Fractional Schrödinger Equations with Mixed Nonlinearities: Asymptotic Profiles, Uniqueness and Nondegeneracy of Ground States

Abstract

We study the fractional Schrodinger equations with a vanishing parameter: { (-Delta)(s) u + u = |u|(p-2 )u + lambda|u|(q-2 )u in R-N u is an element of H-s(R-N), (P-lambda) where s is an element of (0, 1), N > 2s, 2 < q < p <= 2(s)* = 2N/N-2s are fixed parameters and lambda > 0 is a vanishing parameter. We investigate the asymptotic behaviour of positive ground state solutions for A small, when p is subcritical, or critical Sobolev exponent 2(s)*. For p < 2(s)*, the ground state solution asymptotically coincides with unique positive ground state solution of (-Delta)(s )u + u = |u|(p-2 )u, whereas for p = 2(s)* the asymptotic behaviour of the solutions, after a rescaling, is given by the unique positive solution of the nonlocal critical Emden-Fowler type equation. Additionally, for lambda > 0 small, we show the uniqueness and nondegeneracy of the positive ground state solution using these asymptotic profiles of solutions.