Local superlinearity and sublinearity for indefinite semilinear elliptic problems.
type: Item
instance of: scholarly article
title: Local superlinearity and sublinearity for indefinite semilinear elliptic problems.
zbMATH Open document ID: 1034.35042
DOI: 10.1016/S0022-1236(02)00060-5
author: Djairo G. de Figueiredo
author: Pedro Ubilla
author: J.-P. Gossez
published in: Journal of Functional Analysis
publication date: 2003-05-25T00:00:00Z
review text: The authors study the semilinear elliptic boundary value problem (*) \(-\Delta u = f(x,u)\), \(u\geq 0\) and \(u\not\equiv 0\) in \(\Omega\), \(u=0\) on \(\partial\Omega\), where \(\Omega\) is a bounded domain in \({\mathbb R}^{N}\) and \(f\) is sublinear at 0 and superlinear at infinity (in a weaker sense than usual; in particular a superlinearity condition is somewhat weaker than that of Ambrosetti and Rabinowitz). A typical example is the concave-convex nonlinearity \[ f(x,u) = \lambda a(x)u^q+b(x)u^p, \] where \(\lambda>0\), \(0\leq q<1<p\) and \(a\), \(b\) may be \(\leq 0\) on some subsets of \(\Omega\). The first result asserts that if \(f\) is of subcritical growth (in the example above this means \(p<2^{*}-1=(N+2)/(N-2)\)), then, under some additional hypotheses, (*) possesses at least 2 solutions. These solutions are respectively found as a mountain pass point and a local minimum of the associated energy functional. In the second result it is shown that under somewhat different conditions (*) has at least 1 solution. The proof uses the method of upper and lower solutions, and this time \(f\) is allowed to grow supercritically (\(p<+\infty\) in the example). Finally, a nonexistence result for (*) is proved. The results of the paper are related to and extend those of \textit{A. Ambrosetti, H. Brézis} and \textit{G. Cerami} [J. Func. Anal. 122, 519--543 (1994; Zbl 0805.35028)] and [\textit{H. Tehrani}, On elliptic equations with nonlinearities that are sum of a sublinear and superlinear term, volume dedicated to Dr. S. Shahshahani, Iran Dept. Math. Sci., Univ. Nevada, Las Vegas, Nevada 89154, 123--131 (2002)].
reviewed by: Andrzej Szulkin
Mathematics Subject Classification ID: 35J61
Mathematics Subject Classification ID: 35J20
Mathematics Subject Classification ID: 35J25
zbMATH DE Number: 1915836
zbMATH Keywords: Superlinearity
zbMATH Keywords: sublinearity
zbMATH Keywords: indefinite nonlinearity
zbMATH Keywords: concave-convex nonlinearity
zbMATH Keywords: semilinear elliptic problem
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 1915836