Some Bifurcation Results for Fractional Laplacian Problems

We use bifurcation theory to establish the existence of connected set of solutions of a fractional Laplacian problem satisfying Dirichlet type boundary condition on the exterior of the domain. We discuss the nodal properties of solutions on these connected sets and determine the direction of bifurcation of these connected sets. Under additional assumptions, we establish the multiplicity of solutions near the resonance and the existence of solution in the resonant case. We also discuss anti-maximum principle, and solvability for the resonant case satisfying the so called Landesman-Lazer type condition