Multiplicity results and sign changing solutions of non-local equations with concave-convex nonlinearities

Abstract

In this paper, we prove the existence of infinitely many nontrivial solutions of the following equations driven by a nonlocal integro-differential operator LK with concave-convex nonlinearities and homogeneous Dirichlet boundary conditions LKu+μ|u|q−1u+λ|u|p−1uu=0inΩ,=0inRN∖Ω, where Ω is a smooth bounded domain in RN, N>2s, s∈(0,1), 0<q<1<p≤N+2sN−2s. Moreover, when LK reduces to the fractional laplacian operator −(−Δ)s, p=N+2sN−2s, 12(N+2sN−2s)<q<1, N>6s, λ=1, we find μ∗>0 such that for any μ∈(0,μ∗), there exists at least one sign changing solution.