Nehari Manifold For A Class Of Elliptic Problems

Résumé: In this work, we study some minimization techniques to solve semilinear elliptic equation, posed on a bounded open set Ω ⊂ RN. the first problem considered is  −∆u + q(x)u = |u|p−2u in Ω, u = 0 on ∂Ω. We prove the existence of a nontrivial solution by two different methods: Minimization on Sphere and Minimization on the Nehari Manifold. Also, we consider the perturbed problem, for h ∈ L2(Ω) and p ∈]2, 2∗[, the Dirichlet problem  −∆u + q(x)u = |u|p−2u + h(x) in Ω, u = 0 on ∂Ω. By using the method of Minimization on the Nehari Manifold we proved that this problem admits at least two nontrivial solution. Using the Nehari manifold and fibering maps, we prove the existence theorem of the nonlinear boundary problem  −∆u + u = |u|p−2u in Ω, ∂u ∂n = λ|u|q−2u on ∂Ω. on a bounded domain Ω ⊆ RN. We discuss how the Nehari manifold changes as λ changes, and show how the existence of solutions depends on the properties of the Nehari manifold.

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