The existence of sign-changing solutions for Schrödinger-Kirchhoff problems in $\mathbb{R}^3$

1.   Introduction

In this paper, we study the existence of sign-changing solution to the following Kirchhoff equation by using a direct method

where $a$, $b > 0$, $p\in(1, 5)$. In recent years, problem (1.1) has been extensively researched by many mathematicians. Therefore, there are a large number of results for the existence of nontrivial solutions, positive solutions, ground state solutions, sign-changing solutions, nodal solutions for problem (1.1). Please see ^{[1,2,3,4,5,6]} and the references therein. It is worth noting that Chen, Fu and Wu [4] established the existence of a positive ground state solution to problem (1.1) for any $b > 0$ and $p \in (1, 5)$. However, there is a question: whether problem (1.1) has sign-changing solutions for any $p \in (1, 5)$?

Recently, Wang, Zhang and Cheng ^[7] established the existence results of sign-changing solutions to the following problem

where $f(t)$ satisfies the following crucial conditions:

(f1) $\lim_{t \to \infty}\frac{F(t)}{t^4} = \infty$, where $F(t) = \int_{0}^{t} f(s)d{s}$;

(f2) $\frac{f(t)}{t^3}$ is nondecreasing for $|t| > 0$.

Obviously, when $p \in (1, 3]$, $f(t) = |t|^{p-1}t$ does not satisfy (f1) and (f2). Qian ^[8] researched the existence of a ground state sign-changing solution to the following problem

where $a$ is a positive constant, $q \in (2, 2^*)$($2^* = +\infty$ for $N = 1, 2$, $2^* = \frac{2N}{N-2}$ for $N \ge 3$), $\Omega \subset \mathbb{R}^3$ is a bounded domain and $\lambda > 0$ is a parameter. They mainly obtained that problem (1.3) has at least one sign-changing solution for small enough $\lambda$, thanks to truncated technique and constraint variational method. Besides, some similar problems have also been extensively researched. For more relevant results, please refer to ^[9,10] and the references therein.

Motivated by the above mentioned results, our result is given in the following.

Theorem 1.1 For any $a$, $b > 0$ and $p \in (1, 5)$, problem (1.1) has at least one sign-changing solution.

Remark 1.2 When $p \in (3, 5)$, the existence of one sign-changing solution to (1.1) is obtained by ^[7]. But when $p \in (1, 3]$, it is difficult to prove the existence of sign-changing solutions. The main difficulty lies in proving the functional of problem (1.1) satisfies (PS)-conditions. To overcome this difficulty, we will apply some new tricks. Moreover, $f(t)\triangleq |t|^{p-1}t$ does not satisfy (f1)-(f2) when $p \in (1, 3]$. We must point out that our result holds for any $b > 0$. Therefore, our result can be seen as an improvement and extension of ^[7,8]. Our result can also extent to more general $f(u)$.

In this paper, we shall work on the space

with the inner product and norm

$L^q(\mathbb{R}^3)(1 \le q < \infty)$ denotes Lebesgue space with norm $\|u\|_q = \left(\int_{ \mathbb{R}^3}|u|^q dx\right)^{1/q}$. It is well known that the weak solution of problem (1.1) corresponds to the critical point of

Clearly, $I \in C^1(E, \mathbb{R})$ and we have

Setting $u^+ = \max\{u, 0\}$, $u^- = \min\{u, 0\}$, $A(u^+, u^-) = \frac{b}{2}\int_{ \mathbb{R}^3}|\nabla u^+|^2dx\int_{ \mathbb{R}^3}|\nabla u^-|^2dx$. To state our result, we establish the following minimization problem

where

Obviously, the set $\mathcal{M}$ is a subset of the following special manifold:

Remark 1.3 Clearly, the manifold $\mathcal{M}$ has not been used in the existing literature. The usual manifold $\mathcal{M}_1$ has been used in previous literature is a subset of manifold $\mathcal{N}_1$, where

As we all know, the manifold $\mathcal{N}_1$ is a commonly used manifold in the study of positive solutions. But the manifold $\mathcal{M}_1$ is not enough for us to prove our result when $p \in (1, 3]$. Thus, we need to find an another manifold. For researching positive solutions, one can also use a special manifold $\mathcal{N}$, which is a combination of the Nehari manifold and Pohožaev manifold for power $p \in (1, 5)$. In order to prove our result, we choose the manifold $\mathcal{M}$.

2.   Preliminaries

Comparing with the 4-superlinear condition in ^[7], we meet some new difficulties. We need to show that the constraint set $\mathcal{M}$ is nonempty and the minimizing sequence on $\mathcal{M}$ is a (PS)-sequence of $I$ in $E$ by using some new tricks.

Lemma 2.1 If $p \in (1, 5)$, then $\mathcal{M} \neq \varnothing$.

Proof. For any $u \in E$ and $u^{\pm} \neq 0$, we set $u_t \triangleq t^{\frac{1}{2}}u(\frac{x}{t})$. In the following, we shall prove that there are positive constants $s_1$ and $t_1$ such that

which implies that $u_{s_1}^{+}+u_{t_1}^{-} \in \mathcal{M}$. Actually, equation (2.1) holds if and only if

where

In the other words, we only need to show that there exists $m \in (0, M)$ such that

where $M$ is a positive constant. Since $p \in (1, 5)$, then $\frac{p+7}{2} > 4$. By (2.2), we can derive that $r(s, t) < 0$ as $s$ enough large, $r(s, t) > 0$ as $s$ enough small. And $l(s, t) < 0$ as $t$ enough large, $l(s, t) > 0$ as $t$ enough small. Consequently, (2.4)-(2.5) hold. Then from the Miranda's Theorem ^[11], there exist two positive constants $s_1$ and $t_1$ such that

Hence, (2.1) holds, which shows that $u_{s_1}^{+}+u_{t_1}^{-} \in \mathcal{M}$, i.e., $\mathcal{M} \neq \varnothing$. The proof is completed.

Lemma 2.2 The pair $(s_1, t_1)$ with positive numbers in Lemma 2.1 is unique.

Proof. In view of Lemma 2.1, there exists a pair $(s_1, t_1)$ such that $u_{s_1}^{+}+u_{t_1}^{-} \in \mathcal{M}$ for any $u \in E$ and $u^{\pm} \neq 0$. Next, we shall prove the uniqueness of $(s_1, t_1)$ by two steps.

Step 1. If $u \in \mathcal{M}$, then $(s_1, t_1) = (1, 1)$.

Since $u \in \mathcal{M}$, then we have

Assume that $s_1\le t_1$. By (2.2), we have

It follows from (2.7) and (2.8) that

If $s_1 < 1$, the negative right side of inequality (2.10) contradicts the positive left side. So $1 \le s_1 \le t_1$. Moveover, combining (2.7) and (2.9), $t_1 \le 1$ can be also obtained. Then $(s_1, t_1) = (1, 1)$.

Step 2. If $u \notin \mathcal{M}$, then there exists a unique $u_1$ such that $u_1^++u_1^- \in \mathcal{M}$.

Suppose that there is an another pair $(s_2, t_2)$ such that $u_{s_2}^{+}+u_{t_2}^{-} \in \mathcal{M}$. We set $v_1 \triangleq u_{s_1}^{+}+u_{t_1}^{-}$ and $v_2 \triangleq u_{s_2}^{+}+u_{t_2}^{-}$. By a simple calculation, we have

Thanks to $v_{2} \in \mathcal{M}$ and step 1, we deduce that $(s_1, t_1) = (s_2, t_2)$. The proof is completed.

Similar to ^[7], we can prove that $I(u_{s_{1}}^++u_{t_1}^{-}) = \max_{s, t\ge0}I(u_{s}^{+}+u_{t}^{-})$. From Lemma 2.2, we consider the minimization problem

Lemma 2.3 $c_{\mathcal{M}}$ is achieved.

Proof. For each $u \in \mathcal{M}$, we have $G(u)\triangleq \frac{1}{2}\langle I^{\prime}(u), u\rangle+P(u) = 0$. Then for any $p \in (1, 5)$, we have

That is $c_{\mathcal{M}} > 0$. Letting $\{u_n\} \subset \mathcal{M}$ such that $I(u_n) \to c_{\mathcal{M}}$. From (2.13), we know that $\{|\nabla u_n|_2\}$ is bounded in $E$. Since $G(u_n) = 0$, then

From Hölder and Sobolev inequalities, we have

where $\frac{1}{p+1} = \frac{\vartheta}{2}+\frac{1-\vartheta}{6}$. Then $(p+1)\vartheta < 2$. According to Young's inequality, we obtain that for any $\varepsilon > 0$, there exists $C_{\varepsilon} > 0$ such that

Set $\varepsilon = 1$, from (2.14) and (2.16), we have that $\{\|u_n\|_{2}\}$ is bounded. Hence, $\{u_n\}$ is bounded. Then, there exists $u$ such that $u_{n}^{\pm} \rightharpoonup u^{\pm}$ in $E$. From (2.13), we can find a constant $\theta$ such that $\|u_{n}^{\pm}\| > \theta > 0$ for every $n \in \mathbb{N}$.

Since $\{u_n\} \subset \mathcal{M}$, we have that

Therefore, we have

Then $\int_{ \mathbb{R}^3}|u_{n}^{\pm}|^{p+1}dx > \frac{\theta^2}{C_1} > 0$. Since the embedding $E \hookrightarrow L^q(\mathbb{R}^3)$ is compact for $2 < q < 6$, (2.18) shows that $u^{\pm} \neq 0$. Combining the compactness lemma of Strauss ^[11] and the weak semicontinuity of norm, we obtain

and

Then from (2.17) and (2.19)–(2.21), we have that

Thus, there exists $(s_u, t_u)$ such that $u_{s_u}^{+}+u_{t_u}^{-} \in \mathcal{M}$. Suppose that $0 < t_u\le s_u$, then we obtain

From (2.19) and (2.22), we have

By (2.23) and (2.24), we obtain

which shows $s_u \le 1$. Then $0 < t_u \le s_u \le 1$. Setting $\bar{u} = u_{s_u}^{+}+u_{t_u}^{-}$. Therefore, we can deduce that

(2.25) implies that $s_u = t_u = 1$. That is $u = \bar{u}$ and $I(u) = c_{\mathcal{M}}$. The proof is completed.

3.   Proof of Theorem 1.1

Lemma 3.1. Assume $c_{\mathcal{M}}$ attained in $\mathcal{M}$, then $u$ is a critical point of $I$.

Proof. Since $u \in \mathcal{M}$, $u^{\pm} \neq 0$. Then for any fixed $v \in H^{1}(\mathbb{R}^3)$, there exists $\varepsilon > 0$ such that $(u+wv)^{\pm} \neq 0$ for all $w\in (-\varepsilon, \varepsilon)$. Arguing by a contradiction, there is a sequence $\{w_i\}_{i = 1}^{\infty}$ such that

Letting $i \to \infty$, we have $u = 0$ a.e. on $\mathbb{R}^3$. Which is a contradiction with $u^{\pm} \neq 0$.

From Lemma 2.1, there exists a unique pair $(s(w), t(w))$ such that $s(w)(u+wv)^++t(w)(u+wv)^- \in \mathcal{M}$. Next, we prove some standard properties of $(s(w), t(w))$ as Nehari manifold. For our purpose, we consider the function

defined for $(s, t, w)\in (0, +\infty)\times (0, +\infty) \times (-\varepsilon, \varepsilon)$. Since $u \in \mathcal{M}$, we have $\varphi(1, 1, 0) = 0$. Moveover, $\varphi$ is a $C^1$ function and

Similarly, we can deduce that $\frac{\partial \varphi(s, t, w)}{\partial t}|_{(s, t, w) = (1, 1, 0)} \le 0$. From the Implicit Function Theorem, the functions $s(w)$, $t(w)$ are $C^1$. And $t(0) = s(0) = 1$. Moveover, $s(w)$, $t(w) \neq 0$ near 0. We define

Then we obtain that $\Upsilon$ is differentiable for small $w$ and attains its minimum at $w = 0$. Hence, we derive that

Since $v \in E$ is arbitrary, we have that $I^{\prime}(u) = 0$.

Proof of Theorem 1.1. From Lemma 2.3 and 3.1, there is a $u \in \mathcal{M}$ such that $I(u) = c_{\mathcal{M}}$ and $I^{\prime}(u) = 0$. Then problem (1.1) has at least one sign-changing solution. The proof is completed.

Acknowledgments

The authors would like to thank the editors and referees for their useful suggestions which have significantly improved the paper. This work is supported by the National Natural Science Foundation of China (No. 11961014, No. 61563013) and Guangxi Natural Science Foundation (2016GXNSFAA380082, 2018GXNSFAA281021).

Conflict of interest

The authors declare that they have no competing interests.