Elliptic Schrödinger Equations with Gradient-Dependent Nonlinearity and Hardy Potential Singular on Manifolds

Result code in IS VaVaI
<a href="https://www.isvavai.cz/riv?ss=detail&h=RIV%2F00216224%3A14310%2F25%3A00144360" target="_blank" >RIV/00216224:14310/25:00144360 - isvavai.cz</a>
Result on the web
<a href="https://doi.org/10.1007/s12220-025-02046-9" target="_blank" >https://doi.org/10.1007/s12220-025-02046-9</a>
DOI - Digital Object Identifier
<a href="http://dx.doi.org/10.1007/s12220-025-02046-9" target="_blank" >10.1007/s12220-025-02046-9</a>

Result language
angličtina
Original language name
Elliptic Schrödinger Equations with Gradient-Dependent Nonlinearity and Hardy Potential Singular on Manifolds
Original language description
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, 0le 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.
Czech name
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Czech description
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Type
J<sub>imp</sub> - Article in a specialist periodical, which is included in the Web of Science database
CEP classification
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OECD FORD branch
10101 - Pure mathematics

Project
<a href="/en/project/GA22-17403S" target="_blank" >GA22-17403S: Nonlinear Schrödinger equations and systems with singular potentials</a><br>
Continuities
P - Projekt vyzkumu a vyvoje financovany z verejnych zdroju (s odkazem do CEP)

Publication year
2025
Confidentiality
S - Úplné a pravdivé údaje o projektu nepodléhají ochraně podle zvláštních právních předpisů

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