Semilinear Schrödinger equations with Hardy potentials involving the distance to a boundary submanifold and gradient source nonlinearities
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| Formato: | Preprint |
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2024
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| author | Gkikas, Konstantinos T. Paschalis, Miltiadis |
| author_facet | Gkikas, Konstantinos T. Paschalis, Miltiadis |
| contents | Let $Ω\subset\mathbb{R}^N$ ($N\geq 3$) be a bounded $C^2$ domain and $Σ\subset\partialΩ$ be a compact $C^2$ submanifold of dimension $k$. Denote the distance from $Σ$ by $d_Σ$. In this paper, we study positive solutions of the equation $(*)\, -Δu -μu/d_Σ^2 = g(u,|\nabla u|)$ in $Ω$, where $μ\leq \big( \frac{N-k}{2} \big)^2$ and the source term $g:\mathbb{R}\times\mathbb{R}_+ \rightarrow \mathbb{R}_+$ is continuous and non-decreasing in its arguments with $g(0,0)=0$. In particular, we prove the existence of solutions of $(*)$ with boundary measure data $u=ν$ in two main cases, provided that the total mass of $ν$ is small. In the first case $g$ satisfies some subcriticality conditions that always ensure the existence of solutions. In the second case we examine power type nonlinearity $g(u,|\nabla u|) = |u|^p|\nabla u|^q$, where the problem may not possess a solution for exponents in the supercritical range. Nevertheless we obtain criteria for existence under the assumption that $ν$ is absolutely continuous with respect to some appropriate capacity or the Bessel capacity of $Σ$, or under other equivalent conditions. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2406_00354 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Semilinear Schrödinger equations with Hardy potentials involving the distance to a boundary submanifold and gradient source nonlinearities Gkikas, Konstantinos T. Paschalis, Miltiadis Analysis of PDEs 35J10, 35J25, 35J60 Let $Ω\subset\mathbb{R}^N$ ($N\geq 3$) be a bounded $C^2$ domain and $Σ\subset\partialΩ$ be a compact $C^2$ submanifold of dimension $k$. Denote the distance from $Σ$ by $d_Σ$. In this paper, we study positive solutions of the equation $(*)\, -Δu -μu/d_Σ^2 = g(u,|\nabla u|)$ in $Ω$, where $μ\leq \big( \frac{N-k}{2} \big)^2$ and the source term $g:\mathbb{R}\times\mathbb{R}_+ \rightarrow \mathbb{R}_+$ is continuous and non-decreasing in its arguments with $g(0,0)=0$. In particular, we prove the existence of solutions of $(*)$ with boundary measure data $u=ν$ in two main cases, provided that the total mass of $ν$ is small. In the first case $g$ satisfies some subcriticality conditions that always ensure the existence of solutions. In the second case we examine power type nonlinearity $g(u,|\nabla u|) = |u|^p|\nabla u|^q$, where the problem may not possess a solution for exponents in the supercritical range. Nevertheless we obtain criteria for existence under the assumption that $ν$ is absolutely continuous with respect to some appropriate capacity or the Bessel capacity of $Σ$, or under other equivalent conditions. |
| title | Semilinear Schrödinger equations with Hardy potentials involving the distance to a boundary submanifold and gradient source nonlinearities |
| topic | Analysis of PDEs 35J10, 35J25, 35J60 |
| url | https://arxiv.org/abs/2406.00354 |