Iterative solutions via some variants of extragradient approximants in Hilbert spaces

1.   Introduction

In this paper, we consider the existence of nontrivial positive solutions for the following generalized quasilinear elliptic equations

where $N\geq3$, $1 < p < N$, $g: \mathbb{R}\rightarrow \mathbb{R}^{+}$ is an even differential function with $g'(t)\geq 0$ for all $t\geq 0$ and $g(0) = 1$, $V: \mathbb{R}^{N}\rightarrow \mathbb{R}$ and $h: \mathbb{R}^{N}\times \mathbb{R}\rightarrow \mathbb{R}$ are continuous functions.

Notice that if we take $p = 2$ and $g^{2}(u) = 1+ \frac{[(l(u^{2}))']^{2}}{2}$, where $l$ is a given real function, then (1.1) turns into

Equation (1.2) is related to the existence of solitary wave solutions for quasilinear Schrödinger equations

where $z: \mathbb{R} \times \mathbb{R}^{N}\rightarrow \mathbb{C}$, $W: \mathbb{R}^{N} \rightarrow \mathbb{R}$ is a given potential, $h: \mathbb{R}^{N}\times \mathbb{R}\rightarrow \mathbb{R}$ and $l: \mathbb{R}\rightarrow \mathbb{R}$ are suitable functions. The form of (1.3) has many applications in physics. For instance, the case $l(s) = s$ was used to model the time evolution of the condensate wave function in superfluid film ^[16,17]. In the case of $l(s) = \sqrt{1+s}$, Eq (1.3) was used as a model of the self-channeling of a high-power ultrashort laser in matter ^[3,29]. For more details on the physical background, we can refer to ^[2,4] and references therein.

Putting $z(t, x) = \exp(-iEt)u(x)$ in (1.3), where $E\in \mathbb{R}$ and $u$ is a real function, we obtain a corresponding equation of elliptic type (1.2).

If we take $p = 2$, then (1.1) turns into

In recent years, many researchers have studied (1.4) under various hypotheses on the potential and nonlinearity, for example ^{[6,7,8,9,24,28,38]}.

If we set $p = 2, g^{2}(u) = 1+ 2u^{2}$, i.e., $l(s) = s$, we can get the superfluid film equation in plasma physics

Equation (1.5) has been extensively studied, see ^{[5,20,27,30,31]}.

If we take $p = 2, g^{2}(u) = 1+ \frac{u^{2}}{2(1+u^{2})}$, i.e., $l(s) = \sqrt{1+s}$, (1.2) derives the following equation

which models of the self-channeling of a high-power ultrashort laser in matter. For (1.6), there are many papers studying the existence of solutions, see ^[10,13,33] and references therein.

Furthermore, if we set $g^{p}(u) = 1+ 2^{p-1}u^{p}$ in (1.1), then we get the following quasilinear elliptic equations

where $\triangle_{p} = {\rm{div}}(|\nabla u|^{p-2}\nabla u)$ is the p-Laplacian operator with $1 < p\leq N$. In ^[11], where $h(x, u) = h(u)$, the authors constructed infinitely many nodal solutions for (1.7) under suitable assumptions.

We point out that the related semilinear elliptic equations with the asymptotically periodic condition have been extensively researched, see ^{[12,19,22,23,25,34,35,36,37]} and their references.

Especially, in ^[12], Lins and Silva considered the following asymptotically periodic p-laplacian equations

Assume that the potential $V$ satisfies

$(V_{0})$ there exist a constant $a_{0} > 0$ and a function $\bar{V}\in C(\mathbb{R}^{N}, \mathbb{R})$, 1-periodic in $x_{i}$, $1\leq i\leq N$, such that $\bar{V}(x)\geq V(x)\geq a_{0} > 0$ and $\bar{V}-V \in \mathcal{K}$, where

and the asymptotically periodic of $g$ at infinity was assumed to the following condition

$(g_{0})$ there exist a constant $p < q_{1} < p^{*}$ and functions $h\in \mathcal{K}$, $g_{0}\in C(\mathbb{R}^{N}, \mathbb{R})$, 1-periodic in $x_{i}$, $1\leq i\leq N$, such that

For the other conditions on $g$, please see ^[12].

In recent paper ^[36], Xue and Tang studied the following quasilinear Schrodinger equation

they proposed reformative conditions, which unify the asymptotic process of the potential and nonlinear term at infinity, see the below condition $(V_{1})$ and $({\rm{i}})$ of $(f_{5})$. It is easy to see that this conditions contains more elements than those in ^[12]. To the best of our knowledge, there is no work concerning with the unified asymptotic process of the potential and nonlinear term at infinity for general quasilinear elliptic equations.

Motivated by above papers, under the asymptotically periodic conditions, we establish the existence of a nontrivial positive solution for Eq (1.1) with critical nonlinearity. We assume $h(x, u) = f(x, u) + K(x)g(u)|G(u)|^{p^{*}-2}G(u)$. Equation (1.1) can be rewritten in the following form:

where $p^{*} = \frac{Np}{N-p}$ for $N\geq3$, $G(t) = \int^{t}_{0}g(\tau)d\tau$ and $f: \mathbb{R}^{N}\times \mathbb{R }\rightarrow \mathbb{R }$ is continuous function.

We observe that the energy functional associate with (1.10) is given by

where $F(x, u) = \int_{0}^{u}f(x, \tau)d\tau$. However, $\mathcal{I}$ may be not well defined in $W^{1, p}(\mathbb{R}^{N})$ because of the term $\int_{\mathbb{R}^{N}}g^{p}(u)|\nabla u|^{p}dx$. To overcome this difficulty, we make use of a change of variable constructed by ^[32],

Then we obtain the following functional

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 clear that $\mathcal{J}$ is well defined in $W^{1, p}(\mathbb{R}^{N})$ and $\mathcal{J}\in C^{1}$.

If $u$ is said to be a weak solution for Eq (1.10), then it should satisfy

Let $\psi = \frac{1}{g(u)}\varphi$, we know that (1.11) is equivalent to

for any $\varphi \in C_{0}^{\infty}(\mathbb{R}^{N})$.

Therefore, in order to obtain a nontrivial solution of (1.1), it suffices to study the following equations

Obviously, if $v$ is a nontrivial critical point of the functional $\mathcal{J}$, then $u = G^{-1}(v)$ is a nontrivial critical point of the functional $\mathcal{I}$, i.e., $u = G^{-1}(v)$ is a nontrivial solution of equation (1.1).

In the asymptotically periodic potential case, the functional $\mathcal{J}$ loses the $\mathbb{Z}^{N}$-translation invariance due to the asymptotically periodic potential. For this reason, many effective methods applied in periodic problems become invalid in asymptotically periodic problems. In this paper, we adopt some tricks to overcome the difficulties caused by the dropping of periodicity of $V(x)$.

Before stating our results, we introduce some hypotheses on the potential $V, K$:

$(K_{1})$ The function $K\in C(\mathbb{R}^{N}, \mathbb{R})$ is 1-periodic in $x_{i}, 1\leq i\leq N$ and there exists a point $x_{0}\in \mathbb{R}^{N}$ such that

Moreover, the nonlinear term $f\in C(\mathbb{R}^{N}\times \mathbb{R}, \mathbb{R})$ should satisfy the following assumptions:

In the asymptotically periodic case, we establish the following theorem.

Theorem 1.1 Assume that $(V_{1}), (K_{1})$ and $(f_{1})-(f_{5})$ hold. Then Eq (1.1) has a nontrivial positive solution.

In the special case: $V = V_{0}, f = f_{0}$, we can get a nontrivial positive solution for the periodic equation from Theorem 1.1. Indeed, considering the periodic equation

under the hypothesis:

$(V_{2})$ The function $V_{0}(x)$ is 1-periodic in $x_{i}, 1\leq i\leq N$ and there exists a constant $V_{min} > 0$ such that

In the periodic case, we obtain the following theorem.

Theorem 1.2 Assume that $(V_{2}), (K_{1})$ hold, $f_{0}$ satisfies $(f_{1})-(f_{4})$. Then Eq (1.12) has a nontrivial positive solution.

Remark 1.3 Compared with the results obtained by ^{[12,21,23,36]}, our results are new and different due to the following some facts:

(i) Compared with Eq (1.8) and Eq (1.9), Eq (1.10) is more general. In our results, there is no need to assume $f(x, t)\in C^{1}(\mathbb{R}^{N}, \mathbb{R})$. To some extent, our results extends the results of the work ^{[12,23,36,38]}.

(ii) We choose condition $(f_{3})$ to be weaker than Ambrosetti-Rabinowitz type condition (see ^[21]).

(iii) The aim of $(f_{3})$ is to ensure that the Cerami sequence is bounded, which is different from the conditions of ^[12].

The rest of this paper is organized as follows: in Section 2, we present some preliminary lemmas. We will prove Theorems 1.1 and 1.2 in Sections 3 and 4, respectively.

2.   Some preliminary lemmas

In this section, we present some useful lemmas.

Let us recall some basic notions. $W: = W^{1, p}(\mathbb{R}^{N})$ is the usual Sobolev space endowed with the norm $||u||_{W} = \Big(\int_{\mathbb{R}^{N}}(|\nabla u|^{p} +|u|^{p})dx\Big)^{\frac{1}{p}}$, we denote by $L^{s}(\mathbb{R}^{N})$ the usual Lebesgue space endowed with the norm $||u||_{s} = \Big(\int_{\mathbb{R}^{N}}|u|^{s}dx \Big)^{\frac{1}{s}}, \forall s\in[1, +\infty)$ and let $C$ denote positive constants. Next, we define the following working space

endowed with the norm $||u|| = \Big(\int_{\mathbb{R}^{N}}(|\nabla u|^{p} +V(x)|u|^{p})dx\Big)^{\frac{1}{p}}$. According to ^[22], it is easy to verify that the norms $||\cdot||$ and $||\cdot||_{W}$ are equivalent under the assumption $(V_{1})$.

Next, we summarize some properties of $g, G$ and $G^{-1}$.

Lemma 2.1 ^[32] The functions $g, G$ and $G^{-1}$ satisfy the following properties:

Denote

Then

Therefore,

Lemma 2.2 The functions $\bar{f}(x, s)$ and $\bar{F}(x, s)$ satisfy the following properties under $(f_{1})-(f_{3}), (f_{5})$:

(i) $\lim_{s\rightarrow 0^{+}}\frac{\bar{f}(x, s)}{|s|^{p-1}}$ = 0 and $\lim_{s\rightarrow 0^{+}}\frac{\bar{F}(x, s)}{|s|^{p}} = 0$ uniformly in $x\in \mathbb{R}^{N}$;

(ii) $\lim_{s\rightarrow +\infty}\frac{\bar{f}(x, s)}{|s|^{p^{*}-1}}$ = 0 and $\lim_{s\rightarrow +\infty}\frac{\bar{F}(x, s)}{|s|^{p^{*}}} = 0$ uniformly in $x\in \mathbb{R}^{N}$;

(iii) $t\bar{f}(x, t) - p\bar{F}(x, t) \geq st\bar{f}(x, st) - p\bar{F}(x, st)$ for all $t\in \mathbb{R^{+}}$ and $s\in [0, 1]$;

(iv) $\bar{F}(x, s)\geq 0$ for all $(x, s)\in \mathbb{R}^{N}\times \mathbb{R^{+}}$.

Proof From $(f_{1})$ and Lemma 2.1-(8), we have

uniformly in $x\in \mathbb{R}^{N}$. Moreover, by $(f_{2})$ and Lemma 2.1-(8), one has

uniformly in $x\in \mathbb{R}^{N}$. Then using the L'Hospital rule, we obtain

uniformly in $x\in \mathbb{R}^{N}$. Hence, $({\rm{i}})$ and $({\rm{ii}})$ hold.

Let $\mathcal{H}(x, t) = t\bar{f}(x, t) - p\bar{F}(x, t)$ for $(x, t)\in(\mathbb{R}^{N}, \mathbb{R^{+}})$.

We claim that $\mathcal{H}(x, t)$ is an increasing function with respect to $t$.

By (2.1) and (2.2), we directly calculate

Next, set $\eta(t) = G^{-1}(t)^{2} - \frac{G^{-1}(t)t}{g(G^{-1}(t))}$ for any $t\in \mathbb{R^{+}}$. To compete the claim, combining with $(f_{3})$ and Lemma 2.1-(1), We only need to prove that $\eta(t)$ is an increasing function on $\mathbb{R^{+}}$. Please see literature ^[26], for the reader's convenience, we give a brief proof.

By Lemma 2.1-(2) and $g'(t)\geq 0$ for all $t\geq 0$, one has

which deduces that

for all $t\geq 0$. Set $\xi = G(t)$. Then

and thus

for all $\xi\geq0$. Therefore,

for all $t\geq 0$. It follows that $\eta(t)$ is increasing with respect to $t\geq 0$. Thus, $\eta(st)\leq \eta(t)$ for all $s\in [0, 1]$ and $t\geq 0$, and then

for all $s\in [0, 1]$ and $t\geq 0$. So $({\rm{iii}})$ holds. Moreover, Lemma 2.1-(5) and $(f_{5})-(i)$ imply that $\bar{F}(x, s)\geq 0$ for all $(x, s)\in \mathbb{R}^{N}\times \mathbb{R^{+}}$. This completes the proof.

Lemma 2.3 Assume that $(V_{1}), (K_{1})$ and $(f_{1})-(f_{2})$ are satisfied. Then the functional $\mathcal{J}$ satisfies the following mountain pass geometry structure:

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

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

Proof By $(f_{1}), (f_{2})$, Lemma 2.2-(1), (2), for any $\varepsilon > 0$, there exists $C_{\varepsilon} > 0$ and $q\in (p, p^{*})$ such that

for all $(x, s)\in \mathbb{R}^{N}\times \mathbb{R^{+}}$. Therefore, by (2.3) and (2.4), we have

where $\varepsilon$ is small enough, thus (i) is proved because $p < q < p^{*}$.

It follows from $(2.3)$ that Lemma 2.2-(4), for any fixed $v\in X$ with $v\geq 0$ and $v\not\equiv 0$, we obtain

Obviously, $\mathcal{J}(tv)\rightarrow -\infty$ as $t\rightarrow + \infty$. Thus there exists a $t_{0} > 0$ large enough such that $\mathcal{J}(t_{0}v) < 0$ with $||t_{0}v|| > \rho$. Hence, we take $v_{0} = t_{0}v$, (ii) is proved.

Lemma 2.4 Assume that $(V_{1}), (K_{1})$ and $(f_{1})-(f_{3})$ hold, then there exists a bounded Cerami sequence $\{v_{n}\}\subset X$ for $\mathcal{J}$.

Proof From Lemma 2.3, we know that $\mathcal{J}$ satisfies the mountain pass geometry structure. By the mountain pass theorem (see ^[1]), there exists a Cerami sequence $\{v_{n}\}\subset X$ such that

where

As in ^[14], we can take a sequence $\{t_{n}\}\subset [0, 1]$ satisfying

We claim that $\{\mathcal{J}(t_{n}v_{n})\}$ is bounded from above.

Indeed, without loss of the generality, we may assume that $t_{n}\in (0, 1)$ for all $n\in \mathbb{N}$. Thus, by Lemma 2.2-(3), we have

which implies that $\{\mathcal{J}(t_{n}v_{n})\}$ is bounded from above.

Now, we prove that $\{v_{n}\}$ is bounded in $X$. Assume by contradiction that $\{v_{n}\}$ is unbounded, then up to a subsequence, we may assume that $||v_{n}||\rightarrow +\infty$. Set $w_{n} = \frac{v_{n}}{||v_{n}||}$. Clearly, $w_{n}$ is bounded in $X$ and $||w_{n}|| = 1$. Then, there exists $w\in X$ such that $w_{n}\rightharpoonup w$ in $X$. Set $\Lambda = \{x\in \mathbb{R}^{N}: w(x)\neq 0\}$. If meas$(\Lambda) > 0$, the Fatou lemma and Lemma 2.2-(4) implies

which is a contradiction. Thus $w = 0$. For any $B > 0$, by $||v_{n}||\rightarrow +\infty$ and (2.5), we have

By $(f_{1}), (f_{2})$, Lemma 2.2-(1), (2), for $\varepsilon > 0$, there exists $C_{\varepsilon} > 0$ and $q\in(p, p^{*})$ such that

for all $(x, s)\in \mathbb{R}^{N}\times \mathbb{R^{+}}$. Hence, by (2.6), we get

as $n\rightarrow \infty$, namely, $\int_{\mathbb{R}^{N}}|w_{n}|^{p^{*}}dx\rightarrow 0$ as $n\rightarrow \infty$. Then by interpolation inequality, we have $\int_{\mathbb{R}^{N}}|w_{n}|^{q}dx\rightarrow 0$ as $n\rightarrow \infty$. Moreover, from (2.4), we have

By the arbitrariness of $\varepsilon$, we can get $\int_{\mathbb{R}^{N}}\bar{F}(x, Bw_{n})dx\rightarrow 0$ as $n\rightarrow \infty$. Consequently,

This contradicts the fact that $\mathcal{J}(t_{n}v_{n})$ is bounded above. Therefore, $\{v_{n}\}$ is bounded in $X$. The proof of Lemma 2.4 is complete

Next, we do an estimate on $c$ and follow the approach presented in ^[12]. Given $\varepsilon > 0$, we consider the function

which is a solution of the following equation

and

can be achieved at $U_{\varepsilon}$.

Let $\phi\in C^{\infty}_{0}(\mathbb{R}^{N}, [0, 1])$ be a cut-off function such that $\phi\equiv 1$ in $B_{1}(0)$, $\phi\equiv 0$ in $\mathbb{R}^{N}\backslash B_{2}(0)$. Define

then by a direct computation, there exist positive constants $l_{1}, l_{2}$ and $\varepsilon_{0}$ such that

and as $\varepsilon\rightarrow 0$, we have

Lemma 2.5 Assume that $(V_{1}), (K_{1})$ and $(f_{1})-(f_{2}), (f_{4})$ are satisfied. Then $c < \frac{1}{N}||K||^{\frac{p-N}{p}}_{\infty}S^{\frac{N}{p}}$.

Proof For $t > 0$, $v_{\varepsilon}$ defined by above (2.7), we have

Lemma 2.3 implies that there exists $t_{\varepsilon} > 0$ such that $\mathcal{J}(t_{\varepsilon}v_{\varepsilon}) = \max_{t\geq0}\mathcal{J}(tv_{\varepsilon})$.

We claim that there exist $T_{1}, T_{2} > 0$ such that $T_{1}\leq t_{\varepsilon} \leq T_{2}$ for $\varepsilon$ small enough.

Indeed, if $t_{\varepsilon}\rightarrow 0$ as $\varepsilon\rightarrow 0$, we have $0 < \mathcal{J}(t_{\varepsilon}v_{\varepsilon})\rightarrow \mathcal{J}(0) = 0$, which is a contradiction.

On the other hand, from (2.11) and $(f_{4})$, one has

as $t_{\varepsilon}\rightarrow + \infty$, a contradiction, which implies that the claim holds.

To complete the proof, it suffices to show that $\mathcal{J}(t_{\varepsilon}v_{\varepsilon}) < \frac{1}{N}||K||^{\frac{p-N}{p}}_{\infty}S^{\frac{N}{p}}$. Therefore,

for some constant $C = \frac{T^{p}_{2}||V_{0}||_{\infty}}{p} > 0$.

Indeed, for $t > 0$, define $l(t) = \frac{t^{p}}{p}\int_{\mathbb{R}^{N}}|\nabla v_{\varepsilon}|^{p}dx - \frac{t^{p^{*}}}{p^{*}}$, we have that $t_{0} = \Big(\int_{\mathbb{R}^{N}}|\nabla v_{\varepsilon}|^{p}dx \Big)^{\frac{1}{p^{*}- p}}$ is a maximum point of $l$ and $l(t_{0}) = \frac{1}{N}\Big(\int_{\mathbb{R}^{N}}|\nabla v_{\varepsilon}|^{p}dx \Big)^{\frac{N}{p}}$. Applying the inequality

By (2.9) and (2.12), we have

Now consider

From (2.13) and (2.14), we have

Next we claim that

It follows $(f_{4})$ that any $A > 0$, there exists $R = R_{A} > 0$ such that for all $(x, s)\in \Omega \times [R_{A}, +\infty)$,

where $\mu = p^{*} - \frac{p}{p-1}$. Now consider the function $\eta_{\varepsilon} : [0, +\infty)\rightarrow \mathbb{R}$ defined by

Since $\phi\equiv 1$ in $B_{1}(0)$, due to (2.8), we choose a constant $C > 0$ such that $v_{\varepsilon}(x)\geq C\eta_{\varepsilon}(|x|)$ for $|x| < 1$. Note that $\eta_{\varepsilon}$ is decreasing and $G^{-1}$ is increasing, there exists a positive constant $C$ such that, for $|x| < \varepsilon^{\frac{p-1}{p}}$,

Then we can choose $\varepsilon_{1} > 0$ such that

for $|x| < \varepsilon^{\frac{p-1}{p}}$, $0 < \varepsilon < \varepsilon_{1}$. It follows from (2.16) and (2.17) that

for $|x| < \varepsilon^{\frac{p-1}{p}}$, $0 < \varepsilon < \varepsilon_{1}$.

Using $(f_{5})$-(i), one has

Since $B_{2}(0)\subset \Omega$, by (2.18) and (2.19), for $0 < \varepsilon < \varepsilon_{1}$, we have

where

Consequently, by (2.20), we obtain

Choosing $A > 0$ sufficiently large, (2.21) establishes (2.15). Lemma 2.5 is proved.

3.   The asymptotically period case

In this section, in order to overcome the difficulties caused by the loss of translation invariance due to the asymptotically periodic potential, we need to state the following technical convergence results. The detailed proofs can be found in ^[23,36], where $p = 2$.

Lemma 3.1 Assume that $(V_{1})$, $(K_{1})$, $(f_{1})$ and $({\rm{i}})$ of $(f_{5})$ hold. Suppose that $\{v_{n}\}$ is bounded in $X$, $\{y_{n}\}\subset \mathbb{Z}^{N}$ with $|y_{n}|\rightarrow +\infty$ and $v_{n}\rightarrow 0$ in $L^{\alpha}_{loc}(\mathbb{R}^{N})$, for any $\alpha\in [p, p^{*})$. Then up to a subsequence, one has

Proof of Theorem 1.1 Lemma 2.3 implies the existence of a Cerami sequence $\{v_{n}\}\subset X$. By Lemma 2.4, $\{v_{n}\}$ is bounded in $X$. Thus, there exists $v\in X$ such that $v_{n}\rightharpoonup v$ in $X$, $v_{n}\rightarrow v$ in $L^{p}_{loc}(\mathbb{R}^{N})$, $v_{n}(x)\rightarrow v(x)$ a.e. in $\mathbb{R}^{N}$. For any $\varphi\in C^{\infty}_{0}(\mathbb{R}^{N})$, one has

that is, $v$ is a weak solution of Eq (1.10).

Now we prove that $v$ is nontrivial. By contradiction, we assume that $v = 0$. We divide the proof four steps.

Step 1: We claim that $\{v_{n}\}\subset X$ is also a Cerami sequence for the functional $\mathcal{J}_{0}: X\rightarrow \mathbb{R}$, where

From (3.1) and (3.3), we can deduce that

and taking $\varphi \in X$ with $||\varphi|| = 1$, by (3.2) and (3.4), we obtain that

From (3.5) and (3.6), we can get that $\{v_{n}\}$ is also a Cerami sequence for $\mathcal{J}_{0}$.

Step 2: We prove that $\{v_{n}\}$ is non-vanishing i.e.,

If $\beta = 0$, the Lions lemma ^[18], we have $v_{n}\rightarrow 0$ in $L^{q}(\mathbb{R}^{N})$ for all $q\in (p, p^{*})$.

Note that

which combining with (2.6) leads to

Therefore, there exists a constant $l\geq 0$ such that

Obviously, $l > 0$. Otherwise, $\mathcal{J}(v_{n})\rightarrow 0$ as $n\rightarrow \infty$, which contradicts with $c > 0$. Since

that is, $l\geq ||K||^{\frac{p-N}{p}}_{\infty}S^{\frac{N}{p}}$. Consequently, (2.4) implies that

as $n\rightarrow \infty$, which deduces that $c\geq \frac{1}{N}||K||^{\frac{p-N}{p}}_{\infty}S^{\frac{N}{p}}$, a contradiction.

Step 3: After a translation of $\{v_{n}\}$ denoted $\{w_{n}\}$, then $\{w_{n}\}$ converges weakly to a nonzero critical point of $\mathcal{J}_{0}$.

Choose $\{y_{n}\}\subset {\mathbb{Z}}^{N}$ such that $|y_{n}|\rightarrow +\infty$ as $n\rightarrow \infty$ and denote $w_{n}(x) = v_{n}(x + y_{n})$. Then

Thus, $\{w_{n}\}$ is a bounded $(PS)_{c_{0}}$ of $\mathcal{J}_{0}$, where $c_{0}$ is defined below. Going if necessary to a subsequence, we get that $w_{n}\rightharpoonup w$ in $X$ and $\mathcal{J}_{0}'(w) = 0$. So by Step 2 we get $w\neq 0$. Therefore, by $(f_{5})$-(ii), Lemma 2.1-(8) and Fatou Lemma, we have

which implies that $\mathcal{J}_{0}(w)\leq c$.

Step 4: We use $w$ to construct a path which allows us to get a contradiction with the definition of mountain pass level $c$.

Define the mountain pass level

where $\bar{\Gamma}: = \{\gamma\in C([0, 1], X): \gamma(0) = 0, \mathcal{J}_{0}(\gamma(1)) < 0\}$. Applying similar arguments used in ^[15], we can construct a path $\gamma: [0, 1]\rightarrow X$ such that

Then $c_{0}\leq \max_{t\in [0, 1]}\mathcal{J}_{0}(\gamma(t)) = \mathcal{J}_{0}(w)$. Due to the fact that $V(x)\leq V_{0}(x)$ but $V(x)\not\equiv V_{0}(x)$, we take the path $\gamma$ given by above and by $\gamma\in \bar{\Gamma}\subset \Gamma$, we have

which is a contradiction. Consequently, $v$ is a nontrivial solution of Eq (1.10), then using the strong maximun principle, we obtain $v > 0$, namely, Eq (1.1) possesses a nontrivial positive solution $u = G^{-1}(v)$. This completes the proof of Theorem 1.1.

4.   The period case

In this section, we give the proof of Theorem 1.2.

Proof of Theorem 1.2 By Lemma 2.3, there exists a Cerami sequence $\{v_{n}\}\subset X$ such that

Applying Lemma 2.4, the Cerami sequence $\{v_{n}\}$ is bounded in $X$. Similar to (3.7), it is easy to verify that $\{v_{n}\}$ is non-vanishing.

As in the step 3 of Theorem 1.1, set $w_{n}(x) = v_{n}(x + y_{n})$. It is easy to know that $||w_{n}|| = ||v_{n}||$ and $\{w_{n}\}$ is bounded and non-vanishing. Going if necessary to a subsequence, we have

Moreover, since $V_{0}(x), K(x)$ and $f_{0}(x, u)$ are periodic on $X$, we see that $\{w_{n}\}$ is also a a Cerami sequence of $\mathcal{J}_{0}$. Then for any $\varphi\in C^{\infty}_{0}(\mathbb{R}^{N})$,

That is $\mathcal{J}_{0}'(w) = 0$ and $w$ is a nontrivial solution to (1.12). By the strong maximun principle, we obtain $w > 0$. This completes the proof of Theorem 1.2.

5.   Conclusions

In ^[38], we discussed a class of generalized quasilinear Schrödinger equations with asymptotically periodic potential, where $p = 2$ and the nonlinear term is subcritical. In this current work, we have established the existence of nontrivial positive solutions for a class of generalized quasilinear elliptic equations with critical growth. In the next work, we will extend the study to the case of variable exponent $p = p(t)$.

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 declare no conflicts of interest in this paper.