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 with homogeneous Dirichlet boundary conditions. In particular, using fibering maps and Nehari manifold, we obtain multiple solutions to the following fractional elliptic problem

{(-Delta)(8)u(x) = lambda u(q) + u(p), u > 0 in Omega;

u = 0, in R-N\Omega,

where Omega is a smooth bounded set in R-n, n > 2s with s is an element of(0, 1), lambda is a positive parameter, the exponents p and q satisfy 0 < q < 1 < p <= 2(s)* - 1 with 2(s)* = 2n/n-2s.