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By using the mountain pass lemma, we study the existence ofpositive solutions for the equation −Δu(x)=λ(u|u|+u)(x) for x∈Ω together with Dirichletboundary 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 equationhas a positive solution while no positive solution exists forλ≥λ1

The existence of positive solutions for an elliptic boundary value problem