Perturbation methods and semilinear elliptic problems on \(\mathbb R^n\)
type: Item
instance of: scholarly article
title: Perturbation methods and semilinear elliptic problems on \(\mathbb R^n\)
zbMATH Open document ID: 1115.35004
author: A. Ambrosetti
author: Andrea Malchiodi
published in: Progress in Mathematics
publication date: 2006-03-20T00:00:00Z
review text: The introductory chapter 1 is devoted to the statements of the main nonlinear variational problems which are studied in more detail in the monograph. These are the following. 1\(^\circ\). The existence of solutions problem of elliptic equation on \(\mathbb R^n\), \(-\Delta u+u=b(x)u^p\), \(0<u\in W^{1,2}(\mathbb R^n)\), \(n\geq 3\), \(1<p<\frac{n+2}{n-2}\) subcritical case and \(p=\frac{n+2}{n-2}\) -- critical case, where the main difficulties is the lack of the Palais-Smale (PS) compactness condition; 2\(^\circ\). Bifurcation from the essential spectrum for the equation \(F(\lambda,x)=0\), \(F:\mathbb R^n\times H\to H\), \(F(\lambda,0)\equiv 0\), \(\forall\lambda\in\mathbb R\), in a Hilbert space \(H\), when unlike to the compact case, in the presence of the essential spectrum of \(F'(\lambda,0)\) one tires to show that the infimum of such a spectrum is still a bifurcation point. Typical example of such a situation is the problem \(\psi''+\lambda\psi+h(x)|\psi|^{p-1}\psi=0\), \(\lim_{|x|\to 0}\psi(x)=0\), \(p>1\), where from the bottom \(\lambda=0\) of the essential spectrum of \(\psi''+\lambda\psi\), \(\psi\in W^{1,2}(\mathbb R)\), bifurcates a family of solutions \((\lambda,\psi_\lambda)\), \(\lambda<0\) of the nonlinear equation with \((\lambda,\psi_\lambda)\to (0,0)\) as \(\lambda\downarrow 0\). 3\(^\circ\). Semiclassical standing waves of nonlinear Schrödinger equation (NLS) \(\frac{\partial\psi}{\partial t}=-\hbar^2\Delta\psi+(a_0+Q(x))\psi+a_1|\psi|^{p-1}\psi\), \(a_1<0\), arising in plasma physics, in nonlinear optics at the presence of self-focusing material and in the Ginzburg-Landau theory with the nonlinearity of the form \(|\psi|^2\psi-|\psi|^4\psi\) etc. Here various perturbation problems with solutions concentrating at higher-dimensional manifolds arise. 4\(^\circ\). Other problems whose main feature is that they possess solutions concentrating at points or at manifolds, such as Neumann singularly perturbed problems and radial problems with concentra…
Mathematics Subject Classification ID: 35-02
Mathematics Subject Classification ID: 35B20
Mathematics Subject Classification ID: 35J60
Mathematics Subject Classification ID: 35B25
Mathematics Subject Classification ID: 53C43
Mathematics Subject Classification ID: 58E05
zbMATH DE Number: 5013399
zbMATH Keywords: semilinear variational elliptic equation on \(\mathbb R^n\)
zbMATH Keywords: perturbation problems
zbMATH Keywords: critical point theory
zbMATH Keywords: Yamabe problem and other problems in conformal geometry
zbMATH Keywords: nonlinear Schrödinger equation
zbMATH Keywords: singularly perturbed Neumann problems
reviewed by: Boris V. Loginov
MaRDI profile type: Publication
description: scientific article; zbMATH DE number 5013399