Symmetry and uniqueness of minimizers of Hartree type equations with external Coulomb potential

In the present article we study the radial symmetry of minimizers of the energy functional, corresponding to the repulsive Hartree equation in external Coulomb potential. To overcome the difficulties, resulting from the "bad" sign of the nonlocal term, we modify the reflection method and then, by using Pohozaev integral identities we get the symmetry result

Blow Up for the Semilinear Wave Equation in Schwarzschild Metric

We study the semilinear wave equation in Schwarzschild metric (3+1 dimensional space--time). First, we establish that the problem is locally well--posed in \cs H^\sigma for any $\sigma \geq 1$; then we prove the blow up of the solution for every real $p \in ]1,1+\sqrt{2}[$ and non--negative non--trivial initial data.Comment: some typos are corrected and some references are adde

Solitary waves for Maxwell-Schrodinger equations

In this paper we study the solitary waves for the coupled Schr\"odinger - Maxwell equations in three-dimensional space. We prove the existence of a sequence of radial solitary waves for these equations with a fixed $L^2$ norm. We study the asymptotic behavior and the smoothness of these solutions. We show also the fact that the eigenvalues are negative and the first one is isolated.Comment: 31 page