BIFURCATION PROBLEM FOR A CLASS OF QUASILINEAR FRACTIONAL SCHRÖDINGER EQUATIONS

Abstract

We study bifurcation for the following fractional Schrödinger equation  (−∆)su+ V (x)u = λ f(u) in Ω u > 0 in Ω u = 0 inRn \ Ω where 0 < s < 1, n > 2s, Ω is a bounded smooth domain of Rn, (−∆)s is the fractional Laplacian of order s, V is the potential energy satisfying suitable assumptions and λ is a positive real parameter. The nonlinear term f is a positive nondecreasing convex function, asymptotically linear that is lim t→+∞ f(t) t = a ∈ (0,+∞). We discuss the existence, uniqueness and stability of a positive solution and we also prove the existence of critical value and the uniqueness of extremal solutions. We take into account the types of Bifurcation problem for a class of quasilinear fractional Schrödinger equations, we also establish the asymptotic behavior of the solution around the bifurcation point.