n})$. Since $||w_{n}|| = ||v_{n}||$, we see that $\{w_{n}\}$ is bounded. Going if necessary to a subsequence, we have $w_{n}\rightharpoonup w$ in $X$, $w_{n}\rightarrow w$ in $L_{loc}^{\gamma}(\mathbb{R}^{N}), 2\leq \gamma < 2^{*}$ and $w_{n}(x)\rightarrow w(x)$ a.e. on $\mathbb{R}^{N}$. Thus, (3.13) implies that $w\neq 0$. For any $\varphi\in C^{\infty}_{0}(\mathbb{R}^{N})$, we have

Hence $\Phi'_{p}(w) = 0$. Next, our task is to verify that $\Phi_{p}(w)\leq c$. Since

From (2.1), (3.10), (3.14), Lemma 2.1-(8), (1) of $(H_{4})$ and the Fatou lemma, we obtain

By (3.15), (1) of $(H_{4})$ and the definition of $c$

This implies that $\Phi(t_{w}w) = c$. Let $w_{0} = t_{w}w$, then $w_{0}\in \mathcal{N}$ and $\Phi(w_{0}) = c$. In view of Lemma 3.1, $\Phi'(w_{0}) = 0$. This shows that $w_{0}$ is a ground state solution of problem (1.8).

(ii) The case $v\neq 0$. Since $v$ is a weak solution of problem (1.8), $\Phi(v)\geq c$. It follows from (2.1), Lemma 2.1-(8) and Fatou's lemma that

This shows that $\Phi(v) = c$ and $\Phi'(v) = 0$, which implies that $v$ is a ground state solution of problem (1.8).

From (i) and (ii), we can obtain that Eq (1.8) has a ground state solution $v\in X$. By using the strong maximum principle ^[36], we can get that $v$ is a positive ground state solution of (1.8). Namely, Eq (1.1) possesses a positive ground state solution $u = G^{-1}(v)$ and the proof is completed.

4.   Conclusions

In this paper, we have established the existence of a positive ground state solution for Eqs (1.1) and (1.10) by the variational method. In comparison with previous works, this paper has several new features. Firstly, we consider a more general model than the literature ^[18,34,37]. Secondly, compared to the literature ^[34], we give a new asymptotic process of potential and nonlinearity term. Finally, we choose the more general nonlinear term than Ambrosetti-Rabinowitz condition. Therefore, to some extent, we have improved and extended the results of the existing literature.

Acknowledgments

The author expresses his appreciation to the reviewers and the handling editor whose careful reading of the manuscript and valuable comments greatly improved the original manuscript.

Conflict of interest

The author declares that he has no competing interests concerning the publication of this article.