Normalized solutions for an horizontal transmission problem. Issue 15 (18th November 2021) - Figueiredo, Giovany M.; Siciliano, Gaetano
ABSTRACT: Let Ω be a bounded and smooth domain in R N,   N ≥ 3 . We study in this paper the following nonlinear transmission problem 1 − Δ u + λ u = | u | p − 2 u in   Ω, − Δ v + μ v = a ( x ) | v | q − 2 v in   R N ∖ Ω, u = v   a n d   ∂ u ∂ n = ∂ v ∂ n = 0 on   ∂ Ω, where the unknowns are u : Ω → R, v : R N ∖ Ω → R and the real numbers λ, μ . Actually we are looking for solutions u, v with prescribed L 2 -norm. The problem has a variational formulation and indeed, under minimal assumptions, we are able to find infinitely many solutions by using the Krasnoselkii genus, whatever the L 2 − norm of u and v a priori is.