## Differential and Integral Equations

- Differential Integral Equations
- Volume 30, Number 9/10 (2017), 641-666.

### On multiple solutions for nonlocal fractional problems via $\nabla$-theorems

Giovanni Molica Bisci, Dimitri Mugnai, and Raffaella Servadei

#### Abstract

The aim of this paper is to prove multiplicity of solutions for nonlocal fractional equations modeled by $$ \left\{ \begin{array}{ll} (-\Delta)^s u-\lambda u=f(x,u) & {\mbox{ in }} \Omega\\ u=0 & {\mbox{ in }} \mathbb R^n\setminus \Omega\,, \end{array} \right. $$ where $s\in (0,1)$ is fixed, $(-\Delta)^s$ is the fractional Laplace operator, $\lambda$ is a real parameter, $\Omega\subset \mathbb R^n$, $n>2s$, is an open bounded set with continuous boundary and nonlinearity $f$ satisfies natural superlinear and subcritical growth assumptions. Precisely, along the paper, we prove the existence of at least three non-trivial solutions for this problem in a suitable left neighborhood of any eigenvalue of $(-\Delta)^s$. For this purpose, we employ a variational theorem of mixed type (one of the so-called $\nabla$-theorems).

#### Article information

**Source**

Differential Integral Equations, Volume 30, Number 9/10 (2017), 641-666.

**Dates**

First available in Project Euclid: 27 May 2017

**Permanent link to this document**

https://projecteuclid.org/euclid.die/1495850422

**Mathematical Reviews number (MathSciNet)**

MR3656482

**Zentralblatt MATH identifier**

06770137

**Subjects**

Primary: 35J20: Variational methods for second-order elliptic equations 35S15: Boundary value problems for pseudodifferential operators 47G20: Integro-differential operators [See also 34K30, 35R09, 35R10, 45Jxx, 45Kxx] 45G05: Singular nonlinear integral equations

#### Citation

Molica Bisci, Giovanni; Mugnai, Dimitri; Servadei, Raffaella. On multiple solutions for nonlocal fractional problems via $\nabla$-theorems. Differential Integral Equations 30 (2017), no. 9/10, 641--666. https://projecteuclid.org/euclid.die/1495850422