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arXiv · 2510.12484

Classification and qualitative properties of positive solutions to double-power nonlinear stationary Schr\"{o}dinger equations

Takafumi Akahori·Slim Ibrahim·Hiroaki Kikuchi·Masataka Shibata·Juncheng Wei
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Abstract

In this paper, we investigate positive radial solutions to double-power nonlinear stationary Schrodinger equations in three space dimensions. It is now known that the non-uniqueness of H^{1}-positive solutions can occur in three dimensions when the frequency is sufficiently small. Under suitable conditions, in addition to the ground state solution (whose L^{\infty} norm vanishes as the frequency tends to zero), there exists another positive solution that minimizes a different constrained variational problem, with an L^{\infty} norm diverging as the frequency tends to zero (see Theorem 1.4). We classify all positive solutions with small frequency into two categories: the ground state and the Aubin-Talenti type solution. As a consequence, we establish the multiplicity of positive solutions. Finally, we also examine the non-degeneracy and Morse index of each positive solution.

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Takafumi Akahori, Slim Ibrahim, Hiroaki Kikuchi, Masataka Shibata, Juncheng Wei. 2025-10-14. Classification and qualitative properties of positive solutions to double-power nonlinear stationary Schr\"{o}dinger equations. https://arxiv.org/abs/2510.12484

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