Informace o publikaci
Elliptic Schrödinger Equations with Gradient-Dependent Nonlinearity and Hardy Potential Singular on Manifolds
| Autoři | |
|---|---|
| Rok publikování | 2025 |
| Druh | Článek v odborném periodiku |
| Časopis / Zdroj | Journal of Geometric Analysis |
| Fakulta / Pracoviště MU | |
| Citace | |
| www | https://doi.org/10.1007/s12220-025-02046-9 |
| Doi | https://doi.org/10.1007/s12220-025-02046-9 |
| Klíčová slova | Hardy potentials; Gradient-dependent nonlinearities; Boundary trace; Capacities |
| Popis | Let \Omega \subset {\mathbb {R}}^N N \ge 3 be a C^2 bounded domain and \Sigma \subset \Omega is a C^2 compact boundaryless submanifold in {\mathbb {R}}^N of dimension k, 0\le k < N-2. For \mu \le (\frac{N-k-2}{2})^2, put L_\mu := \Delta + \mu d_{\Sigma }^{-2} where d_{\Sigma }(x) = \textrm{dist}(x,\Sigma ). We study boundary value problems for equation -L_\mu u = g(u,|\nabla u|) in \Omega \setminus \Sigma, subject to the boundary condition u=\nu on \partial \Omega \cup \Sigma, where g: {\mathbb {R}} \times {\mathbb {R}}_+ \rightarrow {\mathbb {R}}_+ is a continuous and nondecreasing function with g(0,0)=0, \nu is a given nonnegative measure on \partial \Omega \cup \Sigma. When g satisfies a so-called subcritical integral condition, we establish an existence result for the problem under a smallness assumption on \nu. If g(u,|\nabla u|) = |u|^p|\nabla u|^q, there are ranges of p, q, called subcritical ranges, for which the subcritical integral condition is satisfied, hence the problem admits a solution. Beyond these ranges, where the subcritical integral condition may be violated, we establish various criteria on \nu for the existence of a solution to the problem expressed in terms of appropriate Bessel capacities. |
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