An $L^1$-theory for $p$-Schrödinger equations with confinement in measure
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arXiv
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| Autores principales: | , |
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| Formato: | Preprint |
| Publicado: |
2026
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| _version_ | 1866908970331930624 |
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| author | Alves, Nuno J. Urbano, José Miguel |
| author_facet | Alves, Nuno J. Urbano, José Miguel |
| contents | We consider stationary $p$-Schrödinger equations on the whole space with integrable data and potentials that are confining in measure. We introduce asymptotic energy solutions in an asymptotic $L^p$ framework and establish existence and uniqueness in the degenerate range $p\ge2$. The proof relies on a new Rellich$\unicode{x2013}$Kondrachov-type compactness theorem of independent interest, which provides sufficient conditions for families of Sobolev functions to be precompact in asymptotic $L^p$ spaces, without any dimension-dependent restriction on the exponent. For data in the duality regime $L^1(\mathbb{R}^n)\cap L^{p'}(\mathbb{R}^n)$, asymptotic energy solutions coincide with weak energy solutions. We also show that additional compactness assumptions yield localized entropy-type solutions and, under suitable local regularity, distributional solutions. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2604_14916 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | An $L^1$-theory for $p$-Schrödinger equations with confinement in measure Alves, Nuno J. Urbano, José Miguel Analysis of PDEs Functional Analysis We consider stationary $p$-Schrödinger equations on the whole space with integrable data and potentials that are confining in measure. We introduce asymptotic energy solutions in an asymptotic $L^p$ framework and establish existence and uniqueness in the degenerate range $p\ge2$. The proof relies on a new Rellich$\unicode{x2013}$Kondrachov-type compactness theorem of independent interest, which provides sufficient conditions for families of Sobolev functions to be precompact in asymptotic $L^p$ spaces, without any dimension-dependent restriction on the exponent. For data in the duality regime $L^1(\mathbb{R}^n)\cap L^{p'}(\mathbb{R}^n)$, asymptotic energy solutions coincide with weak energy solutions. We also show that additional compactness assumptions yield localized entropy-type solutions and, under suitable local regularity, distributional solutions. |
| title | An $L^1$-theory for $p$-Schrödinger equations with confinement in measure |
| topic | Analysis of PDEs Functional Analysis |
| url | https://arxiv.org/abs/2604.14916 |