Boundary value problems for semilinear Schrödinger equations with singular potentials and measure data
type: Item
instance of: scholarly article
title: Boundary value problems for semilinear Schrödinger equations with singular potentials and measure data
author: Mousomi Bhakta
author: Moshe Marcus
author: Phuoc-Tai Nguyen
published in: Mathematische Annalen
publication date: 2024-10-28T00:00:00Z
review text: The authors consider the boundary value problem with measure data in a bounded \(C^2\)-smooth domain \(\Omega\) in \(\mathbb{R}^N\), \(N\ge 3\) for the semilinear equation\N\[\N\begin{aligned}\N-\Delta u-Vu +f(u)&=\tau\quad \text{in }\Omega\\\N\operatorname{tr}_V u&=\nu \quad\text{on }\partial\Omega\,.\N\end{aligned}\tag{1}\N\]\NHere it is assumed that \(V\in C^1(\Omega)\) satisfies \(|V(x)| \leq \bar a \operatorname{dist}(x,\partial\Omega)^{-2}\) in \(\Omega\) for some \(\bar{a}>0\) and \(\gamma_-<1<\gamma_+\), where \(\gamma_-:=\inf E_0\), \(\gamma_+:=\sup E_0\) and \(E_0:=\{\gamma: \exists u_\gamma>0\text{ such that }L_{\gamma V} (u_\gamma)=0\}\).\NThen \(L_{V}:=\Delta + V\) has a (minimal) ground state \(\Phi_V\) in \(\Omega\) and in addition, \(-L_V\) admits a (minimal) Green function \(G_V\) and Martin kernel \(K_V\).\NThe function \(f\in C(R)\) is assumed monotonely increasing with \(f(0)=0\) and \(\operatorname{tr}_V u\) denotes the measure boundary trace of \(u\) associated with \(L_V\). Furthermore, \(\nu\in\mathfrak{M}(\partial \Omega)\) and \(\tau\in\mathfrak{M}(\Omega;\Phi_V)\).\N\NHere \(\mathfrak{M}(\partial \Omega)\) denotes the space of finite Borel measures on \(\partial \Omega\) and \(\mathfrak{M}(\Omega;\Phi_V)\) the space of real Borel measures \(\tau\) on \(\Omega\) such that \(\int_\Omega \Phi_V d |\tau|<\infty\).\NBy convention \(\mathfrak{M}_+(\partial\Omega)\) and \(\mathfrak{M}_+(\Omega;\Phi_V)\) denote the positive cones of these spaces respectively.\N\NThe authors prove the existence for the boundary value problem (1) in the sense of reduced measures for couples \((\tau,\nu)\) of positive measures. Then, such result is extended to the case of signed measures \(\tau\) and \(\nu\), which may require more complex tools and techniques.\NThe idea of `reduced measure' was introduced for \(V=0\) in [\textit{H. Brezis} et al., Ann. Math. Stud. 163, 55--109 (2007; Zbl 1151.35034)].\NThe approach is to provide a reduction process that converges to…
reviewed by: Shijun Zheng
Mathematics Subject Classification ID: 35J61
Mathematics Subject Classification ID: 35J75
Mathematics Subject Classification ID: 35J10
zbMATH DE Number: 7932311
zbMATH Keywords: measure boundary value problem
zbMATH Keywords: semilinear Schrödinger equation
zbMATH Keywords: singular potentials
zbMATH Keywords: reduced measures
MaRDI profile type: Publication
cites work: Theorie du Potentiel sur les Graphes et les Varietes
cites work: Positive solutions of a class of semilinear equations with absorption and Schrödinger equations
cites work: Reduced limit for semilinear boundary value problems with measure data
cites work: Nonlinear elliptic equations with measures revisited
cites work: Kato's inequality when \(\Delta u\) is a measure
cites work: Reduced measures on the boundary
cites work: Weak solutions of semilinear elliptic equations with Leray-Hardy potentials and measure data
cites work: Singularities of positive supersolutions in elliptic PDEs
cites work: Q4830500
cites work: Boundary singularities of solutions of semilinear elliptic equations with critical Hardy potentials
cites work: The fractional Schrödinger equation with singular potential and measure data
cites work: The Brownian snake and solutions of \(\Delta u = u^ 2\) in a domain
cites work: Q4266067
cites work: Complete classification of the positive solutions of \(-\Delta u+u^q\)
cites work: Estimates of Green and Martin Kernels for Schrödinger operators with singular potential in Lipschitz domains
cites work: Moderate solutions of semilinear elliptic equations with Hardy potential under minimal restrictions on the potential
cites work: Moderate solutions of semilinear elliptic equations with Hardy potential
cites work: Schrödinger equations with singular potentials: linear and nonlinear boundary value problems
cites work: Reduced limits for nonlinear equations with measures
cites work: The boundary trace of positive solutions of semilinear elliptic equations: the subcritical case
cites work: The boundary trace of positive solutions of semilinear elliptic equations: The supercritical case
cites work: Classification and probabilistic representation of the positive solutions of a semilinear elliptic equation
DOI: 10.1007/S00208-023-02764-X
description: scientific article; zbMATH DE number 7932311