The Nehari manifold for nonlocal elliptic operators involving concave–convex nonlinearities

Abstract

In this paper, we study the multiplicity of solutions to equations driven by a nonlocal integro-differential operator ${{\mathcal{L}}_K}$ with homogeneous Dirichlet boundary conditions. In particular, using fibering maps and Nehari manifold, we obtain multiple solutions to the following fractional elliptic problem

$$\left\{\begin{array}{ll}(-\triangle)^su(x)=\lambda u^q+ u^p,\quad u > 0 \; {\rm in}\; \Omega;\\ u=0, \qquad\qquad\qquad\qquad\quad \,\,\,{\rm in}\; {\mathbb{R}}^N\backslash\Omega,\end{array}\right.$$

where Ω is a smooth bounded set in ${{\mathbb{R}}^n}$, n >  2s with ${s \in (0,1)}$, λ is a positive parameter, the exponents p and q satisfy ${0 < q < 1 < p\; \leqslant \; 2_s^\ast-1}$ with ${2_s^\ast=\frac{2n}{n-2s}}$.