Normalized solutions for an horizontal transmission problem

Let Ω be a bounded and smooth domain in (Formula presented.). We study in this paper the following nonlinear transmission problem (Formula presented.) where the unknowns are (Formula presented.) and the real numbers (Formula presented.). Actually we are looking for solutions u, v with prescribed (Formula presented.) -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 (Formula presented.) norm of u and v a priori is.