Concentration of solutions for some singularly perturbed mixed problems: existence results
type: Item
instance of: scholarly article
title: Concentration of solutions for some singularly perturbed mixed problems: existence results
zbMATH Open document ID: 1214.35024
author: Andrea Malchiodi
author: Luigi Montoro
author: Jesús Garcia-Azorero
author: Ireneo Peral
published in: Archive for Rational Mechanics and Analysis
publication date: 2010-07-06T00:00:00Z
review text: Let \(\Omega\) be a bounded open subset of \(\mathbb R^n\) with smooth boundary. Let \(p\in \,]1,\frac{n+2}{n-2}[\). Let \(\nu\) be the outward unit normal to \(\partial\Omega\). Let \(\partial_{\mathcal D}\Omega\), \(\partial_{\mathcal N}\Omega\) be disjoint open subsets of the boundary \(\partial\Omega\) of \(\Omega\) such that the union of the closures of \(\partial_{\mathcal D}\Omega\) and of \(\partial_{\mathcal N}\Omega\) equals \(\partial\Omega\). The authors show that under suitable assumptions on the geometry of \(\partial\Omega\), if \(\overline{Q}\) belongs to the intersection of the closures of \(\partial_{\mathcal D}\Omega\) and of \(\partial_{\mathcal N}\Omega\), then the mixed boundary value problem \[ -\varepsilon^2\Delta u + u=u^p\quad\text{in }\Omega, \qquad u>0 \quad\text{on }\Omega, \qquad \frac{\partial u}{\partial \nu}=0\quad\text{on }\partial_{\mathcal N}\Omega, \qquad u=0\quad\text{on }\partial_{\mathcal D}\Omega, \] admits a family of solutions \(u_\varepsilon\) which concentrate at \(\overline{Q}\) as the parameter \(\varepsilon>0\) shrinks to \(0\).
reviewed by: Massimo Lanza de Cristoforis
Mathematics Subject Classification ID: 35J61
Mathematics Subject Classification ID: 35J25
Mathematics Subject Classification ID: 35B25
Mathematics Subject Classification ID: 35A01
zbMATH DE Number: 5731030
zbMATH Keywords: singularly perturbed nonlinear Poisson equation
zbMATH Keywords: mixed problems
zbMATH Keywords: concentration
MaRDI profile type: Publication
full work available at URL: s00205-009-0259-0
OpenAlex ID: W2034624307
cites work: Semiclassical states of nonlinear Schrödinger equations
cites work: Perturbation methods and semilinear elliptic problems on \(\mathbb R^n\)
cites work: Q3431230
cites work: Singularly perturbed elliptic equations with symmetry: existence of solutions concentrating on spheres. I
cites work: Singularly perturbed elliptic equations with symmetry: Existence of solutions concentrating on spheres, Part II
cites work: Multiplicity results for some nonlinear Schrödinger equations with potentials
cites work: Dual variational methods in critical point theory and applications
cites work: Concentration around a sphere for a singularly perturbed Schrödinger equation
cites work: Semiclassical symmetric Schrödinger equations: existence of solutions concentrating simultaneously on several spheres
cites work: The semiclassical limit of the nonlinear Schrödinger equation in a radial potential
cites work: Multipeak solutions for a singularly perturbed Neumann problem
cites work: On a class of solutions with non-vanishing angular momentum for nonlinear Schrödinger equations
cites work: Boundary spikes in the Gierer-Meinhardt system
cites work: On the Role of Mean Curvature in Some Singularly Perturbed Neumann Problems
cites work: Multi-bump ground states of the Gierer-Meinhardt system in \(\mathbb R^2\).
cites work: Nonspreading wave packets for the cubic Schrödinger equation with a bounded potential
cites work: Concentration of solutions for some singularly perturbed mixed problems: asymptotics of minimal energy solutions
cites work: Q4051967
cites work: Q3470840
cites work: Existence of multipeak solutions for a semilinear Neumann problem via nonsmooth critical point theory
cites work: Multipeak solutions for a semilinear Neumann problem
cites work: On Multiple Mixed Interior and Boundary Peak Solutions for Some Singularly Perturbed Neumann Problems
cites work: Multiple boundary peak solutions for some singularly perturbed Neumann problems
cites work: Uniqueness of positive solutions of \(\Delta u-u+u^ p=0\) in \(R^ n\)
cites work: On a singularly perturbed equation with neumann boundary condition
cites work: Q4266425
cites work: Large amplitude stationary solutions to a chemotaxis system
cites work: Concentration on minimal submanifolds for a singularly perturbed Neumann problem
cites work: Concentration at curves for a singularly perturbed Neumann problem in three-dimensional domains
cites work: Boundary concentration phenomena for a singularly perturbed elliptic problem
cites work: Multidimensional boundary layers for a singularly perturbed Neumann problem
cites work: Multiple clustered layer solutions for semilinear Neumann problems on a ball
cites work: On the location and profile of spike‐layer solutions to singularly perturbed semilinear dirichlet problems
cites work: Asymptotic behavior and stability of solutions of semilinear diffusion equations
cites work: Concentration phenomena for solutions of superlinear elliptic problems
cites work: Q4221182
cites work: On the shape of least‐energy solutions to a semilinear Neumann problem
cites work: Locating the peaks of least-energy solutions to a semilinear Neumann problem
cites work: On positive multi-lump bound states of nonlinear Schrödinger equations under multiple well potential
cites work: Q5607484
cites work: The chemical basis of morphogenesis
cites work: On the existence of multiple, single-peaked solutions for a semilinear Neumann problem
cites work: On the boundary spike layer solutions to a singularly perturbed Neumann problem
cites work: On the effect of domain geometry in singular perturbation problems
DOI: 10.1007/S00205-009-0259-0
description: scientific article; zbMATH DE number 5731030