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
Department of Mathematics, Faculty of Basic Science, University of Mazandaran, Pasdaran Street, P.O. Box 416, Babolsar 47415, Iran
Received 28 November 2004; Revised 4 July 2005
Copyright © 2005 G. A. Afrouzi. 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
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.