The existence of positive solutions for an elliptic boundary value problem

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.