International Journal of Mathematics and Mathematical Sciences
Volume 2005 (2005), Issue 13, Pages 2005-2010
doi:10.1155/IJMMS.2005.2005
The existence of positive solutions for an elliptic boundary value problem
G.A. Afrouzi
Department of Mathematics, Faculty of Basic Science, University of Mazandaran, Pasdaran Street, P.O. Box 416, Babolsar 47415, Iran
Abstract
By using the mountain pass lemma, we study the existence of positive solutions for the equation −Δu(x)=λ(u|u|+u)(x) for x∈Ω together with Dirichlet boundary conditions and show that for every λ<λ1, where λ1 is the first eigenvalue of −Δu=λu in Ω with the Dirichlet boundary conditions, the equation has a positive solution while no positive solution exists for λ≥λ1.