Positive and nodal limiting profiles for a semilinear elliptic equation with a shrinking region of attraction

Mónica Clapp; Víctor Hernández-Santamaría; Alberto Saldaña

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arXiv · 2407.04830

Positive and nodal limiting profiles for a semilinear elliptic equation with a shrinking region of attraction

Mónica Clapp·Víctor Hernández-Santamaría·Alberto Saldaña

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Abstract

We study the existence and concentration of positive and nodal solutions to a Schr\"odinger equation in the presence of a shrinking self-focusing core of arbitrary shape. Via a suitable rescaling, the concentration gives rise to a limiting profile that solves a nonautonomous elliptic semilinear equation with a sharp sign change in the nonlinearity. We characterize the (radial or foliated Schwarz) symmetries and the (polynomial) decay of the least-energy positive and nodal limiting profiles.

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Mónica Clapp, Víctor Hernández-Santamaría, Alberto Saldaña. 2024-07-05. Positive and nodal limiting profiles for a semilinear elliptic equation with a shrinking region of attraction. https://arxiv.org/abs/2407.04830

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