Low energy solutions for singularly perturbed coupled nonlinear systems on a Riemannian manifold with boundary

Let (M,g) be asmooth, compact Riemannian manifold with smooth boundary, with n= dim M= 2,3. We suppose the boundary of M to be a smooth submanifold of M with dimension n-1. We consider a singularly perturbed nonlinear system, namely Klein-Gordon-Maxwell-Proca system, or Klein-Gordon-Maxwell system of Scrhoedinger-Maxwell system on M. We prove that the number of low energy solutions, when the perturbation parameter is small, depends on the topological properties of the boundary of M, by means of the Lusternik Schnirelmann category. Also, these solutions have a unique maximum point that lies on the boundary

Non degeneracy of critical points of the Robin function with respect to deformations of the domain

We show a result of genericity for non degenerate critical points of the Robin function with respect to deformations of the domai

Positive solutions for singularly perturbed nonlinear elliptic problem on manifolds via Morse theory

Given (M, g0) we consider the problem -{\epsilon}^2Delta_{g0+h}u + u = (u+)^{p-1} with ({\epsilon}, h) \in (0, {\epsilon}0) \times B{\rho}. Here B{\rho} is a ball centered at 0 with radius {\rho} in the Banach space of all Ck symmetric covariant 2-tensors on M. Using the Poincar\'e polynomial of M, we give an estimate on the number of nonconstant solutions with low energy for ({\epsilon}, h) belonging to a residual subset of (0, {\epsilon}0) \times B{\rho}, for ({\epsilon}0, {\rho}) small enough

On Yamabe type problems on Riemannian manifolds with boundary

Let $(M,g)$ be a $n-$dimensional compact Riemannian manifold with boundary. We consider the Yamabe type problem \begin{equation} \left\{ \begin{array}{ll} -\Delta_{g}u+au=0 & \text{ on }M \\ \partial_\nu u+\frac{n-2}{2}bu= u^{{n\over n-2}\pm\varepsilon} & \text{ on }\partial M \end{array}\right. \end{equation} where $a\in C^1(M),$ $b\in C^1(\partial M)$, $\nu$ is the outward pointing unit normal to $\partial M$ and $\varepsilon$ is a small positive parameter. We build solutions which blow-up at a point of the boundary as $\varepsilon$ goes to zero. The blowing-up behavior is ruled by the function $b-H_g ,$ where $H_g$ is the boundary mean curvature

Blow-up phenomena for linearly perturbed Yamabe problem on manifolds with umbilic boundary

We build blowing-up solutions for linear perturbation of the Yamabe problem on manifolds with umbilic boundary, provided the Weyl tensor is nonzero everywhere on the boundary and the dimension of the manifold is n>10

The role of the scalar curvature in some singularly perturbed coupled elliptic systems on Riemannian manifolds

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 perturbation 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