34 research outputs found
Nonlinear Klein-Gordon-Maxwell systems with Neumann boundary conditions on a Riemannian manifold with boundary
Let (M,g) be a smooth compact, n dimensional Riemannian manifold, n=3,4 with
smooth n-1 dimensional boundary. We search the positive solutions of the
singularly perturbed Klein Gordon Maxwell Proca system with homogeneous Neumann
boundary conditions or for the singularly perturbed Klein Gordon Maxwell system
with mixed Dirichlet Neumann homogeneous boundary conditions. We prove that
stable critical points of the mean curvature of the boundary generates
solutions when the perturbation parameter is sufficiently small.Comment: arXiv admin note: text overlap with arXiv:1410.884
Stable standing waves for a class of nonlinear Schroedinger-Poisson equations
We prove the existence of orbitally stable standing waves with prescribed
-norm for the following Schr\"odinger-Poisson type equation \label{intro}
%{%{ll} i\psi_{t}+ \Delta \psi - (|x|^{-1}*|\psi|^{2}) \psi+|\psi|^{p-2}\psi=0
\text{in} \R^{3}, %-\Delta\phi= |\psi|^{2}& \text{in} \R^{3},%. when . In the case we prove the existence and
stability only for sufficiently large -norm. In case our approach
recovers the result of Sanchez and Soler \cite{SS} %concerning the existence
and stability for sufficiently small charges. The main point is the analysis of
the compactness of minimizing sequences for the related constrained
minimization problem. In a final section a further application to the
Schr\"odinger equation involving the biharmonic operator is given
A guide to the Choquard equation
We survey old and recent results dealing with the existence and properties of
solutions to the Choquard type equations and some of its variants and extensions.Comment: 39 page