We study the asymptotic behavior of ground state energy for Schrodinger-Poisson-Slater energy functional. We show that ground state energy restricted to radially symmetric functions is above the ground state energy when the number of particles is sufficiently large.
We prove $L^p$ lower bounds for Coulomb energy for radially symmetric functions in $dot H^s(R^3)$ with $frac 12 <s<frac{3}{2}$. In case $frac 12 <s leq 1$ we show that the lower bounds are sharp.
Let $M$ be a compact smooth Riemannian manifold of finite dimension $n+1$ with boundary $partial M$and $partial M$ is a compact $n$-dimensional submanifold of $M$. We show that for generic Riemannian metric $g$, all the critical points of the mean curvature of $partial M$ are nondegenerate.
We show that the number of solutions of Schroedinger Maxwell system on a smooth bounded domain in R^3 depends on the topological properties of the domain. In particular we consider the Lusternik-Schnirelmann category and the Poincare polynomial of the domain.
Given a 3-dimensional Riemannian manifold (M,g), we investigate the existence of positive solutions of the nonlinear Klein-Gordon-Maxwell system and nonlinear Schroedinger-Maxwell system with subcritical nonlinearity. We prove that the number of one
peak solutions depends on the topological properties of the manifold M, by means of the Lusternik Schnirelmann category.
Given a 3-dimensional Riemannian manifold (M,g), we investigate the existence of positive solutions of singularly perturbed Klein-Gordon-Maxwell systems and Schroedinger-Maxwell systems on M, with a subcritical nonlinearity. We prove that when the pe
rturbation parameter epsilon is small enough, any stable critical point x_0 of the scalar curvature of the manifold (M,g) generates a positive solution (u_eps,v_eps) to both the systems such that u_eps concentrates at xi_0 as epsilon goes to zero.
We consider the nonlinear Klein-Gordon equation in $R^d$. We call multi-solitary waves a solution behaving at large time as a sum of boosted standing waves. Our main result is the existence of such multi-solitary waves, provided the composing boosted
standing waves are stable. It is obtained by solving the equation backward in time around a sequence of approximate multi-solitary waves and showing convergence to a solution with the desired property. The main ingredients of the proof are finite speed of propagation, variational characterizations of the profiles, modulation theory and energy estimates.
We investigate the validity of a soliton dynamics behavior in the semi-relativistic limit for the nonlinear Schrodinger equation in $R^{N}, Nge 3$, in presence of a singular external potential.
We investigate the orbital stability and instability of standing waves for two classes of Klein-Gordon equations in the semi-classical regime.
We give some generic properties of non degeneracy for critical points of functionals. We apply these results, obtaining some theorems of multiplicity of solutions for the equation -{epsilon}^2Delta_g u+u=|u|p-2u in M, u in H_g^1(M) where M is a compa
ct Riemannian manifold of dimension n and 2< p<2n/(n-2).
