Normalized solutions for an horizontal transmission problem

ABSTRACT

Let Ω be a bounded and smooth domain in ${\mathbb{R}}^N,N\ge 3$. We study in this paper the following nonlinear transmission problem $\left\{\begin{array}{ll}-\mathrm{\Delta }u+\lambda u=|u{|}^{p-2}u& \text{in}\mathrm{\Omega },\\ -\mathrm{\Delta }v+\mu v=a\left(x\right)|v{|}^{q-2}v& \text{in}{\mathbb{R}}^N\setminus \mathrm{\Omega },\\ u=v\mathrm{a}\mathrm{n}\mathrm{d}\frac{\mathrm{\partial }u}{\mathrm{\partial }\mathbf{n}}=\frac{\mathrm{\partial }v}{\mathrm{\partial }\mathbf{n}}=0& \text{on}\mathrm{\partial }\mathrm{\Omega },\end{array}\right.$ where the unknowns are $u:\mathrm{\Omega }\to \mathbb{R},v:{\mathbb{R}}^N\setminus \mathrm{\Omega }\to \mathbb{R}$ and the real numbers $\lambda ,\mu$. 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.

KEYWORDS:

• Nonlinear transmission problem
• variational methods
• Krasnoselskii genus
• infinitely many solutions

2010 MATHEMATICS SUBJECT CLASSIFICATIONS:

• 35J50
• 35J57
• 35D30

Disclosure statement

No potential conflict of interest was reported by the authors.

ORCID

Gaetano Siciliano  http://orcid.org/0000-0001-5823-0392

Additional information

Funding

G. M. Figueiredo is partially supported by Fundação de Apoio à Pesquisa do Distrito Federal (FAPDF), CNPq and Capes, Brazil. G. Siciliano is partially supported by Fundação de Amparo à Pesquisa do Estado de São Paulo (FAPESP) grant 2018/17264-4, Conselho Nacional de Desenvolvimento Científico e Tecnológico (CNPq) grant 304660/2018-3 and Capes, Brazil.