On indefinite superlinear elliptic equations
type: Item
instance of: scholarly article
title: On indefinite superlinear elliptic equations
zbMATH Open document ID: 0852.35052
DOI: 10.1007/BF01189951
author: Hossein Tehrani
published in: Calculus of Variations and Partial Differential Equations
publication date: 1996-03-07T00:00:00Z
review text: The author studies the solvability of the semilinear elliptic Neumann problem \[ - \Delta u+ m(x) u= a(x) u^p,\quad u> 0\quad \text{in } \Omega,\quad \partial u/\partial n= 0\quad \text{on }\partial\Omega,\tag{1} \] \(\Omega\subset \mathbb{R}^n\) a bounded, smooth domain. He assumes that \(p\) is subcritical, \(m\) is Hölder continuous and \(a\) is continuous and may change sign. Denote by \(\lambda_1\) the first eigenvalue of the linear operator \(-\Delta+ m(x)\) with homogeneous Neumann boundary condition, and by \(\phi\) the corresponding positive eigenfunction normalized to \(\int \phi^2= 1\). A typical result of this paper is the following: Assume, \(m\) is bounded in the Hölder norm, \(\int a(x) dx< 0\), \(\int m(x) dx\leq 0\); then there exists a constant \(\Lambda\) (only depending on the norm of \(m\)), such that if \(0> \lambda_1> \Lambda\) and \(\int a\phi^{p+ 1} dx< 0\), problem (1) has a solution. It is shown by a counterexample, that the condition \(\int a(x) dx< 0\) is essential for this existence result. The same problem was studied recently by \textit{H. Berestycki}, \textit{I. Capuzzo-Dolcetta} and \textit{L. Nirenberg} [NoDEA 2, No. 4, 553-572 (1995; Zbl 0840.35035)] in the case \(\lambda_1\geq 0\).
Mathematics Subject Classification ID: 35J65
Mathematics Subject Classification ID: 35J20
zbMATH DE Number: 852857
zbMATH Keywords: indefinite nonlinearity
zbMATH Keywords: existence of positive solutions
zbMATH Keywords: nonexistence
zbMATH Keywords: first eigenvalue
MaRDI profile type: Publication
cites work: On semilinear elliptic equations with indefinite nonlinearities
cites work: Variational methods for indefinite superlinear homogeneous elliptic problems
cites work: Superlinear indefinite elliptic problems and nonlinear Liouville theorems
cites work: Q3992017
full work available at URL: https://doi.org/10.1007/bf01189951
OpenAlex ID: W2074922612
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description: scientific article; zbMATH DE number 852857