Positive solutions for a critical quasilinear Schrödinger equation

1.   Introduction

In this work we discuss the critical quasilinear Schrödinger equation

where $N\geq 3$, $\mathcal V, \mathcal K \in C(\mathbb{R}^N, \mathbb{R}^+)$, $g \in C(\mathbb{R}, \mathbb{R})$ is a quasilinear growth function, $2^* = 2N/(N-2)$ and $22^*$ is the critical exponent for (1.1).

Quasilinear equations are often involved in studying standing wave solutions for the quasilinear Schrödinger equation

where $\mathcal W$ is a potential, $\kappa \in \mathbb R$, $\rho$, $l: \mathbb R\to \mathbb R$. The form of (1.2) has many applications in physics, for example see ^[1,2,3,4]. In ^[5] the authors used the method of Nehari manifold to discuss the concentration behavior and the exponential decay phenomenon of ground state solutions for the equation

where $N \in [3, \infty), \varepsilon\in (0, \infty), p\in (4, 22^*), g \in {C}^1\left(\mathbb{R}, \mathbb{R}^{+}\right), \mathcal V, \mathcal K \in {C}\left(\mathbb{R}^N\right) \cap L^{\infty}\left(\mathbb{R}^N\right)$. In ^[6] the authors studied the equation

and obtained that the above problem has infinitely many high energy solutions, where $1 \leq \alpha < 2, \widetilde{f} \in C\left(\mathbb{R}^N \times \mathbb{R}, \mathbb{R}\right)$. For some related papers we refer the reader to ^{[7,8,9,10,11,12,13,14,15]} and the references cited therein.

Because of the quasilinear term $\triangle(\Theta^2)\Theta$, we note that the quasilinear case is much more complicated than the semilinear case. Moreover, the main difficulty of $(1.1)$ is there is no suitable space on which the energy functional is well-defined and of the $C^1$-class except for $N = 1$ (see ^[4]). Also an important problem of $(1.1)$ is the zero mass case, which appears when $\mathcal V$ vanishes at infinity, i.e.,

In ^[16], when $(1.1)$ has no critical term and a quasilinear term, the authors studied the zero mass case with

and $\mathcal V, \mathcal K$ satisfy the assumption:

(VK) $\mathcal V, \mathcal K\in C^1(\mathbb{R}^N, \mathbb{R})$, and there exist $\tau, \xi, a_i > 0(i = 1, 2, 3)$ such that

where $\tau, \ \xi$ satisfy

In ^[17], using (VK) the authors established the following result: $E$ is compactly embedded into the Lebesgue space

where $1 < p < 2^*-1$ and

and the norm on $E$ is defined as follows:

In ^[18] the authors also considered the condition (VK), when the inequality of $\mathcal V$ is only imposed outside of a ball centered at origin. In ^[19] the authors introduced some new hypotheses for $\mathcal K$, using the Marcinkiewicz spaces $L^{r, \infty}(\mathbb{R}^N)(r > 1)$, which ensures that the embedding $E\hookrightarrow L_K^q(\mathbb{R}^N)$ is continuous and compact for $q > 1$. The space $L^{r, \infty}(\mathbb{R}^N)$ is formed by measurable functions $h: \ \mathbb{R}^N\rightarrow \mathbb{R}$ verifying

We will consider the subspace $L_0^{r, \infty}(\mathbb{R}^N)$ of $L^{r, \infty}(\mathbb{R}^N)$, which is the closure of $L^{\infty}(\mathbb{R}^N)\cap L^1(\mathbb{R}^N)$ in $L^{r, \infty}(\mathbb{R}^N)$. In that paper, it was proved that the embedding

is continuous for $p \in [2, 2^*]$ if $\mathcal K \in L^{r, \infty}(\mathbb{R}^N)$. If $\mathcal K \in L_0^{r, \infty}(\mathbb{R}^N)$, the above embedding is compact for all $p \in [2, 2^*)$.

In order to study our problem, we first give some assumptions:

We say that $(\mathcal V, \mathcal K) \in \mathbb {K}$ if the following conditions are satisfied:

($\mathbb K$I) $\mathcal K \in L^{\infty}(\mathbb{R}^N)$, $\mathcal V(x), \ \mathcal K(x) > 0, x \in \mathbb{R}^N$.

($\mathbb K$II) There is a sequence of Borel sets $\{A_n\}\subset \mathbb{R}^N$ such that $|A_n|\leq R$ for some $R > 0$, $n\in \mathbb N$ and we have

($\mathbb K$III) One of the following two conditions occurs:

or there exists a $p \in (2, 22^*)$ such that

Moreover, for the function $g$, we assume that:

$(g_1) \; \begin{array}{ll} \left\{ \begin{array}{ll} \lim\limits_{t\rightarrow 0}\frac{g(t)}{t} = 0 & \hbox{ if }\ (K_2) \ \hbox{ holds, }\\ \lim\limits_{t\rightarrow 0}\frac{|g(t)|}{|t|^{p-1}} < +\infty & \hbox{ if } \ (K_3) \ \hbox{ holds.} \end{array} \right. \end{array}$

$(g_2) \; \lim\limits_{|t|\rightarrow +\infty}\frac{g(t)}{|t|^{22^*-1}} = 0.$

$(g_3) \; tg(t)-4G(t)\geq 0$, $t \in \mathbb{R}$.

$(g_4) \; g(t)t > 0$, $t\neq 0$.

Now we give the main theorem:

Theorem 1.1. Let $(\mathcal V, \mathcal K) \in \mathbb {K}$ and suppose $(g_1)-(g_4)$ are true. Then $(1.1)$ has a positive solution for large $\lambda$.

In our paper, $C$ and $C_i$ are utilized in various places to denote different positive constants.

2.   Preliminaries

The energy functional of (1.1) is defined as

where $G(\Theta): = \int_{0}^{\Theta}g(s)ds$. Since $J_\lambda(\Theta)$ is not well-defined on $E$, we cannot adopt directly the variational method to study $(1.1)$. Motivated by ^[20,21], let $\Theta = \mathscr{S}(\mathscr{V})$, where $\mathscr{S}$ is defined by

and

By variable transform, we obtain the modified energy functional

We easily obtain $I_\lambda \in C^1(E, \mathbb{R})$, and its Gateaux derivative is given by

for all $\mathscr{V}, \varphi \in E$.

For completeness we provide some properties for $\mathscr{S}$.

Lemma 2.1. (see ^[22,23,24]) $\mathscr{S}(t)$ has the following properties:

$(\mathscr{S}_1) \; \mathscr{S}$ is of class $C^\infty$, and invertible;

$(\mathscr{S}_2) \; 0 < \mathscr{S}^{\prime}(t)\leq 1, \ t\in \mathbb{R}$;

$(\mathscr{S}_3) \; |\mathscr{S}(t)|\leq |t|, \ t\in \mathbb{R}$;

$(\mathscr{S}_4) \; \lim_{t\to0}\frac{\mathscr{S}(t)}{t} = 1$;

$(\mathscr{S}_5) \; \lim_{t\to+\infty}\frac{\mathscr{S}^2(t)}{t} = \sqrt{2}, \ \lim_{t\to-\infty}\frac{\mathscr{S}^2(t)}{t} = -\sqrt{2}$;

$(\mathscr{S}_6) \; \frac{\mathscr{S}(t)}{2}\leq t\mathscr{S}^{\prime}(t)\leq \mathscr{S}(t), \ \ \forall t\geq0$; $\ \mathscr{S}(t)\leq t\mathscr{S}^{\prime}(t)\leq \frac{\mathscr{S}(t)}{2}, \ \ t\leq 0$;

$(\mathscr{S}_7) \; \mathscr{S}^2(t)\leq \sqrt{2}|t|, \ t\in \mathbb{R}$;

$(\mathscr{S}_8) \; \mathscr{S}^2(t)$ is strictly convex;

$(\mathscr{S}_9)$ There exists $\theta > 0$ such that

$(\mathscr{S}_{10})$ There exist $C_1, C_2 > 0$ such that

$(\mathscr{S}_{11}) \; |\mathscr{S}(t)\mathscr{S}^{\prime}(t)|\leq \frac{1}{\sqrt{2}}, \ t\in \mathbb{R}$;

$(\mathscr{S}_{12}) \; \mathscr{S}(t)$ is odd, $\mathscr{S}^2(t)$ is even;

$(\mathscr{S}_{13}) \; \forall \xi > 0$, there exists $C(\xi) > 0$ such that

$(\mathscr{S}_{14}) \; \mathscr{S}(t)\mathscr{S}^\prime(t)t^{-1}$ is strictly decreasing for $t > 0$;

$(\mathscr{S}_{15}) \; \mathscr{S}^q(t)\mathscr{S}^\prime(t)t^{-1}$ is strictly increasing for $q \geq 3$, $t > 0$;

$(\mathscr{S}_{16}) \; \mathscr{S}^2(\lambda t)\leq \lambda^2\mathscr{S}^2(t), \lambda > 1, t \in \mathbb{R}$;

$(\mathscr{S}_{17}) \; \mathscr{S}^2(\frac{1}{\lambda} t)\leq \frac{1}{\lambda} \mathscr{S}^2(t), \lambda\ge 1, t \in \mathbb{R}$.

From Lemma 2.1, Proposition 2.1 in ^[25] or Lemma 2.2 in ^[26] we can obtain the result:

Lemma 2.2. Let $(\mathcal V, \mathcal K) \in \mathbb {K}$ and $(K_2)$ or $(K_3)$ be satisfied. Then $\mathscr{V}_n\rightharpoonup \mathscr{V}$ in $E$ implies that

From Lemmas 2.1 and 2.2, Lemma 2.2 in ^[25] we get the result:

Lemma 2.3. Let $(\mathcal V, \mathcal K) \in \mathbb {K}$ and $(g_1)-(g_2)$ hold. If $\mathscr{V}_n \rightharpoonup \mathscr{V}$ in $E$, then we have

and

Proof. (i) If $(K_2)$ holds, then Lemma 2.2 implies that

Therefore, for all $\varepsilon > 0$, there exists $r > 0$ such that $\int_{B_r^c(0)}\mathcal K(x)|\mathscr{S}(\mathscr{V}_n)|^qdx < \varepsilon$ for large $n$. Moreover, $(g_1)$ and $(g_2)$ imply that

Hence, from $(\mathscr{S}_3)$ and $(\mathscr{S}_7)$ we have

for large $n$. By the compactness lemma of Strauss (see ^[27]) we have

Consequently, from $(2.1)$ and $(2.2)$ we obtain

(ii) If $(K_3)$ holds, then for any $\varepsilon > 0$, there exists a sufficient large $r$ such that

Hence, we have

Combining $(2.3)$ with $(g_1)$ and $(g_2)$, there exist $0 < s_0 < s_1$ such that

where $I: = \{s \in \mathbb{R}: |s| < s_0 \ \hbox{ or }\ |s| > s_1\}$. Consequently, we obtain

Moreover, there exists a $M_1 > 0$ such that

Let

Then $s_0^{2^*}|A_n|\leq \int_{A_n}|\mathscr{V}_n|^{2^*}dx\leq M_1, \ n \in \mathbb{N}$. This implies that $\sup\limits_{n \in \mathbb{N}}|A_n| < +\infty$. Thus $(K_1)$ implies that

because $r$ is big enough. Hence, $(\mathscr{S}_3)$, $(\mathscr{S}_7)$ and $(2.4)$ imply that

Similarly, by the compactness lemma of Strauss we have

Consequently,

Similarly, we get

and

The proof is completed. □

Lemma 2.4. (^[28,29]) Let $X$ be a real Banach space and $I\in C^1(X, \mathbb{R})$. Let $\Sigma$ be a closed subset of $X$ which disconnects (arcwise) $X$ into distinct connected components $X_1$ and $X_2$. Suppose that $I(0) = 0$ and

$(I_1) \; 0 \in X_1$, and there exists $\alpha > 0$ such that $I|_\Sigma\geq\alpha > 0$.

$(I_2)$ there exists a $e \in X_2$ such that $I(e) < 0$.

Then $I$ possesses a Cerami sequence with $c\geq\alpha > 0$ given by

where

Now, we prove that $I_\lambda$ has the mountain pass geometry.

Lemma 2.5. Suppose that $(V, K) \in \mathbb {K}$ and $g$ satisfies $(g_1)$, $(g_2)$ and $(g_4)$. Then $I_\lambda$ satisfies the conditions in Lemma 2.4 $(I_1)$ and $(I_2)$.

Proof. For any $\rho > 0$, let $S_\rho = \{\mathscr{V} \in E: \mathcal Q(\mathscr{V}) = \rho^2\}$, where $\mathcal Q:E\rightarrow \mathbb{R}$ is given by

Since $\mathcal Q(\mathscr{V})$ is continuous on $E$, $S_\rho$ is a closed subset of $E$ and it disconnects $E$ into distinct connected components $E_1$ and $E_2$.

If either $(K_2)$ or $(K_3)$ hold, $(g_1)$ and $(g_2)$ imply that for any $\varepsilon > 0$, there exists $C_\varepsilon > 0$ such that

for all $s \in \mathbb{R}$. Hence, by an inequality (see [30, (4.5)]) we have

Moreover, by $(\mathscr{S}_7)$ we have

Consequently, for $\mathscr{V} \in S_\rho$, we obtain

for $\rho > 0$ and $\varepsilon > 0$ small enough.

In what follows, we take a function $\varphi \in C_0^{\infty}(\mathbb{R}^N)$ with $\text{supp}\varphi = \overline{B_1}$ and $\varphi\in [0, 1]$, $x\in B_1$. For any $t > 0$, since $\frac{\mathscr{S}(t)}{t}$ is decreasing about $t\geq 0$, we get $\mathscr{S}(t)\varphi(x)\leq \mathscr{S}(t\varphi(x))$, for $x \in B_1$ and $t > 0$. Hence by $(\mathscr{S}_3)$, $(\mathscr{S}_5)$ and $(g_4)$ we have

as $t\rightarrow +\infty$, i.e., $I_\lambda(t\varphi)\rightarrow -\infty$ as $t\rightarrow +\infty$. Consequently, let $e: = t^*\varphi$ be such that $I_\lambda(e) < 0$($t^*$ large enough). The proof is completed. □

Lemma 2.6. Let $(V, K) \in \mathbb {K}$ and $(g_1)-(g_3)$ hold. Then there is a bounded Cerami sequence $\{\mathscr{V}_n\}\subset E$ with $I_\lambda(\mathscr{V}_n)\rightarrow c_\lambda\geq \alpha > 0$, where

and $\alpha$ is found in Lemma $2.5$.

Proof. Step 1: We prove that the sequence $\{\mathcal Q(\mathscr{V}_n)\}$ is bounded. Let $\varphi_n = \frac{\mathscr{S}(\mathscr{V}_n)}{\mathscr{S}^\prime(\mathscr{V}_n)}$. Then $\|\varphi_n\|_E\leq 2\|\mathscr{V}_n\|_E$. Consequently, we get

Hence, from $(g_3)$ we have $\{\|\mathscr{S}(\mathscr{V}_n)\|_E\}$ and $\{\|\mathscr{S}(\mathscr{V}_n)\|_{22^*}\}$ are bounded, and

Note that $(g_1)$ and $(g_2)$ imply that

Therefore, we have

and thus $\{\mathcal Q(\mathscr{V}_n)\}$ is bounded.

Step 2: We verify that there exists $C > 0$ such that $\mathcal Q(\mathscr{V}_n)+[\mathcal Q(\mathscr{V}_n)]^\frac{2^*}{2}\geq C\|\mathscr{V}_n\|_E^2$. In view of $(\mathscr{S}_9)$, the Sobolev embedding theorem implies that

and

Hence, we obtain

Consequently,

i.e., $C\|\mathscr{V}_n\|_E^2\leq \mathcal Q(\mathscr{V}_n)+[\mathcal Q(\mathscr{V}_n)]^\frac{2^*}{2}$.

Combining with the two steps we have $\{\|\mathscr{V}_n\|_E\}$ is bounded. The proof is completed. □

Lemma 2.7. Let $(V, K) \in \mathbb {K}$ and $(g_1)$, $(g_2)$, $(g_4)$ hold. Then there exists $\lambda^* > 0$ such that $0 < c_\lambda < \frac{1}{2N}S^\frac{N}{2}$ for all $\lambda > \lambda^*$.

Proof. Suppose the contrary. Then there exists a sequence $\{\lambda_n\}$ with $\lambda_n\rightarrow +\infty$ such that $c_{\lambda_n}\geq \frac{1}{2N}S^\frac{N}{2}$. Choosing $\mathscr{V} \in E\setminus\{0\}$, then from $(g_1)$, $(g_2)$ and $(g_4)$ there exists a unique $t_{\lambda_n} > 0$ such that $\max\limits_{t\geq0}I_{\lambda_n}(t\mathscr{V}) = I_{\lambda_n}(t_{\lambda_n} \mathscr{V})$. From $(\mathscr{S}_3)$ and $(\mathscr{S}_6)$ we have

From $(\mathscr{S}_5)$ and $(\mathscr{S}_6)$ we have $\{t_{\lambda_n}\}$ is bounded. Hence, there exists $t_0\geq 0$ such that $t_{\lambda_n}\rightarrow t_0$ as $n\rightarrow \infty$. If $t_0 > 0$, then by $(g_4)$ and Fatou's Lemma we have

However we note that

and this contradicts (2.5). Hence $t_0 = 0$. Let $\mathscr{W} = t_{\lambda_n} \mathscr{V}$. Then we have

Consequently, by $(\mathscr{S}_3)$ and $(g_4)$ we have

as $n \rightarrow \infty$. Hence

and this is a contradiction. Therefore, we have our conclusion in this lemma. The proof is completed. □

3.   Proof of Theorem 1.1

Lemma 2.7 indicates that there is a $\lambda^* > 0$ such that $0 < c_\lambda < \frac{1}{2N}S^\frac{N}{2}$ for all $\lambda > \lambda^*$. For a fixed $\lambda > \lambda^*$, by Lemma 2.6 there exists a bounded Cerami sequence $\{\mathscr{V}_n\}\subset E$ with $I_\lambda(\mathscr{V}_n)\rightarrow c_\lambda\geq \alpha > 0$, where

and $\alpha$ is found in Lemma $2.5$. Hence, there is a $\mathscr{V} \in E$ such that

By a standard argument we obtain $I_\lambda^\prime(\mathscr{V}) = 0$, i.e., $\mathscr{V}$ is a weak solution of $(1.1)$. Indeed, for any $\psi \in C_0^{\infty}(\mathbb{R}^N)$, we have

From (3.1) we have

and

Consequently, we obtain

For any $\varphi \in E$, there exists a sequence $\{\psi_n\}\subset C_0^{\infty}(\mathbb{R}^N)$ such that $\psi_n\rightarrow \varphi$ in $E$. Hence

Let $n\rightarrow \infty$ and we get

i.e., $\langle I_\lambda^\prime(\mathscr{V}), \varphi\rangle = 0$ for all $\varphi \in E$. Hence $I_\lambda^\prime(\mathscr{V}) = 0$. Now, let $\mathscr{V}^+: = \max\{\mathscr{V}, 0\}$ and $\mathscr{V}^-: = \min\{\mathscr{V}, 0\}$. Then we replace $I_\lambda$ with the functional

Consequently, we obtain that $\mathscr{V}$ is a solution for the equation

Let $\mathscr{V}^-$ be a test function, and we get

Consequently, $\int_{\mathbb{R}^N}\mathcal V(x)\mathscr{S}(\mathscr{V})\mathscr{S}^\prime(\mathscr{V})\mathscr{V}^-dx = 0$, i.e., $\mathscr{V}\geq 0$. By Lemma 2.3 we find

and

From $\|\frac{\mathscr{S}(\mathscr{V}_n)}{\mathscr{S}^\prime(\mathscr{V}_n)}\|_E\leq 2\|\mathscr{V}_n\|_E$ we obtain

and

Let

Then $c_\lambda > 0$ implies that $\Lambda(\mathscr{V}_n)$ has a positive lower bound. Otherwise, $\Lambda(\mathscr{V}_n)\to 0(n\to \infty)$, and we have

This contradicts $c_\lambda > 0$. Therefore, we have

If $\mathscr{V}\equiv 0$, then

Moreover, Lemma 2.7 and $(3.2)$ imply that

From $c_\lambda > 0$ we have

i.e.,

Consequently,

and this has a contradiction. Hence, $\mathscr{V}\neq 0$, and by the maximum principle we have $\mathscr{V} > 0$. The proof is completed. □

4.   Conclusions

In this paper we use the dual approach and the mountain pass theorem to investigate the existence of positive solutions for the critical quasilinear Schrödinger equation (1.1) considering suitable conditions about nonlinearity $g$ and the potential $V$. It is interesting to notice that our work gives some weaker conditions than those in the cited works, and generalizes the corresponding ones in the literature.

Use of AI tools declaration

The authors declare they have not used Artificial Intelligence (AI) tools in the creation of this article.

Acknowledgments

The authors are grateful to the anonymous referee for carefully reading, valuable comments and suggestions to improve the earlier version of the paper.

This research was supported by Natural Science Foundation of Chongqing (grant No. cstc2020jcyj-msxmX0123), and Technology Research Foundation of Chongqing Educational Committee (grant No. KJQN202000528).

Conflict of interest

The authors declare no conflict of interest.