On existence, uniqueness and radiality of normalized solutions to Schrödinger-Poisson equations with non-autonomous nonlinearity

2023/12/27 by Chengcheng Wu, Wu, Chengcheng
Mathematics · #Analysis of PDEs (math.AP) #FOS: Mathematics #math.AP

paper · pdf · doi:10.48550/arxiv.2312.16368

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arxiv created 2026/08/02 · arxiv updated 2026/08/04

Abstract

We investigate the existence, uniqueness, and radial symmetry of normalized solutions to the Schrödinger Poisson equation with non-autonomous nonlinearity f(x,u): -\triangle u+(|x|^-1*|u|²)u=f(x,u)+λu, subject to the constraint S_c=\u∈ H¹(ℝ³)|∫_ℝ³u²=c>0 \. We consider three cases based on the behavior of f(x,u): the L² supercritical case, the L² subcritical case with growth speed less than three power times, and the L² subcritical case with growth speed more than three power times. We establish the existence of solutions using three different methods depending on f(x,u). Furthermore, we demonstrate the uniqueness and radial symmetry of normalized solutions using an implicit function framework when c is small.

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