The existence of positive solutions for an elliptic
boundary value problem
G. A. Afrouzi1
Received28 Nov 2004
Revised04 Jul 2005
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.