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

1.   Introduction

In this article, we are consider the existence of the ground state sign-changing solution for the following $p$-Laplacian Kirchhoff-type equation

where $\Omega\subset \mathbb{R}^{N}$ is a smooth bounded domain, $a, b > 0$, $4\leq2p < q < p^{*}$ and $N > p$, $\Delta_p$ denote the $p$-Laplacian operator defined by $\Delta_p u = div(|\nabla u|^{p-2}\cdot\nabla u)$.

Problem (1.1) stem from following Kirchhoff equations

where $\Omega\subset \mathbb{R}^{N}$ is a bounded domain or $\Omega = \mathbb{R}^{N}$, $a > 0, b > 0$ and $u$ satisfies some boundary conditions.

Problem (1.2) is related to the following stationary analogue of the equation of Kirchhoff type

which was introduced by Kirchhoff ^[1] as a generalization of the well-known D'Alembert wave equation

for free vibration of elastic strings.

After the pioneer work of Lions ^[2], where a functional analysis approach was proposed to (1.3) with Dirichlet boundary condition, a lot of interesting results for (1.2) or similar problems are obtained in last decades.

Recently, many authors pay their attentions to find sign-changing solutions to problem (1.2) or similar equations, and indeed some interesting results were obtained, see for examples, ^{[3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28]} and the references therein.

On the other hand, the problem (1.1) derive from the following Logarithmic Schrödinger equation

Recently, there are many results about Logarithmic Schrödinger equation like (1.5), see ^{[26,29,30,31,32,33,34,35,36,37]} and references therein. Moreover, some scholars considered sign-changing solutions to Logarithmic Schrödinger equation like (1.5) ^[26,33].

Motivated by the works mentioned above, especially by ^[9,10,26], in the present paper, we consider the existence of ground state sign-changing solutions for problem (1.1).

Denote ${W}^{1, p}_{0}(\Omega)$ the usual Sobolev space equipped with the norm

The usual $L^p(\Omega)$ norm is denote by $|u|_{p}^p = \int_{\Omega}|u|^{p}dx$. And we define the energy functional of problem (1.1) as follow:

for any $u\in {W}^{1, p}_{0}(\Omega)$.

Moreover, under our conditions, $\Phi(u)$ belongs to $C^{1}$, and the Fréchet derivative of $\Phi$ is

for any $u, v\in {W}^{1, p}_{0}(\Omega)$.

The solution of problem (1.1) is the critical point of the functional $\Phi(u)$. Furthermore, if $u\in {W}^{1, p}_{0}(\Omega)$ is a solution of problem (1.1) and $u^{\pm}\neq0$, then $u$ is a sign-changing solution of problem (1.1), where

It is noticed that

The main results can be stated as follows.

Theorem 1.1. The problem (1.1) has a sign-changing $u_{0}\in\mathcal{M}$ with precisely two nodal domains such that $\Phi(u_{0}) = inf_{\mathcal{M}}\Phi: = m,$ where

Theorem 1.2. The problem (1.1) has a solutions $u_{0}\in\mathcal{N}$ such that $\Phi(u_{0}) = inf_{\mathcal{N}}\Phi: = c,$ where $\mathcal{N} = \{u\in {W}^{1, p}_{0}(\Omega), u\neq 0,$ and $\langle \Phi'(u), u\rangle = 0\}$. Moreover, $m\geq2c$.

Remark 1.1. The Kirchhoff function $M(t) = a+bt^{n} (n > 0, a > 0, b\geq0)$ can be regarded as the special case of the function $M:\mathbb{R}^{+}\rightarrow\mathbb{R}^{+}$ satisfying the following conditions:

$(M1)$ $M\in C(\mathbb{R}^{+})$ satisfies $\inf_{t\in\mathbb{R}^{+}}M(t)\geq m_{0} > 0$, where $m_{0}$ is a constant;

$(M2)$ There exists $\theta\geq1$ such that $\theta \mathbb{M} = \theta\int_{0}^{t}M(\tau)d\tau\geq M(t)t$ for any $t\geq0.$

In this paper, we consider this problem under condition $M(t) = a+bt.$ From details in proof of Lemma 2.1 (see later), we can not obtain result similar as Lemma 2.1 for the case of Kirchhoff equations of the general forms. However, Lemma 2.1 play an important role in proof of main resuls. So, for the case of Kirchhoff equations of the general forms, the methods in this paper seems not valid.

2.   Technical lemmas

Lemma 2.1. For all $u\in {W}^{1, p}_{0}(\Omega)$ and $s, t\geq 0$ there holds

Proof. Since (2.1) holds when $u = 0$, in the following, we always assume that $u\neq0$. It is easy to see that

Let $\Omega^{+} = \{x\in\Omega:u(x) > 0\}$ and $\Omega^{-} = \{x\in\Omega:u(x) < 0\}$. For any $u\in {W}^{1, p}_{0}(\Omega)\setminus\{0\}$, we have that

Combining (2.2) with (2.3), we obtain

which implies that (2.2) holds.

According to Lemma 2.1, we can obtain the following corollaries.

Corollary 2.1. For all $u\in {W}^{1, p}_{0}(\Omega)$ and $t\geq 0$, we have that

Corollary 2.2. For any $u\in \mathcal{M}$, we have that

Corollary 2.3. For any $u\in \mathcal{N}$, we have that

Lemma 2.2. For any $u\in W^{1, p}_{0}(\Omega)$ with $u^{\pm}\neq 0$, there exists an unique pair $(s_{u}, t_{u})$ of positive numbers such that $s_{u}u^{+}+t_{u}u^{-}\in\mathcal{M}$.

Proof. Firstly, for any $u\in W^{1, p}_{0}(\Omega)$ with $u^{\pm}\neq 0$, we prove the existence of $(s_{u}, t_{u})$. Let

and

From assumptions, we have that

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

Since $4\leq 2p < q < p^{*}$, it follows from (2.8) that

Thus, there exist $0 < \alpha < \beta$ such that

Thanks to (2.6), (2.7) and (2.9), we have that

and

So, combining (2.10), (2.11) with Miranda's Theorem ^[38], there exists some point $(s_{u}, t_{u})$ with $\alpha < s_{u}, t_{u} < \beta$ such that

That is, there exists a pair $(s_{u}, t_{u})$ of positive numbers such that $s_{u}u^{+}+t_{u}u^{-}\in\mathcal{M}$.

Secondly, we prove that $(s_{u}, t_{u})$ is unique.

Arguing by contradiction, we assume that there exist two pair $(s_{i}, t_{i})$, $i = 1, 2$ such that $s_{1}u^{+}+t_{1}u^{-}\in\mathcal{M}$ and $s_{2}u^{+}+t_{2}u^{-}\in\mathcal{M}$.

According to corollary 2.2, we have that

It is noticed that

Therefore, by (2.12) and (2.13), we have that $(s_{1}, t_{1}) = (s_{2}, t_{2}).$ That is, $(s_{u}, t_{u})$ is unique.

Lemma 2.3. For any $u\in {W}^{1, p}_{0}(\Omega)$ with $u\neq0$, there exists an unique $t_{u} > 0$ such that $t_{u}u\in\mathcal{N}$.

Proof. Since the proof is similar to that of Lemma 2.2, we omit detail here.

Through the standard discussions, we have following result.

Lemma 2.4. The following minimax characterization hold

Lemma 2.5. $m > 0$ is achieved.

Proof. For any $u\subset\mathcal{M}$, we have $\langle \Phi'(u), u\rangle = 0$ and then

Since $q, r > p$, there exists a constant $\rho > 0$ such that $\|u\|\geq \rho$ for any $u\subset\mathcal{M}$.

Let ${u_{n}}\subset\mathcal{M}$ be such that $\Phi(u_{n})\rightarrow m$, then ${u_{n}}$ is bounded in $W^{1, p}_{0}$. Thus, there exists $u_{0}$, in subsequence sense, such that

Since $u_{n}\subset\mathcal{M}$, one has $\langle \Phi'(u_{n}), u_{n}^{\pm}\rangle = 0$, that is

By the compactness of the embedding $W^{1, p}_{0}(\Omega)\hookrightarrow L^{r}(\Omega)$, we get

which implies $u_{0}^{\pm}\neq0$.

By the Lebesgue dominated convergence theorem and the weak semicontinuity of norm, we have

that is,

Since $u_{0}^{\pm}\neq0$, it follows from Lemma 2.2 that there exist constants $s, t > 0$ such that $su_{0}^{+}+tu_{0}^{-}\in\mathcal{M}$.

From corollary 2.2, corollary 2.3 and the weak semicontinuity of norm, we have that

which asserts

Furthermore, thanks to $u_{0}^{\pm}\neq0$ and (2.1), we have that

3.   Proof main results

$\mathbf{Proof\; of\; Theorem\; 1.1:}$

Proof. Firstly, thanks to Lemma 2.5, we prove the minimizer $u_{0}$ of $\inf_{\mathcal{M}}\Phi$ is critical point of $\Phi$.

Arguing by contradiction, we assume that $\Phi'(u_{0})\neq0$. Then there exist $\delta > 0$ and $\varsigma > 0$ such that $\|(\Phi'(u)\|\geq\varsigma$, for all $\|u-u_{0}\|\leq3\delta$ and $u\in {W}^{1, p}_{0}(\Omega)$.

Let $D: = (\frac{1}{2}, \frac{3}{2})\times(\frac{1}{2}, \frac{3}{2})$, by Lemma 2.1, one has

For $\varepsilon: = \min{(m-\epsilon)/3}, \delta\varsigma/8\}$ and $S_{\delta}: = B(u_{0}, \delta)$, according to Lemma 2.3 in ^[39], there exists a deformation $\eta\in C([0, 1]\times {W}^{1, p}_{0}(\Omega), {W}^{1, p}_{0}(\Omega))$ such that

$(a)$ $\eta(1, v) = v$ if $v \notin\Phi^{-1}([m-2\varepsilon, m+2\varepsilon])\cap S_{2\delta}$;

$(b)$ $\eta(1, (\Phi^{m+\varepsilon}\cap S_{\delta})\subset \Phi^{m-\varepsilon}$, where $\Phi^{c} = \{u\in {W}^{1, p}_{0}(\Omega):\Phi(u)\leq c\}$;

$(c)$ $\Phi(\eta(1, v))\leq \Phi(v)$ for all $v\in {W}^{1, p}_{0}(\Omega)$.

By Lemma 2.1 and $(c)$, we have

On the other hand, by Corollary 2.3, we can obtain that $\Phi(su_{0}^{+}+tu_{0}^{-})\leq \Phi(u_{0}) = m$ for $s, t > 0$. Then it follows from $(b)$ that

So, thanks to (3.2) and (3.3), one has

Let $k(s, t) = su_{0}^{+}+tu_{0}^{-}$, we now prove that $\eta(1, k(D))\cap \mathcal{M}\neq\varnothing$. Let $\gamma(s, t): = \eta(1, k(s, t))$

and

Clearly, $\Psi_{0}$ is a $C^{1}$ functions and by a direct calculation, we have

Similarly, we have

Let

then we have that

Therefore, by topological degree theory ^[40,41,42], we conclude that $\Psi_{1}(s_{0}, t_{0}) = 0$ for some $(s_{0}, t_{0})\in D$, so that $\eta(1, k(s_{0}, t_{0})) = \gamma(s_{0}, t_{0})\in \mathcal{M}$, which is contradicted to (3.4).

Next, we prove $u_{0}$ has two nodal domains.

We assume that

where

are two connected open subsets of $\Omega$.

Setting $v: = u_{1}+u_{2}$, we see that $v^{+} = u_{1}$ and $v^{-} = u_{2}$, i.e., $v^{\pm}\neq0$. According to $\langle \Phi'(u_{0}), v^{\pm}\rangle = 0$, we have that

Thanks to (2.1) and (3.5), we have that

which implies $u_{3} = 0$. That is, $u_{0}$ has two nodal domains.

$\mathbf{Proof\; of\; Theorem\; 1.2:}$

Proof. Similar as the proof of Lemma 2.5, there exists $v_{0}\in \mathcal{N}$ such that $\Phi(v_{0}) = c > 0$. By arguments similar to that of Theorem 1.1, the critical points of the functional $\Phi$ on $\mathcal{N}$ are critical points of $\Phi$ in $W^{1, p}_{0}(\Omega)$ and we obtain $(\Phi)'(v_{0}) = 0$. It follows from definitions of $c = inf_{\mathcal{N}}\Phi$ and $\mathcal{N} = \{u\in {W}^{1, p}_{0}(\Omega), u\neq 0,$ and $\langle \Phi'(u), u\rangle = 0\}$ that $v_{0}$ is a ground state solution of (1.1).

According to Theorem 1.1, there exists a $u_{0}\in\mathcal{M}$ such that $\Phi(u_{0}) = m, \Phi'(u_{0}) = 0$. Therefore, by Corollary 2.2 and Lemma 2.4, we have that

The proof of Theorem 2.2 is completed.

4.   Conclusions

In this paper, by using constraint variational method, topological degree theory and the quantitative deformation lemma, we prove the existence of ground state sign-changing solutions with precisely two nodal domains for a class of $p$-Laplacian Kirchhoff-type problem (special form) with logarithmic nonlinearity. However, for the case of Kirchhoff equations of the general forms, the methods in this paper seems not valid. So, we will continuous discuss sign-changing solutions for general Kirchhoff equations with logarithmic nonlinearity in the follow-up work.

Acknowledgments

This research was supported by the Natural Science Foundation of China (No.11561043, 11961043). Authors are grateful to the reviewers and editors for their suggestions and comments to improve the manuscript.

Conflict of interest

The authors declare no conflict of interest in this paper.