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 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).
Given a symmetric Riemannian manifold (M, g), we show some results of genericity for non degenerate sign changing solutions of singularly perturbed nonlinear elliptic problems with respect to the parameters: the positive number {epsilon} and the symm
etric metric g. Using these results we obtain a lower bound on the number of non degenerate solutions which change sign exactly once.
We study the behavior of the soliton solutions of the equation i((partial{psi})/(partialt))=-(1/(2m)){Delta}{psi}+(1/2)W_{{epsilon}}({psi})+V(x){psi} where W_{{epsilon}} is a suitable nonlinear term which is singular for {epsilon}=0. We use the stron
g nonlinearity to obtain results on existence, shape, stability and dynamics of the soliton. The main result of this paper (Theorem 1) shows that for {epsilon}to0 the orbit of our soliton approaches the orbit of a classical particle in a potential V(x).
