Multiplicity and orbital stability of normalized solutions to non-autonomous Schrödinger equation with mixed nonlinearities

2022/07/17 by Li, Xinfu, Xu, Li, Zhu, Meiling
#Analysis of PDEs (math.AP) #FOS: Mathematics

paper · doi:10.48550/arxiv.2207.08167

Abstract

This paper studies the multiplicity of normalized solutions to the Schrödinger equation with mixed nonlinearities \begincases -Δu=λu+h(εx)|u|^q-2u+η|u|^p-2u, x∈ ℝ^N,
∫_ℝ^N|u|²dx=a², \endcases where a, ε, η>0, q is L²-subcritical, p is L²-supercritical, λ∈ ℝ is an unknown parameter that appears as a Lagrange multiplier, h is a positive and continuous function. It is proved that the numbers of normalized solutions are at least the numbers of global maximum points of h when ε is small enough. Moreover, the orbital stability of the solutions obtained is analyzed as well. In particular, our results cover the Sobolev critical case p=2N/(N-2).

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