Existence of Ground State Sign-Changing Solutions of Fractional Kirchhoff-Type Equation with Critical Growth

Abstract

In this paper, we study the following fractional Kirchhoff-type equation $$\begin{aligned}{\left\{ \begin{array}{ll} -(a+ b\int _{{\mathbb {R}}^{N}}\int _{{\mathbb {R}}^{N}}|u(x)-u(y)|^{2}K(x-y)dxdy){\mathcal {L}}_{K}u=|u|^{2_{\alpha }^{*}-2}u+\mu f(u), ~\ x\in \Omega ,\\ u=0, ~\ x\in {\mathbb {R}}^{N}\backslash \Omega , \end{array}\right. } \end{aligned}$$ where $$\Omega \subset {\mathbb {R}}^{N}$$ is a bounded domain with a smooth boundary, $$\alpha \in (0,1)$$ , $$2\alpha<N<4\alpha $$ , $$2_{\alpha }^{*}$$ is the fractional critical Sobolev exponent and $$\mu , a, b>0$$ ; $${\mathcal {L}}_{K}$$ is non-local integrodifferential operator. Under suitable conditions on f, for $$\mu $$ large enough, by using constraint variational method and the quantitative deformation lemma, we obtain a ground state sign-changing (or nodal) solution to this problem, and its energy is strictly larger than twice that of the ground state solutions.