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

J 2025

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

GKIKAS, Konstantinos T. and Phuoc Tai NGUYEN

Basic information

Original name

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

Authors

GKIKAS, Konstantinos T. and Phuoc Tai NGUYEN (704 Viet Nam, guarantor, belonging to the institution)

Edition

Journal of Geometric Analysis, Springer, 2025, 1050-6926

Other information

Language

English

Type of outcome

Article in a journal

Field of Study

10101 Pure mathematics

Country of publisher

United States of America

Confidentiality degree

is not subject to a state or trade secret

References:

URL

Impact factor

Impact factor: 1.200 in 2023

Organization unit

Faculty of Science

DOI

http://dx.doi.org/10.1007/s12220-025-02046-9

UT WoS

001503680400002

EID Scopus

2-s2.0-105007446706

Keywords in English

Hardy potentials; Gradient-dependent nonlinearities; Boundary trace; Capacities

Abstract

In the original language

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.

Links

GA22-17403S, research and development project

Name: Nelineární Schrödingerovy rovnice a systémy se singulárním potenciálem (Acronym: NSESSP)

Investor: Czech Science Foundation

Cite

GKIKAS, Konstantinos T. and Phuoc Tai NGUYEN. Elliptic Schrödinger Equations with Gradient-Dependent Nonlinearity and Hardy Potential Singular on Manifolds. Journal of Geometric Analysis. Springer, 2025, vol. 35, No 7, p. 212-257. ISSN 1050-6926. Available from: https://dx.doi.org/10.1007/s12220-025-02046-9.

@article{2505727,
   author = {Gkikas, Konstantinos T. and Nguyen, Phuoc Tai},
   article_number = {7},
   doi = {http://dx.doi.org/10.1007/s12220-025-02046-9},
   keywords = {Hardy potentials; Gradient-dependent nonlinearities; Boundary trace; Capacities},
   language = {eng},
   issn = {1050-6926},
   journal = {Journal of Geometric Analysis},
   title = {Elliptic Schrödinger Equations with Gradient-Dependent Nonlinearity and Hardy Potential Singular on Manifolds},
   url = {https://doi.org/10.1007/s12220-025-02046-9},
   volume = {35},
   year = {2025}
}

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AU  - Gkikas, Konstantinos T. - Nguyen, Phuoc Tai
PY  - 2025
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JF  - Journal of Geometric Analysis
VL  - 35
IS  - 7
SP  - 212
EP  - 212
PB  - Springer
SN  - 10506926
KW  - Hardy potentials
KW  - Gradient-dependent nonlinearities
KW  - Boundary trace
KW  - Capacities
UR  - https://doi.org/10.1007/s12220-025-02046-9
N2  - 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.
ER  -

GKIKAS, Konstantinos T. and Phuoc Tai NGUYEN. Elliptic Schrödinger Equations with Gradient-Dependent Nonlinearity and Hardy Potential Singular on Manifolds. \textit{Journal of Geometric Analysis}. Springer, 2025, vol.~35, No~7, p.~212-257. ISSN~1050-6926. Available from: https://dx.doi.org/10.1007/s12220-025-02046-9.

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