Concentrated solutions to fractional Schrödinger equations with prescribed $L^2$-norm

Abstract

In this work, we investigate the existence and local uniqueness of normalized k-peak solutions for the fractional Schrödinger equations with attractive interactions with a class of degenerated trapping potential with non-isolated critical points.Precisely, applying the finite dimensional reduction method, we first obtain the existence of k-peak concentrated solutions and especially describe the relationship between the chemical potential µ and the attractive interaction a. Second, after precise analysis of the concentrated points and the Lagrange multiplier, we prove the local uniqueness of the kpeak solutions with prescribed L 2 -norm, by use of the local Pohozaev identities, the blow-up analysis and the maximum principle associated to the nonlocal operator (−∆) s .To our best knowledge, there is few results on the excited normalized solutions of the fractional Schrödinger equations before this present work. The main difficulty lies in the non-local property of the operator (−∆) s . First, it makes the standard comparison argument in the ODE theory invalid to use in our analysis. Second, because of the algebraic decay involving the approximate solutions, the estimates, on the Lagrange multiplier for example, would become more subtle. Moreover, when studying the corresponding harmonic extension problem, several local Pohozaev identities are constructed and we have to estimate several kinds of integrals that never appear in the classic local Schrödinger problems. In addition, throughout our discussion, we need to distinct the different cases of p − 1 < 4s N , p − 1 = 4s N , and p − 1 > 4s N , which are called respectively that the mass-subcritical, the mass-critical, and the mass-supercritical case, due to the mass-constraint condition. Another difficulty comes from the influence of the different degenerate rates along different directions at the critical points of the potential V (x).