Copyright © 2011 Giovanni Anello. 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 study the existence of positive solutions to the following nonlocal boundary value problem -K(∥u∥2)Δu=λus-1+f(x,u) in Ω, u=0 on ∂Ω, where s∈]1,2[, f:Ω×ℝ+→ℝ is a Carathéodory function, K:ℝ+→ℝ is a positive continuous function, and λ is a real parameter. Direct variational methods are used. In particular, the proof of the main result is based on a property of the infimum on certain spheres of the energy functional associated to problem -K(∥u∥2)Δu=λus-1 in Ω, u|∂Ω=0.