870 research outputs found
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 be a 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 , is the outward pointing unit
normal to and is a small positive parameter. We
build solutions which blow-up at a point of the boundary as goes
to zero. The blowing-up behavior is ruled by the function where
is the boundary mean curvature
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
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
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