Positive solutions for a class of supercritical quasilinear Schrödinger equations

1.   Introduction

It is well-known that the generalized quasilinear Schrödinger equations of the form

serves as models for several physical phenomena corresponding to various forms of the given potential $W(x)$ and the given nonlinearity $\rho,$ where $z:\mathbb{R}^{N}\times \mathbb{R}\rightarrow \mathbb{C}\; and\; W, \; l, \; \rho$ are real functions, $\kappa,$ $\lambda$ are real constants. For example, the case $\rho(s) = s$ was studied in ^[1] for the superfluid film equations in plasma physics. The Eq (1.1) is also related to the condensed matter theory, see ^[2].

In this paper, we consider the case ${ }\rho(s) = (1+s)^{\frac{1}{2}}$ which could be used to describe the self-channeling of a high-power ultrashort laser in matter, cf. e.g., ^[3,4]. Let $z(t, x) = \exp(-iEt)u(x)$ in (1.1), where $E\in \mathbb{R}$ and $u$ is a real function. Then we know that $z$ satisfies (1.1) if and only if the function $u$ solves the following equation

where $V(x) = W(x)-E$ and $f : \mathbb{R}\rightarrow \mathbb{R}$ given by $f(t): = l(|t|^{2})t$ is a new nonlinear term. We shall give the precise hypotheses on $V$ and $f$ latter.

In recent years, the Eq (1.2) with $\kappa < 0$ has already been investigated extensively, for example, ^[5,6,7]. But the results for $\kappa > 0$ is rarely studied, see ^[8,9,10,11]. In ^[8], when $\kappa = 2, \; N = 2,$ Colin studied the existence of ground state solutions for the Eq (1.2) with $V(x) = 2w, \; f(s) = s-\dfrac{s}{\sqrt{1+s^{2}}}$, where $w$ is a fixed positive parameter. In ^[9], with well defined $V(x)$ and improved (AR) condition, Shen and Wang got the better results which obtained the existence of solutions for (1.2) when $\kappa < 2.$ In that paper, a change of variable was used to reduce the quasilinear problem to a semilinear one and the mountain pass theorem, the concentration compactness theorem were used to get the main existence results.

To sum up, in the past, researches of the (1.2) have mostly focused on $\kappa < 2.$ Now, different from the above mentioned results, a natural question for us to pose is how about the existence of solutions for the case $\kappa\geq2$ and $N\geq 3.$ We would like to mention that the work ^[12] which obtained the existence of positive solutions for the supercritical quasilinear Schrödinger equations. In ^[12], Huang and Jia studied the following quasilinear Schrödinger equation

where $N\geq3, \; f(t):\mathbb{R}\rightarrow \mathbb{R}$ is continuous and only superlinear in a neighborhood of $t = 0,$ by using the truncation methods and modifying the functional. Meanwhile, in the literatures ^[12,13,14], the authors have studied the asymptotically periodic quasilinear Schrödinger equations. So motivated by above discussions, we study the Eq (1.2) for the periodic and asymptotically periodic potentials when $\kappa\geq2,$ $\lambda > 0$ and $N\geq 3.$

Hereafter, we give the conditions of $V(x)$ and $f(t).$

Hypothesis 1.1. Suppose that the potential $V(x)$ satisfies assumptions $(v_{0})\; and \; (v_{1})$

$(v_{0})$: $V(x)\in C(\mathbb{R}^N, \mathbb{R}),$ $V(x)\geq V_{0} > 0$ for all $x\in \mathbb{R}^N;$

$(v_{1})$: $V(x) = V(x+y), \; \forall x\in\mathbb{R}^N, \; y\in\mathbb{Z}^N.$

Hypothesis 1.2. Suppose that the potential $V(x)$ satisfies the following assumption

$(v_{2})$: $V(x) = V_{1}(x)-m(x)\geq m_{0} > 0, \; \forall x\in\mathbb{R}^N,$ where $V_{1}(x)$ satisfies Hypothesis 1.1 and $m(x)\in\mathcal{F}$ with $m(x)\geq0,$

Here, we assume the inequality $m(x) > 0$ is strict on asubset of positive measures in $\mathbb{R}^{N}.$

We call the $V(x)$ is periodic if it satisfies Hypothesis 1.1 and is the asymptotically periodic at infinity if it satisfies the Hypothesis 1.2. In particular, if $m(x) = 0,$ the asymptotically periodic problem is reduced to its corresponding periodic problem. Because the periodic potentials and the asymptotically periodic potentials are both bounded, we set $V_{M} = \text{max}\{V(x)\}.$

Hypothesis 1.3. For the nonlinearity $f$, we suppose that it is continuous and satisfies the followingconditions which give its behavior only in a neighborhood of the origin:

$(f_{1})$:$f(t) = 0, \; \mathit{\text{for}} \; t\leq 0$ and there exists $\alpha\in(2, 2^{*})$ such that

$(f_{2})$:there exists $\beta\in(2, 2^{*})$ with $\beta > \alpha$ such that

$(f_{3})$: for $t > 0$ small, there exists $\theta\in(2, 2^{*})$ such that $0 < \theta F(t)\leq t f(t),$

where $F(t) = { }\int_{0}^{t}f(s)ds.$

Remark 1.1. An example of the nonlinearity satisfying Hypothesis 1.3 can be taken as

with $2 < \alpha < 2^{*} < q,$ $2^{*} = \dfrac{2N}{N-2}$ and $C_{0}, \; C_{1}$ are positive constants.

Obviously, (1.2) is the Euler-Lagrange equation associated with the natural energy functional

which is not well defined in $H^{1}(\mathbb{R}^{N}).$ From the variational point of view, the first difficulty is to guarantee the positiveness of the principal part, that is, $\left(1-\dfrac{\kappa u^{2}}{2(1+u^{2})}\right) > 0.$ And then, the change of variable applied in ^[9] loses its meaning when $\kappa\geq2$. Besides these, since there are no conditions imposed on $f$ at infinity, the term ${ }\int_{\mathbb{R}^{N}} {F(u)}dx$ may not be well-defined in $H^{1}(\mathbb{R}^{N})$. Due to these facts, we can't employ the usual variational methods directly. To overcome these difficulties, we use some variational methods to solve (1.2).

We conclude the main features of this paper as follows.

● We study the Eq (1.2) for the periodic and asymptotically periodic potentials when $\kappa\geq2,$ $\lambda > 0$ and $N\geq 3.$

● We will first establish the existence of positive solutions for a modified quasilinear Schrödinger equation which will be given more precisely in (2.3).

● Using Moser iteration we get an $L^{\infty}$-estimate for the weak solutions, which depends on the parameter $\lambda$. And for $\lambda$ large enough, the solutions obtained of the modified problem are solutions of the original Eq (1.2).

Now, we turn to the statement of our results.

Theorem 1.1. Under Hypothesis 1.1 and Hypothesis 1.3, when$\kappa\geq2$ the problem (1.2) has at least one positive solution $u\in H^{1}(\mathbb{R}^{N})$ for $\lambda$ sufficiently large.

Theorem 1.2. Assume that $\kappa\geq2$, under Hypothesis 1.2 and Hypothesis 1.3, theproblem (1.2) has at least one positive solution $u\in H^{1}(\mathbb{R}^{N})$ for $\lambda$ sufficiently large.

The paper is organized as follows. In Section 2, we give a modified problem and the variational setting of the problem. In Section 3, we complete the proof of Theorem 1.1. And the proof of Theorem 1.2 is given in Section 4.

$\bf{Notation}$

● $B_{\varrho}(x_{0})$ denotes a ball centered at $x_{0}$ with radius $\varrho > 0;$

● $o_{n}(1)$ denotes $o_{n}(1)\rightarrow0$ as $n\rightarrow \infty;$

● the strong (respectively weak) convergence is denoted by $\rightarrow$ (respectively$\rightharpoonup$);

● $C, \; C_{0}, ...,$ denote suitable positive constants;

● The notation $|u|_{p}$ denotes the usual $L^{p}(\mathbb{R}^{N})$ norm of the function $u;$

● The working space is $H^{1}(\mathbb{R}^{N})$ endowed with the norm $\|u\| = \left({ }\int_{\mathbb{R}^{N}}(u^{2}+|\nabla u|^{2})dx\right)^{\frac{1}{2}}.$

2.   Variational setting and preliminaries results

First, we give some discussions on the nonlinearity $f(t).$ Note that from $(f_{1})$ there exist two positive constants $\delta\in(0, \dfrac{1}{2}),$ $C_{2}$ such that

For the fixed $\delta > 0$ in the above, we consider a cut-off function $a(t)\in C^{1}(\mathbb{R}, \mathbb{R})$ satisfying

$|a'(t)|\leq \dfrac{2}{\delta}\; \text{and}\; 0\leq a(t)\leq 1\; \text{for}\; t\in \mathbb{R}.$ Define

where

By Hypothesis 1.3 and the definition of $a(t)$, it is easy to see that $\tilde{f}(t)$ has the following properties (see ^[15]).

Lemma 2.1. Let $\tilde{f}(t)$ and $\tilde{F}(t)$ be defined in (2.2). Assume that Hypothesis 1.3 hold, then we have

$(1)$ $\tilde{f}(t)\in C(\mathbb{R}, \mathbb{R}),$ $\tilde{f}(t) = 0$ for all $t\leq0$ and $\tilde{f}(t)\rightarrow0$ as $t\rightarrow 0^{+}.$

$(2)$ ${ } \lim\limits_{t\rightarrow +\infty} \dfrac{\tilde{f}(t)}{t} = +\infty;$

$(3)$ there exists $C > 0$ such that $\tilde{f}(t)\leq Ct^{\alpha-1}$, for all $t\geq0;$

$(4)$ $0 < \theta^{*}\tilde{F}(t)\leq t\tilde{f}(t)$ for all $t > 0,$ where$\theta^{*} = \min\{\alpha, \theta\}.$

Inspired by ^[12], we first consider the following modified quasilinear Schrödinger equation

instead of the Eq (1.2). Here $g(t):[0, +\infty)\rightarrow \mathbb{R}$ in (2.3) is given by

for $\kappa\geq2$. Setting $g(t) = g(-t)$ for all $t\leq 0$, we know that $g\in C^{1}(\mathbb{R}, (\dfrac{1}{\sqrt{2}\kappa}, 1])$ and $g$ is decreasing in $[0, \infty).$

Now, defining a function ${ } G(t) = \int_{0}^{t}g(s)ds,$ we get that $G(t)$ is an odd function, the inverse function $G^{-1}(t)$ exists and the following properties about $G^{-1}(t)$ hold.

Lemma 2.2. For $\kappa\geq2,$ the function $G^{-1}(t)$ satisfies the followingproperties:

$(1)$ ${ } \lim\limits_{t\rightarrow 0^{+}} \dfrac{G^{-1}(t)}{t} = 1;$

$(2)$ ${ } \lim\limits_{t\rightarrow +\infty} \dfrac{G^{-1}(t)}{t} = \sqrt{2}\kappa;$

$(3)$ $t\leq G^{-1}(t)\leq \sqrt{2}\kappa t,$ for all $t\geq0;$

$(4)$ $-1+\dfrac{1}{\kappa}\leq \dfrac{t}{g(t)}g'(t)\leq0,$ for all $t\geq0.$

Proof. By the definition of $g(t)$, we get

and

which show (1) and (2).

Since $g$ is decreasing in $[0, \infty),$ the inequality $\dfrac{1}{\sqrt{2}\kappa}t\leq g(t)t \leq G(t)\leq t$ holds for all $t\geq0.$ Consequently, by replacing $t$ with $G^{-1}(t)$ we gain the conclusion (3).

By a direct calculation, one obtains

Since $\dfrac{t}{g(t)}g'(t)$ reaches the minimum value $-1+\dfrac{1}{\kappa}$ at $t = \sqrt{\dfrac{1}{\kappa-1}}$ and $\dfrac{t}{g(t)}g'(t)\leq0,$ the conclusion (4) holds.

For $\kappa\geq2,$ we observe that the Eq (2.3) is the Euler-Lagrange equation associated with the natural energy functional

In what follows, taking the change of variable

we know that the functional $I_{\lambda}(u)$ can be reformulated in the following way

From Lemma 2.1 and Lemma 2.2, we obtain that the functional $J_{\lambda}(v)$ is well-defined in $H^{1}(\mathbb{R}^{N})$ and $J_{\lambda}(v)\in C^{1}(H^{1}(\mathbb{R}^{N}), \mathbb{R})$. Additionally, for all $\varphi\in H^{1}(\mathbb{R}^{N})$ we have

Lemma 2.3. If $v\in H^{1}(\mathbb{R}^{N})$ is a critical point of $J_{\lambda}(v)$, then$u = G^{-1}(v)\in H^{1}(\mathbb{R}^{N})$ and meanwhile $u$ is a critical point for$I_{\lambda}(u).$

Proof. Suppose that $v$ is a critical point of $J_{\lambda}.$ According to Lemma 2.1 and Lemma 2.2, we have $u = G^{-1}(v)\in H^{1}(\mathbb{R}^{N})$ and

Choosing $\varphi = g(u)\psi$ with $\psi\in C_{0}^{\infty}(\mathbb{R}^{N})$, we obtain

which can be rearranged as

Thus, we complete the proof.

3.   Proof of Theorem 1.1

In this section, we will verify the mountain pass geometry of $J_{\lambda}$ and the boundedness of its $\text{(PS)}$ sequences. Furthermore, we will give the proof of Theorem 1.1.

Lemma 3.1. If Hypothesis 1.1 and Hypothesis 1.3 hold, then for $\kappa\geq2$ there exist $\rho, \; \sigma > 0$ and $e\in H^{1}(\mathbb{R}^{N})\setminus\{0\}$ such that

$(a) \; J_{\lambda}(v) > \sigma,$ for $\|v\| = \rho,$

$(b) \; J_{\lambda}(e) < 0,$ for $\|e\| > \rho.$

Proof. Combining Lemma 2.1, Lemma 2.2 and the Sobolev embedding theorem, we find

Thus, due to the fact $2 < \alpha < 2^{*}$, we conclude that there exists $\sigma > 0$ such that $(a)$ holds for $\rho = \|v\|$ sufficiently small.

In addition, Lemma 2.1 implies $\tilde{F}(t)\geq Ct^{\theta^{*}}$ for all $t > \varepsilon_{0} > 0$. For a fixed $\omega\in C_{0}^{\infty}(\mathbb{R}^{N}),$ we suppose that $\text{supp} \; \omega = \Omega$ and $\omega\geq1$ in $\Omega^{'}\subset\Omega$ with $|\Omega^{'}| > 0.$ Then it turns out that

Since $\theta^{*} > 2,$ it follows that $J_{\lambda}(t\omega)\longrightarrow -\infty$ as $t\longrightarrow \infty.$ Then we will prove the result $(b)$ if we take $e = t\omega$ with $t$ large enough.

In consequence of Lemma 3.1, we can apply the mountain pass theorem without the $\text{(PS)}$ condition (see ^[16]) to get a $\text{(PS)}_{d_{\lambda}}$ sequence $\{v_{n}\}$ of $J_{\lambda},$ where $d_{\lambda}$ is the mountain pass level associated with $J_{\lambda},$ i.e.,

Lemma 3.2. Under the assumptions of Hypothesis 1.1 and Hypothesis 1.3, the $\mathit{\text{(PS)}}$sequence $\{v_{n}\}$ of$J_{\lambda}$ is bounded.

Proof. Let $\{v_{n}\}\subset H^{1}(\mathbb{R}^{N})$ be a $\mathit{\text{(PS)}}$ sequence of the functional$J_{\lambda}.$ By means of (2.5) and (2.6) we know that

and for $\varphi_{n} = G^{-1}(v_{n})g(G^{-1}(v_{n}))\in H^{1}(\mathbb{R}^{N}), \; \langle J'_{\lambda}(v_{n}), \varphi_{n}\rangle = o_{n}(1)\|\varphi_{n}\|,$ that is

From Lemma 2.2 we find that

and

Hence, by (3.3) and (3.4), we get

Additionally, (3.3) and the fact $\langle J'_{\lambda}(v_{n}), \; G^{-1}(v_{n})g(G^{-1}(v_{n}))\rangle = o_{n}(1)\|v_{n}\|$ imply

Then, from (3.1), (3.2), (3.5) and Lemma 2.1 we derive

which indicates $\|v_{n}\| < \infty.$

Remark 3.1. Indeed, in Lemma 3.1 and Lemma 3.2, for the potential $V(x)$ we essentially just need it to be bounded. And there holds $m_{0}\leq V(x)\leq V_{M}$ both in the periodic case and asymptotically periodic case. So if we replace Hypothesis 1.1 with Hypothesis 1.2, the conclusions similar to Lemma 3.1 and Lemma 3.2 still hold, which are about the asymptotically periodic case.

Lemma 3.3. Assume that Hypothesis 1.1 and Hypothesis 1.3 hold. Then $J_{\lambda}$ has apositive critical point.

Proof. With the help of Lemma 3.1 and Lemma 3.2, we get that $J_{\lambda}$ possesses a bounded (PS) sequence $\{v_{n}\}\subset H^{1}(\mathbb{R}_{}^{N}).$ Then, there exists $v\in H^{1}(\mathbb{R}_{}^{N})$ such that

where$\; p\in[2, 2^{*}).$

We claim that $v$ is a critical point of $J_{\lambda},$ that is, $J'_{\lambda}(v) = 0.$ To prove this claim, we only need to show that $\langle J' _{\lambda}(v), \varphi\rangle = 0$ for all $\varphi\in C_{0}^{\infty}(\mathbb{R}_{}^{N})$ owing to the fact that $C_{0}^{\infty }(\mathbb{R}_{}^{N})$ is dense in $H^{1}(\mathbb{R}^{N}).$ Note that from (2.6), one has

We will argue that the right side of (3.7) converges to zero in the following as $n\rightarrow \infty.$ Considering for the $\text{(PS)}$ sequence $\{v_n\}$, we have

$\text{where}\; w_{p}\in L^{p}(K_{\varphi}).$ Hence,

From the condition $(v_{1}),$ we get

Then the Lebesgue Dominated Convergence theorem gives the result

Meanwhile, applying Lemma 2.1, we know that

Making use of the Lebesgue Dominated Convergence theorem again, we deduce that

Thus, (3.8), (3.9) and $v_{n}\rightharpoonup v$ yield $\langle (J'_{\lambda}(v_{n})-J'_{\lambda}(v)), \varphi\rangle\rightarrow 0$ immediately. This limit together with $J'_{\lambda}(v_{n})\rightarrow 0$ shows that $J'_{\lambda}(v) = 0.$ Therefore, $v$ is a critical point of $J_{\lambda}$.

If $v\neq 0$, we can get a nontrivial critical point of $J_{\lambda}$. For the case $v = 0,$ similar as in ^[12], since $\{v_{n}\}$ is bounded in $H^{1}(\mathbb{R}^{N}),$ we can use a standard argument due to Lions (^[16], Lemma 1.21) to prove that there exist a sequence $\{y_{n}\}\subset\mathbb{R}^{N}$ and $r, \; \sigma > 0$ such that $|y_{n}|\rightarrow \infty$ as $n\rightarrow \infty$ and

Without loss of generality, we can assume that $\{y_{n}\}\subset\mathbb{Z}^{N}$. Let us consider the translation $\bar{v}_{n}(x) = v_{n}(x+y_{n}), \; n\in\mathbb{N}.$ In this sense, $\|\bar{v}_{n}(x)\| = \|{v}_{n}(x)\|$ and $\{\bar{v}_{n}\}$ is still a bounded $\text{(PS)}$ sequence of $J_{\lambda}$ in view of the assumption of $(v_{1}).$ Thus, taking a subsequence if necessary, we have a weak limit $\bar{v}\in H^{1}(\mathbb{R}^{N})$ satisfying

By using (3.10) we get the fact

i.e., $\bar{v}\neq0.$ Moreover, by the argument used above, we deduce a further conclusion $J'_{\lambda}(\bar{v})\varphi = 0$ for each $\varphi\in H^{1}(\mathbb{R}^{N})$. Therefore, we have proved that the functional $J_{\lambda}$ has a nontrivial critical point.

Now, assume that $v$ is a nontrivial critical point of $J_{\lambda}.$ Considering $\langle J'_{\lambda}(v), v^{-}\rangle = 0,$ we obtain

where $v^{-}$ = max$\{-v, 0\}.$ By using $(v_{0})$ and the definition of $g(t)$ we get $v^{-} = 0$, i.e., $v\geq0,$ which implies that $v$ is positive through the strong maximum principle. Thus, $J_{\lambda}$ has a positive critical point.

Certainly, now we can't conclude that the origin Eq (1.2) has a positive solution. However, we note that the weak solution of (2.3) whose $L^{\infty}$-norm is not bigger than ${ } \text{min}\{\sqrt{\dfrac{1}{\kappa-1}}, \delta\}$ is also a weak solution of (1.2) for $\kappa\geq2$. So in the following we will show the $L^{\infty}$-estimates for the critical point $v$ of $J_{\lambda}.$

Lemma 3.4. If $(v_{0}), \; (f_{1}), \; (f_{3})$ hold and $v\in H^{1}(\mathbb{R}^{N})$ is a positive critical point of $J_{\lambda}$, then $v\in L^{\infty}(\mathbb{R}^{N}).$Moreover,

where $C > 0$ only depends on $\alpha, \; N.$

Proof. Let $v\in H^{1}(\mathbb{R}^{N})$ be a positive critical point of $J_{\lambda}.$ From (2.6) there holds

On the one hand, for $T > 0,$ we define

Then there has $0\leq v_{T}\leq v$. By taking $\varphi = v_{T}^{2(\gamma-1)}v$ with $\gamma > 1$ in (3.13), one obtains

Since the second and the third terms in the above equation are nonnegative, using Lemma 2.1 we can achieve

On the other hand, the Sobolev inequality implies

Therefore, using the above inequality, (3.14), the Hölder inequality and Sobolev embedding theorem we deduce

From the above inequality, setting $\zeta = \frac{22^{*}}{2^{*}-\alpha+2},$ we have

Then, by the Fatou's lemma, it follows that

Define $\gamma_{n+1}\zeta = 2^{*}\gamma_{n}$ with $n = 0, 1, 2, ...,$ and $\gamma_{0} = \dfrac{2^{*}+2-\alpha}{2}.$ As a consequence of (3.15), we derive the following result

Furthermore, by using the Moser iteration, we obtain

Hence, from the facts that ${ } \sum\limits_{i = 0}^{\infty}(\frac{\zeta}{2^{*}})^{i} = \frac{2^{*}+2-\alpha}{2^{*}-\alpha}$ and ${ } \sum\limits_{i = 0}^{\infty} i(\frac{\zeta}{2^{*}})^{i}$ is convergent, we finally get

Lemma 3.5. Suppose that $(v_{0}), \; (f_{1})$ and $(f_{3})$ hold. Let $v$ be a positivecritical point of $J_{\lambda}$ with $J_{\lambda}(v) = d_{\lambda}.$Then there exists $C > 0$ independent of $\lambda$ such that

Proof. By Lemma 2.2, the inequality (3.3), we get the following result

Thus, from the fact $\theta^{*} = \text{min}\{\alpha, \theta\} > 2,$ we get $\|v\|^{2}\leq Cd_{\lambda}.$

Proof of Theorem 1.1 By Lemma 3.3, there exists a positive critical point $v$ of $J_{\lambda}$ with $J_{\lambda}(v) = d_{\lambda}.$ And it follows from Lemma 2.1, Lemma 2.2 and $(f_{2})$ that

where $e$ is fixed in Lemma 3.1 and $G^{-1}(te) > \varepsilon_{0}$ in $\Omega\subset\mathbb{R}^{N}$. Then, by Lemma 3.4, Lemma 3.5 and (3.16) we have

Since $2 < \theta^{*}\leq\alpha < 2^{*},$ from Lemma 2.2 there exists $\lambda_{0} > 0$ such that for all $\lambda > \lambda_{0},$

where $\delta$ is fixed in (2.1). This means that for $\lambda > \lambda_{0}$ the original Eq (1.2) possesses a positive solution $u = G^{-1}(v)$.

4.   Proof of Theorem 1.2

Different from the preceding section, for the case of asymptotically periodic potential we find that the inequality (3.11) is not valid. In order to overcome this difficulty, in this section, we will achieve the Lemma 4.2 which is a key point to complete the proof of Theorem 1.2. And for convenience, in this section, we give a sign $\bar{J}_{\lambda}(v)$ for the functional of the asymptotically periodic case, while we use $J_{\lambda}(v)$ to represent the functional of the corresponding periodic case. Then, there has

Now, we first give the following two necessary lemmas.

Lemma 4.1. Assume that Hypothesis 1.2 and Hypothesis 1.3 hold. If $\{v_{n}\}$ is bounded and$v_{n}\rightharpoonup\; 0 \; \mathit{\text{in}} \; H^{1}(\mathbb{R}^{N}),$ then

Proof. Firstly, we claim that for any $\varepsilon > 0$ there exists $R_{\varepsilon} > 0$ such that

where $C_{3}, \; C_{4}$ are positive constants and independent on $\varepsilon$.

Clearly, by (1.3), for any $\varepsilon > 0,$ there exists $R_{\varepsilon} > 0$ such that

Now, covering $\mathbb{R}^{N}$ by balls $B_{1}(y_{i}), \; i\in \mathbb{N}, \; y_{i}\in \mathbb{R}^{N},$ in such a way each point of $\mathbb{R}^{N}$ is contained in at most $N+1$ balls. Without loss of generality, we suppose that $|y_{i}| < R_{\varepsilon},$ for $i = 1, 2, ..., n_{\varepsilon}$ and $|y_{i}|\geq R_{\varepsilon},$ for $i = n_{\varepsilon}+1, n_{\varepsilon}+2, ..., +\infty.$ Then we get that $|\Omega_{i}| < \varepsilon,$ for all $|y_{i}|\geq R_{\varepsilon},$ where $\Omega_{i} = \{x\in B_{1}(y_{i}):|m(x)|\geq \varepsilon\}.$ Observe that from the Hölder and Sobolev inequalities one has

Therefore, our claim (4.2) is right.

Next, from Lemma 2.1, the boundedness of $\{v_{n}\}$ and ${v_{n}}\rightarrow 0$ in $L^{p}_{loc}(\mathbb{R}^{N}) \; \text{for \; all}\; p\in[2, 2^{*})$, we arrive at

as $\varepsilon\rightarrow0\; \text{and}\; n\rightarrow \infty.$ Thus, we complete our proof.

Lemma 4.2. Assume that Hypothesis 1.2 and Hypothesis 1.3 all hold. Let $\{v_{n}\}$ be a bounded $\mathit{\text{(PS)}}$sequence of $\bar{J}_{\lambda}$ satisfying $v_{n}\rightharpoonup0$ in $H^{1}(\mathbb{R}^{N}),$ as $n\rightarrow \infty.$Then $\{v_{n}\}$ is also a $\mathit{\text{(PS)}}$sequence for its corresponding periodic case $J_{\lambda}.$

Proof. Since the above Lemma 4.1 guarantees

we have $J_{\lambda}(v_{n})\rightarrow \bar{d}_{\lambda}.$ Taking $\varphi\in H^{1}(\mathbb{R}^{N})$ with $\|\varphi\|\leq1,$ by Hölder inequality and Lemma 4.1, we get

which implies $J'_{\lambda}(v_{n}) = o_{n}(1).$ Hence, we know that $\{v_{n}\}$ is also a (PS) sequence of $J_{\lambda}$.

Proof of Theorem 1.2 Firstly, notice that from the Remark 3.1 we can verify the mountain pass geometry of $\bar{J}_{\lambda}$ and the boundedness of its $\text{(PS)}$ sequence $\{v_{n}\}$ analogously as in Lemma 3.1 and Lemma 3.2. Thus we can get a bounded $\text{(PS)}_{\bar{d}_{\lambda}}$ sequence $\{v_{n}\}$ of $\bar{J}_{\lambda},$ where $\bar{d}_{\lambda}$ is the mountain pass level of $\bar{J}_{\lambda},$ i.e.,

We suppose that $v\in H^{1}(\mathbb{R}^{N})$ is the weak limit for the (PS) sequence $\{v_{n}\}$. Then, arguing exactly like in Lemma 3.3, we could get that $v$ is the critical point of $\bar{J}_{\lambda}.$ However, in the case of asymptotically periodic potential, we can't ensure that $v$ is nontrivial directly. So, the task now is to prove that $v\neq0.$

We suppose, by contradiction, $v\equiv0.$ From Lemma 4.2 we know that the (PS) sequence $\{v_{n}\}$ of $\bar{J}_{\lambda}$ is also a (PS) sequence of $J_{\lambda},$ where $J_{\lambda}$ is the corresponding periodic case of $\bar{J}_{\lambda}.$ Then we can define the translation $\bar{v}_{n}(x) = v_{n}(x+y_{n})$ for $J_{\lambda}$ analogously in Lemma 3.2. Furthermore, there exists a $\bar{v}\neq0$ such that $\bar{v}_{n}\rightharpoonup \bar{v}$ in $H^{1}(\mathbb{R}^{N})$ and $J_{\lambda}'(\bar{v}) = 0.$

Set $Q(x, v, \nabla v) = -\dfrac{G^{-1}(v)g'(G^{-1}(v))}{g(G^{-1}(v))}|\nabla v|^{2}$. Since $g'(v)\leq0$ for all $v\geq0,$ it is easy to see that $Q(x, v, \nabla v)\geq0.$ Moreover, we have $Q(x, v, \nabla v)$ is convex in $\nabla v$ and ${ }\int_{\mathbb{R}^{N}}Q(x, v, \nabla v)dx$ is lower semi-continuous with respect to $v$ by Theorem 1.6 in ^[17]. Then, from the lower semi-continuity of ${ }\int_{\mathbb{R}^{N}}Q(x, v, \nabla v)dx$, Lemma 2.1, Lemma 4.1 and Fatou's Lemma we have

Consequently, $\bar{v}\neq0$ is a critical point of $J_{\lambda}$ satisfying $J_{\lambda}(\bar{v})\leq \bar{d}_{\lambda}$. Setting

and using the similar arguments in ^[18], we get a specific path $\gamma : [0, 1]\rightarrow H^{1}(\mathbb{R}^{N})$ satisfying

Then for the path given by (4.5), there holds $\gamma\in\Gamma\subset \bar{\Gamma}.$ Since $m(x) > 0$ is strict on a subset of positive measures in $\mathbb{R}^{N}$ and $G^{-1}(t)$ is an odd function, we can arrive at

which is a contradiction. Therefore, the above arguments show that the critical point $v$ of $\bar{J}_{\lambda}$ is nontrivial.

Furthermore, we repeat the same arguments used in Section 3 to verify the $L^{\infty}$-estimates of $v.$ Then, under the assumptions of Theorem 1.2 and the change of variable (2.4), we obtain a positive solution of the original Eq (1.2) for $\lambda$ sufficiently large.

5.   Conclusions

In this paper, we investigated a class of quasilinear Schrödinger equations with supercritical growth on the nonlinearity $f(t)$. The nonlinearity $f(t)$ is continuous and only superlinear in a neighborhood of $t = 0$. We supposed the potentials $V(x)$ are periodic and asymptotically periodic. By using variational methods, truncation techniques and Moser iteration, we have shown that the Eq (1.2) has at least one positive solution for the periodic and asymptotically periodic potentials.

Acknowledgments

The authors wish to thank the referees and the editors for their valuable comments and suggestions. This work was supported by the National Natural Science Foundation of China (11171220).

Conflict of interest

All authors declare no conflicts of interest in this paper.