Normalized solutions to a Schr\"odinger-Bopp-Podolsky system under Neumann boundary conditions

In this paper we study a Schr\"odinger-Bopp-Podolsky system of partial differential equations in a bounded and smooth domain of $\mathbb R^3$ with a non constant coupling factor. Under a compatibility condition on the boundary data we deduce existence and multiplicity of solutions by means of the Ljusternik-Schnirelmann theory.Comment: Comments and suggestions are welcom

Energy stability for a class of semilinear elliptic problems

In this paper, we consider semilinear elliptic problems in a bounded domain $\Omega$ contained in a given unbounded Lipschitz domain $\mathcal C \subset \mathbb R^N$. Our aim is to study how the energy of a solution behaves with respect to volume-preserving variations of the domain $\Omega$ inside $\mathcal C$. Once a rigorous variational approach to this question is set, we focus on the cases when $\mathcal C$ is a cone or a cylinder and we consider spherical sectors and radial solutions or bounded cylinders and special one-dimensional solutions, respectively. In these cases, we show both stability and instability results, which have connections with related overdetermined problems.Comment: 33 page

Energy stability for a class of semilinear elliptic problems

In this paper, we consider semilinear elliptic problems in a bounded domain $\Omega$ contained in a given unbounded Lipschitz domain $\mathcal C \subset \mathbb R^N$. Our aim is to study how the energy of a solution behaves with respect to volume-preserving variations of the domain $\Omega$ inside $\mathcal C$. Once a rigorous variational approach to this question is set, we focus on the cases when $\mathcal C$ is a cone or a cylinder and we consider spherical sectors and radial solutions or bounded cylinders and special one-dimensional solutions, respectively. In these cases, we show both stability and instability results, which have connections with related overdetermined problems

Soluções normalizadas para um sistema de Schrödinger-Bopp-Podolsky

The aim is to study a Schrödinger-Bopp-Podolsky system of partial differential equations. We present an original result for the existence and multiplicity of weak solutions to the problem, which consists in the determination of critical points for a functional constrained to a submanifold of a Hilbert space. The calculus in Banach spaces is developed. Krasnoselskii\'s genus theory is discussed, after which the Deformation Lemma and some related notions are presented. Submanifolds of Banach spaces and Lagrange multipliers are discussed. The existence and multiplicity of solutions to the proposed problem is proved.Objetivamos estudar um sistema de tipo Schrödinger-Bopp-Podolsky, que consiste de duas equações diferenciais parciais não lineares. Apresentamos um resultado original de existência e multiplicidade de soluções fracas para o problema, ou seja, de existência de pontos críticos de um funcional restrito a uma subvariedade de um espaço de Hilbert. É desenvoldida a teoria do cálculo em espaços de Banach. Discutimos a teoria do gênero de Krasnoselskii e apresentamos o Lema de Deformação e noções correlatas. Discutimos subvariedades em um espaço de Banach e multiplicadores de Lagrange. Provamos a existência e multiplicidade de soluções fracas para o problema proposto