ABSTRACT
We consider the problem of multiplicity of solutions for a class of nonlinear P.D.E. on a bounded domain Ω⊂ℝ N , with sufficiently smooth boundary. More precisely, I will be concerned with the following strong resonant problem, with Dirichlet boundary conditions: { − Δ u = λ k u + f ( x , u ) − g ( x ) , x ∈ Ω , u = 0 , x ∈ ∂ Ω . https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203744376/a3a6af35-efd2-43dc-a017-2efeb7a27be6/content/eq770.tif"/>
The well-known Lyapunov-Schmidt reduction method and the ideas used by Amann-Ambrosetti-Mancini (Math. Z. 1978) to describe the range of a function denned in a finite dimensional space will be extended when the decay of f at infinity is “like” | u | α u . https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203744376/a3a6af35-efd2-43dc-a017-2efeb7a27be6/content/eq771.tif"/> . Finally inspired by a paper of Pellacci and Villegas [14] we try to extend our result to an equation on ℝN and we obtain an existence result for the Schr¨oedinger equation − Δ u + p ( x ) u = μ k u + f ( x , u ) − g ( x ) , x ∈ ℝ N , u ∈ H 1 ( ℝ N ) . https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203744376/a3a6af35-efd2-43dc-a017-2efeb7a27be6/content/eq772.tif"/>
