## Differential and Integral Equations

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

### 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.

#### Article information

**Source**

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

**Dates**

First available in Project Euclid: 9 March 2016

**Permanent link to this document**

https://projecteuclid.org/euclid.die/1457536889

**Mathematical Reviews number (MathSciNet)**

MR3471971

**Zentralblatt MATH identifier**

1363.35118

**Subjects**

Primary: 35J60: Nonlinear elliptic equations 35R11: Fractional partial differential equations 49J35: Minimax problems 35A15: Variational methods 45G05: Singular nonlinear integral equations 35S15: Boundary value problems for pseudodifferential operators

#### Citation

Fiscella, Alessio. Infinitely many solutions for a critical Kirchhoff type problem involving a fractional operator. Differential Integral Equations 29 (2016), no. 5/6, 513--530. https://projecteuclid.org/euclid.die/1457536889