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International Journal of Mathematics and Mathematical Sciences
Volume 2005, Issue 13, Pages 2005-2010

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 Hindawi Publishing Corporation. 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.


By using the mountain pass lemma, we study the existence of positive solutions for the equation Δu(x)=λ(u|u|+u)(x) for xΩ 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 λλ1.