13 research outputs found
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
Semiclassical limit for the nonlinear Klein Gordon equation in bounded domains
We are interested to the existence of standing waves for the nonlinear Klein
Gordon equation {\epsilon}^2{\box}{\psi} + W'({\psi}) = 0 in a bounded domain
D. The main result of this paper is that, under suitable growth condition on W,
for {\epsilon} sufficiently small, we have at least cat(D) standing wavesfor
the equation ({\dag}), while cat(D) is the Ljusternik-Schnirelmann category
Multiple solutions and profile description for a nonlinear Schr\"odinger-Bopp-Podolsky-Proca system on a manifold
We prove a multiplicity result for \begin{equation*}
\begin{cases}
-\varepsilon^{2}\Delta_g u+\omega u+q^{2}\phi u=|u|^{p-2}u\\[1mm]
-\Delta_g \phi +a^{2}\Delta_g^{2} \phi + m^2 \phi =4\pi u^{2}
\end{cases}
\text{ in }M, \end{equation*} where is a smooth and compact
-dimensional Riemannian manifold without boundary, , , small enough. The proof of this result relies on
Lusternik-Schnirellman category. We also provide a profile description for low
energy solutions.Comment: 26 page
On the stability of standing waves of Klein-Gordon equations in a semiclassical regime
We investigate the orbital stability and instability of standing waves for
two classes of Klein-Gordon equations in the semi-classical regime.Comment: 9 page
Blowing-up solutions for supercritical Yamabe problems on manifolds with umbilic boundary
We build blowing-up solutions for a supercritical perturbation of the Yamabe problem on manifolds with umbilic boundary, provided the dimension of the manifold is n >7 and that the Weyl tensor is not vanishing on the boundary of the manifol