Abstract and Applied Analysis
Volume 2008 (2008), Article ID 578417, 6 pages
Research Article

On Existence of Solution for a Class of Semilinear Elliptic Equations with Nonlinearities That Lies between Different Powers

Claudianor O. Alves and Marco A. S. Souto

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.


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 μtqf(t)Ctp at infinity and behaves like tq near the origin, where 1<q<(N+2)/(N2) if N3 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.