chapter  9
10 Pages

A Class of Strong Resonant Problems via Lyapunov-Schmidt Reduction Method

ByJ. Carmona

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 .

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 . . 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 ) .