Sign-changing solutions for a class of fractional Kirchhoff-type problem with logarithmic nonlinearity

1.   Introduction

In this paper, we consider the following fractional Kirchhoff-Schrödinger-type problem with logarithmic nonlinearity

where $\Omega \subset \mathbb{R}^N$ is a smooth bounded domain, $N > 2s$ ($0 < s < 1$), $(-\Delta)^s$ is the fractional Laplacian, defined for any $u \in \mathcal{C}_c^{\infty}(\mathbb{R}^N)$ by

$a, b > 0$ are constants, $4 < p < 2_s^* : = \frac{2N}{N-2s}$, and $V, Q : \Omega \to \mathbb{R}$ satisfy

(H) $V, Q \in \mathcal{C}(\Omega, [0, \infty))$, and $V, Q \not = 0.$

We know that logarithmic nonlinearities have many applications in quantum optics, quantum mechanics, transport, nuclear physics and diffusion phenomena etc (see ^[1] and the reference therein). Recently, many authors have investigated the following logarithmic Schrödinger equation

Many results about logarithmic Schrödinger equation like (1.2) have been obtained, see ^{[2,3,4,5,6,7]} and reference therein. In ^[8], Chen and Tang studied the ground state sign-changing solutions to elliptic equations with logarithmic nonlinearity of (1.2). The fractional Kirchhoff equation was first introduced in ^[9]. Recently, Li, Wang and Zhang ^[10] considered the existence of ground state sign-changing solutions for following $p$-Laplacian Kirchhoff-type problem with logarithmic nonlinearity

We refer to ^[11,12] for a study of existence of sign-changing solutions to (1.2), or more general problems like (1.2) with a logarithmic nonlinearity. Variational methods for non-local operators of elliptic type was first introduced by Fiscela and Valdinoci in ^[13]. In these years, nonlinear problems involving nonlocal operator have been extent studied, see for instance ^{[14,15,16,17,18,19,20,21,22]} and the references therein. However, to the best of our knowledge, there seem no results on sign-changing solutions for logarithmic fractional Kirchhoff-type problem.

Motivated and inspired by ^[8,10] and the aforementioned works, in this paper, we investigate the existence of sign-changing solutions to logarithmic fractional Kirchhoff-type problem (1.1). The main results we get are based on constraint variational method, some analysis techniques and a quantitative deformation lemma. Our result extends the theorem of Chen and Tang ^[8] from elliptic equations with logarithmic nonlinearity to fractional Kirchhoff-type problem with logarithmic nonlinearity. This article is organized as follows. In Section 2, we give some notations and preliminaries. Section 3 is devoted to the proof of our main result.

2.   Preliminaries

For any $s \in (0, 1)$, we define $W^{s, 2}(\Omega)$ as a linear space of Lebesgue measurable functions from $\mathbb{R}^N$ to $\mathbb{R}$ such that the restriction to $\Omega$ of any function $u$ in $W^{s, 2}(\Omega)$ belongs to $L^p(\Omega)$ and

Equip $W^{s, 2}(\Omega)$ with the norm

Then $W^{s, 2}(\Omega)$ is a Banach space. The space $W_0^{s, 2}(\Omega) = \{ u \in W^{s, 2}(\Omega) : u = 0 \ {\rm in} \ \mathbb{R}^N \setminus \Omega\}$ endowed with the norm

Let

endowed with the norm

Now, we define the energy functional $\mathcal{J} : E \to \mathbb{R}$ associated with problem (1.1) by

For each $q \in (p, 2_s^*)$, one has that

Then for any $\varepsilon > 0$, there exists $C_{\varepsilon} > 0$ such that

By (2.2), we know that $\mathcal{J}$ is well defined and $\mathcal{J} \in C^1(E, \mathbb{R})$ with

Obviously, if $u \in E$ is a critical point of $\mathcal{J}$, then $u$ is a weak solution of (1.1).

If $u \in E$ is a solution of (1.1) and $u^{\pm} \not = 0$, then $u$ is a sign-changing solution of (1.1), where

The Nehari manifold for $\mathcal{J}$ is defined as

Moreover, we define the nodal set

Lemma 2.1. The following inequalities hold :

(1). ${ 2(1-x^p) + p x^p \ln x^2 \ge 0, \quad \forall x \in [0, 1) \cup (1, +\infty), \quad p > 2}$;

(2). ${\frac{1-x^2}{2} - \frac{1-x^p}{p} > 0, \quad \forall x \in [0, 1) \cup (1, +\infty), \quad p > 2}$;

(3). ${ 1- xy - \frac{2-x^p-y^p}{p} \ge 0, \quad \forall x, y \ge 0, \quad p > 2 }$;

(4). ${\frac{1-x^4}{4} - \frac{1-x^p}{p} \ge 0, \quad \forall x \ge 0, \quad p > 4 }$;

(5). ${ \frac{1- x^2 y^2}{2} - \frac{2-x^p-y^p}{p} \ge 0, \quad \forall x, y \ge 0, \quad p > 4 }$;

(6). ${ 1- x^3 y - \frac{4-3 x^p-y^p}{p} \ge 0, \quad \forall x, y \ge 0, \quad p > 4 }$.

Proof. Here we only prove (6) holds, the proof of other cases are similar, we can omit it. Let

The critical points of $f$ must satisfy the system of equations :

Hence, the critical points of $f$ are $(0, 0)$ and $(1, 1)$. Since $A = f_{11}(1, 1) = 3(p-3) > 0$, $B = f_{12}(1, 1) = -3$, $C = f_{22}(1, 1) = p-1$, and $B^2 - AC = 9 - 3(p-3)(p-1) < 0$, which implies that $f$ has a local minimum value at $(1, 1)$, and $f(1, 1) = 0$. Obviously, $f(0, 0) = 1 - \frac{4}{p} > 0$. So, for any $x, y \ge 0$, we have that $f(x, y) \ge \min f(x, y) = f(1, 1) = 0$.

Lemma 2.2. For each $u \in E$ and $\alpha, \beta \ge 0$, we have

Proof. From (2.3) in ^[8], one has

By a direct calculation, we easily obtain that

and

Thus, it follows from (2.5)–(2.8), Lemma 2.1 and $u^+(x) u^-(y) + u^+(y) u^-(x) \le 0$ that

which implies that (2.4) holds for all $u \in E$ and $\alpha, \beta \ge 0$.

According to Lemma 2.2, we have the following corollaries.

Corollary 2.3. For each $u \in E$ and $t \ge 0$, we get that

Corollary 2.4. For each $u \in \mathcal{M}$, there holds

Corollary 2.5. For each $u \in \mathcal{N}$, we have that

Lemma 2.6. Let $4 < p < 2_s^*$. For each $u \in E$, we have

(i) If $u \not = 0$, there exists a unique $t_u > 0$ such that $t_u u \in \mathcal{N}$;

(ii) If $u^{\pm} \not = 0$, there exists a unique pair $(\alpha_u, \beta_u)$ of positive numbers such that $\alpha_u u^+ + \beta_u u^- \in \mathcal{M}$.

Proof. (ⅰ) For any $u \in E \setminus \{0\}$, set

From (2.2), $p > 4$ and (2.9), it is easy to see that $\lim_{t \to 0^+} f_u(t) = 0$, $f_u(t) > 0$ for $t > 0$ small and $f_u(t) < 0$ for $t$ large. Thanks to the continuity of $f_u(t)$, there is $t_u > 0$ such that $f_u(t) = 0$. In the following, we prove that $t_u$ is unique. Arguing by contradiction, we assume that there exist two positive constants $t_1 \not = t_2$ such that $f_u(t_1) = f_u(t_2) = 0$, that is $t_1 u, t_2 u \in \mathcal{N}$. By Corollary 2.3 and Lemma 2.1 (2), we get

and

which is absurd. Thus, $t_u > 0$ is unique.

(ⅱ) For each $u \in E$ with $u^{\pm} \not = 0$, in view of Lemma 2.6 (ⅰ), there exists a pair $(\alpha_u, \beta_u)$ of positive numbers such that $\alpha_u u^+, \beta_u u^- \in \mathcal{N}$. Let

and

Since $4 < p < 2_s^*$, it follows from (2.2) that

So, there exist $0 < t_1 < t_2$ such that

Combining (2.10), (2.11) with (2.12), we obtain that

and

Hence, thanks to (2.13), (2.14) and Miranda's Theorem ^[23], there exists some pair $(\alpha_u, \beta_u)$ with $t_1 < \alpha_u, \beta_u < t_2$ such that

These show that $\alpha_u u^+ + \beta_u u^- \in \mathcal{M}$. The proof of unique of $(\alpha_u, \beta_u)$ is similar to that of (ⅰ), we omit detail here.

From Corollaries 2.4, 2.5, and Lemma 2.6, we can deduce the following lemma.

Lemma 2.7. The following minimax characterization hold

and

Lemma 2.8. $c > 0$ and $m > 0$ are achieved.

Proof. We only prove that $m > 0$ and is achieved since the other case is similar. For each $u \in \mathcal{M}$, one has $\langle \mathcal{J}^{\prime}(u), u \rangle = 0$ and then by (2.2) and fractional Sobolev embedding theorem, there exists a constant $C _1 > 0$ such that

Since $q > p > 4$, by (2.15), there exists a constant $\rho > 0$ such that $[u] \ge \rho$ for each $u \in \mathcal{M}$.

Let $\{u_n\} \subset \mathcal{M}$ be such that $\mathcal{J}(u_n) \to m$. From (2.1) and (2.3), we have

which implies that $\{u_n\}$ is bounded. Thus, there exists $u_*$, in subsequence sense, such that $u_n^{\pm} \rightharpoonup u_*^{\pm}$ in $E$ and $u_n^{\pm} \to u_*^{\pm}$ in $L^r(\Omega)$ for $2 \le r < 2_s^*$. Since $\{u_n\} \subset \mathcal{M}$, we have $\langle \mathcal{J}^{\prime}(u_n), u_n^{\pm} \rangle = 0$, which yields that

By the compactness of the embedding $W_0^{s, 2}(\Omega) \hookrightarrow L^r(\Omega)$, we obtain

which implies $u_*^{\pm} \not = 0$. By the Lebesgue dominated convergence theorem and the weak semicontinuity of norm, one has

which yields that

In view of Lemma 2.6 (ⅱ), there exist constants $\alpha, \beta > 0$ such that $\alpha u_*^+ + \beta u_*^- \in \mathcal{M}$. Thus, from (2.1), (2.3), (2.4), Lemma 2.1 and the weak semicontinuity of norm, we obtain that

which shows

Moreover, it follows from $u_*^{\pm} \not = 0$, $\langle \mathcal{J}^{\prime}(u_*), u_* \rangle = 0$ and (2.6) that

3.   Main result

In this section, we will give the main result and proof.

Lemma 3.1. The minimizers of $\inf_{\mathcal{N}} \mathcal{J}$ and $\inf_{\mathcal{M}} \mathcal{J}$ are critical points of $\mathcal{J}$.

Proof. Thanks to Lemma 2.8, we prove the minimizer $u_*$ of $\inf_{\mathcal{M}} \mathcal{J}$ is critical point of $\mathcal{J}$. Arguing by contradiction, we assume that $u_* = u_*^+ + u_*^- \in \mathcal{M}$, $\mathcal{J}(u_*) = m$ and $\mathcal{J}^{\prime}(u_*) \not = 0$. Then there exist $\delta > 0$ and $\gamma > 0$ such that

Set $D = \left(\frac{1}{2}, \frac{3}{2}\right) \times \left(\frac{1}{2}, \frac{3}{2}\right)$. By Lemma 2.2, one has

Let $\varepsilon : = \min \{(m-\varrho)/3, \delta \gamma/8)$ and $S_{\delta} : = B(u_*, \delta)$. By applying the Lemma 2.3 in Ref. ^[24], there exists a deformation $\eta \in C([0, 1] \times E, E)$ such that

(ⅰ) $\eta(1, \nu) = \nu$ if $\nu \notin \mathcal{J}^{-1}([m-2\varepsilon, m+2\varepsilon]) \cap S_{2\delta}$;

(ⅱ) $\eta(1, \mathcal{J}^{m+\varepsilon} \cap S_{\delta}) \subset \mathcal{J}^{m-\varepsilon}$;

(ⅲ) $\mathcal{J}(\eta(1, \nu)) \le \mathcal{J}(\nu)$, $\forall \nu \in E$.

From (ⅲ) and Lemma 2.2, for each $\alpha, \beta > 0$ with $|\alpha-1|^2 + |\beta-1|^2 \ge \delta^2/\|u_*\|^2$, one has

By Corollary 2.4, we have $\mathcal{J}(\alpha u_*^+ + \beta u_*^-) \le \mathcal{J}(u_*) = m$ for $\alpha, \beta > 0$. According to (ii), one has

Thus, from (3.1) and (3.2), we obtain

Let $h(\alpha, \beta) = \alpha u_*^+ + \beta u_*^-$, we will prove that $\eta(1, h(D)) \cap \mathcal{J} \not = \emptyset$.

Define

Obviously, $\Phi$ is a $C^1$ functions. Moreover, we have by a direct calculation that

and

Similarly, we obtain

and

It is easy to check that

Thus, by degree theory ^[25,26], we can derive that $\Psi(\alpha_0, \beta_0) = 0$ for some $(\alpha_0, \beta_0) \in D$, so that $\eta(1, h(\alpha_0, \beta_0)) = k(\alpha_0, \beta_0) \in \mathcal{M}$. This contradicts (3.3) and shows that $\mathcal{J}^{\prime}(u_*) = 0$. Similarly, we can prove that any minimizer of $\inf_{\mathcal{N}} \mathcal{J}$ is a critical point of $\mathcal{J}$.

Now, we are in a position to prove our main result.

Theorem 3.2. Suppose that condition (H) holds. If $4 < p < 2_s^*$, then problem (1.1) has a solution $u_0 \in \mathcal{N}$ and a sign-changing solution $u_* \in \mathcal{M}$ such that

Proof. By Lemmas 2.8 and 3.1, there exist $u_0 \in \mathcal{N}$ and $u_* \in \mathcal{M}$ such that $\mathcal{J}(u_0) = c$ with $\mathcal{J}^{\prime}(u_0) = 0$, and $\mathcal{J}(u_*) = m$ with $\mathcal{J}^{\prime}(u_*) = 0$. That is, problem (1.1) has a solution $u_0 \in \mathcal{N}$ and a sign-changing solution $u_* \in \mathcal{M}$. Moreover, by (2.5)–(2.7), Corollary 2.4 and Lemma 2.7, we get

Remark 3.3. In ^[8,10], (1.2) and (1.3) has a sign-changing solution with precisely two nodal domains has been proved respectively. By Theorem 3.2, we know that (1.1) has a sign-changing solution. But according to the method is used in ^[8,10], we cannot prove that the sign-changing solution of (1.1) has precisely two nodal domains.

Acknowledgments

The authors thanks editor and anonymous referees for their remarkable comments, suggestion that help to improve this paper. This work is supported by Natural Science Foundation of China (11571136).

Conflict of interest

The authors declare that there are no conflicts of interest in this paper.