Four positive solutions for the semilinear elliptic equation: \(-\Delta u+u=a(x) u^p +f(x)\) in \(\mathbb{R}^N\)
type: Item
instance of: scholarly article
title: Four positive solutions for the semilinear elliptic equation: \(-\Delta u+u=a(x) u^p +f(x)\) in \(\mathbb{R}^N\)
zbMATH Open document ID: 0987.35039
author: Shinji Adachi
author: Kazunaga Tanaka
published in: Calculus of Variations and Partial Differential Equations
publication date: 2002-06-13T00:00:00Z
review text: The authors consider the existence of four positive solutions of the semilinear elliptic problem \[ \begin{cases} -\Delta u+u=a(x)u^p+f(x),\quad u>0\qquad\text{in }\mathbb R^n,\quad n\geq 1,\\ u\in H^1(\mathbb R^n),\end{cases}\tag{1} \] where \(1<p<(n+2)/(n-2)\) if \(n\geq 3\), \(1<p<\infty\) if \(n=1,2\), and under the following structural assumptions: (a) \(a\in C(\mathbb R^n)\), \(0<a(x)\leq 1\) for all \(x\in\mathbb R^n\), \(a(x)\to 1\) as \(|x|\to\infty\), \(a\not\equiv 1\) in \(\mathbb R^n\), and there are \(\delta>0\) and \(C>0\) such that \[ a(x)-1\geq-Ce^{-(2+\delta)|x|}\qquad\text{for all }x\in\mathbb R^n; \] (b) \(f\in H^{-1}(\mathbb R^n)\), \(f\geq 0\) and \(f\not\equiv 0\) in \(\mathbb R^n\). The main result of the paper is: Theorem. There exists a number \(\varepsilon_0>0\) such that if \(0<\|f\|_{H^{-1}(\mathbb R^n)}\leq\varepsilon_0\), then problem \((1)\) admits at least four positive solutions. Let \((f_j)_j\) be a sequence of functions satisfying property (b), with \(\|f_j\|_{H^{-1}(\mathbb R^n)}\to 0\) as \(j\to\infty\). Then there is a subsequence of \((f_j)_j\), still denoted by \((f_j)_j\), and four sequences \((u^{(k)}_j)_j\), \(k=1,2,3,4\), of positive solutions of problem \((1)\), with \(f\) replaced by \(f_j\), such that (i) \(\|u_j^{(1)}\|_{H^{1}(\mathbb R^n)}\to 0\) as \(j\to\infty\); (ii) there are sequences \((y^{(k)}_j)_j\) in \(\mathbb R^n\), \(k=2,3\), such that \[ |y^{(k)}_j|\to\infty\quad\text{and}\quad \|u^{(k)}_j-\omega(\cdot-y^{(k)}_j)\|_{H^{1}(\mathbb R^n)}\to 0 \quad\text{as \(j\to\infty\), for }k=2,3, \] where \(\omega\) is the unique positive radial solution of the problem \[ \begin{cases} -\Delta \omega+\omega=\omega^p,\quad \omega>0\qquad\text{in }\mathbb R^n,\\ \omega\in H^1(\mathbb R^n);\end{cases}\tag{2} \] (iii) there exists a positive solution \(v_0\) of problem \((1)\), with \(f\) replaced by \(f\equiv 0\), such that \[ \|u^{(4)}_j-v_0)\|_{H^{1}(\mathbb R^n)}\to 0 \qquad\text{as }j\to\infty. \] Moreover, if \(a(x)=a…
reviewed by: Patrizia Pucci
Mathematics Subject Classification ID: 35J20
Mathematics Subject Classification ID: 35J60
zbMATH DE Number: 1525160
zbMATH Keywords: radially symmetric solutions
zbMATH Keywords: perturbation
zbMATH Keywords: minimax argument
MaRDI profile type: Publication
DOI: 10.1007/S005260000030
description: scientific article; zbMATH DE number 1525160