## Differential and Integral Equations

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

Alessio Fiscella

#### 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.

#### Article information

Source
Differential Integral Equations Volume 29, Number 5/6 (2016), 513-530.

Dates
First available in Project Euclid: 9 March 2016