Abstract and Applied Analysis
Volume 2008 (2008), Article ID 578417, 6 pages
doi:10.1155/2008/578417
Research Article
On Existence of Solution for a Class of Semilinear Elliptic Equations with Nonlinearities That Lies between Different Powers
Departamento de Matemática e Estatística, Universidade Federal de Campina Grande, CEP 58.109-970, Campina Grande - PB, Brazil
Received 24 August 2007; Revised 9 January 2008; Accepted 14 May 2008
Academic Editor: Mitsuharu Otani
Copyright © 2008 Claudianor O. Alves and Marco A. S. Souto. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
Abstract
We prove that the semilinear elliptic equation −Δu=f(u), in Ω, u=0, on ∂Ω has a positive solution when the nonlinearity f belongs to a class which
satisfies μtq≤f(t)≤Ctp at infinity and behaves like tq near the origin, where 1<q<(N+2)/(N−2) if N≥3 and 1<q<+∞ if N=1,2. In our approach,
we do not need the Ambrosetti-Rabinowitz condition, and the nonlinearity
does not satisfy any hypotheses such those required by the blowup method.
Furthermore, we do not impose any restriction on the growth of p.