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

Abstract

In this paper, we are interested the following fractional Kirchhoff-type problem with logarithmic nonlinearity $ \left\{ \begin{array} {ll} \left(a+b \iint_{\Omega^2} \frac{|u(x)-u(y)|^2}{|x-y|^{N+2s}} dxdy\right)(-\Delta)^s u + V(x)u = Q(x) |u|^{p-2}u \ln u^2, & {\rm in } \ \Omega, \\ u = 0, & {\rm in } \ \mathbb{R}^N \setminus \Omega, \end{array} \right. $ where $\Omega \subset \mathbb{R}^N$ is a smooth bounded domain, $N > 2s$ ($0 < s < 1$), $(-\Delta)^s$ is the fractional Laplacian, $V, Q$ are continuous, $V, Q \ge 0$. $a, b > 0$ are constants, $4 < p < 2_s^* : = \frac{2N}{N-2s}$. By using constraint variational method, a quantitative deformation lemma and some analysis techniques, we obtain the existence of ground state sign-changing solutions for above problem.