Infinitely many solutions for a critical Kirchhoff type problem involving a fractional operator

Abstract

In this paper, we deal with a Kirchhoff type problem driven by a nonlocal fractional integrodifferential operator $\mathcal L_K$, that is, $$ -M( \|u \|^2)\mathcal L_Ku=\lambda f(x,u) \Big [\int_\Omega F(x,u(x))dx \Big ]^r+ \left |u \right |^{2^* -2}u \quad \mbox{in }\Omega, $$ $u=0$ in $ \mathbb{R}^{n}\setminus\Omega, $ where $\Omega$ is an open bounded subset of $\mathbb{R}^n$, $M$ and $f$ are continuous functions, $ \left \|\cdot \right \|$ is a functional norm, $$ F(x,u(x))=\int^{u(x)}_0 f(x,\tau)d\tau , $$ $2^*$ is a fractional Sobolev exponent, $\lambda$ and $r$ are real parameters. For this problem, we prove the existence of infinitely many solutions, through a suitable truncation argument and exploiting the genus theory introduced by Krasnoselskii.