Some bifurcation results for fractional Laplacian problems

Abstract

We consider the following nonlocal problem (−Δ)su=λu+f(λ,x,u)inΩ;u=0inRN∖Ω,where (−Δ)s is the fractional Laplacian operator with fixed 0<s<1, Ω⊂RN(N>2s) is a bounded domain with C1,1 boundary, CN,s>0 is a constant and λ∈R is a bifurcation parameter. Here f:R×Ω×R→R is a Carathéodory function that is sublinear at infinity. We use bifurcation theory to establish the existence of continua of the solution set bifurcating from infinity at the principal eigenvalue of (−Δ)s and discuss the nodal properties of solutions on these continua. We establish the multiplicity of solutions near the resonance and the existence of solution in the resonant case. As corollaries, we obtain anti-maximum principle and solvability for the resonant case satisfying the so called Landesman–Lazer type condition.