Positive and nodal limiting profiles for a semilinear elliptic equation with a shrinking region of attraction
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arXiv
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| Hauptverfasser: | , , |
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| Format: | Preprint |
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2024
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| _version_ | 1866916425714630656 |
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| author | Clapp, Mónica Hernández-Santamaría, Víctor Saldaña, Alberto |
| author_facet | Clapp, Mónica Hernández-Santamaría, Víctor Saldaña, Alberto |
| contents | We study the existence and concentration of positive and nodal solutions to a Schrödinger 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2407_04830 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Positive and nodal limiting profiles for a semilinear elliptic equation with a shrinking region of attraction Clapp, Mónica Hernández-Santamaría, Víctor Saldaña, Alberto Analysis of PDEs 35J61, 35J20, 35B07, 35B40 We study the existence and concentration of positive and nodal solutions to a Schrödinger 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. |
| title | Positive and nodal limiting profiles for a semilinear elliptic equation with a shrinking region of attraction |
| topic | Analysis of PDEs 35J61, 35J20, 35B07, 35B40 |
| url | https://arxiv.org/abs/2407.04830 |