A sign-changing solution for a superlinear Dirichlet problem
type: Item
instance of: scholarly article
title: A sign-changing solution for a superlinear Dirichlet problem
zbMATH Open document ID: 0907.35050
author: Alfonso Castro
author: Jorge Cossio
author: John M. Neuberger
published in: Rocky Mountain Journal of Mathematics
publication date: 1999-02-15T00:00:00Z
full work available at URL: CONT27-4.html
review text: The authors study the problem \[ \Delta u + f(u) = 0 \quad \text{in}\;\Omega, \quad u = 0 \quad \text{on}\;\partial \Omega. \tag{P} \] where \(\Omega\) is a bounded domain in \(\mathbb R^n\), \(f\in C^1\) is superlinear and subcritical and \(f'(0)\) is stricly less than the first eigenvalue of the Laplacian on \(\Omega\). Under these assumptions the existence of three different solutions of (P) is proved: \(u_1\) (positive), \(u_2\) (negative), and \(u_3\) which changes sign exactly once. For the proof, the authors consider the corresponding functional \(J(u) = \int_{\Omega} (1/2) | \nabla u | ^2 - F(u) \,dx\) on the space \(H^1_0(\Omega)\). The sign-changing solution is then obtained by minimizing \(J\) on the subset \[ S_1 = \{ u \neq 0 : (\nabla J(u), u) = 0, \;u_{+}, u_{-} \neq 0, \;(\nabla J(u_{+}), u_{+}) = 0 \}. \] Their proof also shows the inequality \( J(u_3) \geq J(u_1) + J(u_2) \).
Mathematics Subject Classification ID: 35J66
Mathematics Subject Classification ID: 35J20
Mathematics Subject Classification ID: 35J61
zbMATH DE Number: 1206684
zbMATH Keywords: multiple solutions
zbMATH Keywords: semilinear elliptic equation
zbMATH Keywords: superlinear boundary value problem
zbMATH Keywords: critical level of sign-changing solution
zbMATH Keywords: one sign solutions of Morse index 1
zbMATH Keywords: sign-changing solution of Morse index 2
MaRDI profile type: Publication
OpenAlex ID: W2121174441
cites work: Q4073917
cites work: Q3948194
cites work: A minimum-maximum principle for a class of non-linear integral equations
cites work: Q4272029
cites work: Q5187597
cites work: Q4134602
cites work: Q3905599
cites work: Q3749777
cites work: Q3997570
cites work: On a superlinear elliptic equation
DOI: 10.1216/RMJM/1181071858
description: scientific article; zbMATH DE number 1206684