Non degeneracy for solutions of singularly perturbed nonlinear elliptic problems on symmetric Riemannian manifolds
Marco Ghimenti A. M. Micheletti
Communications on Pure & Applied Analysis 2013, 12(2): 679-693 doi: 10.3934/cpaa.2013.12.679
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 $\varepsilon$ and the symmetric metric $g$. Using these results we obtain a lower bound on the number of non degenerate solutions which change sign exactly once.
keywords: non degenerate sign changing solutions Symmetric Riemannian manifolds singularly perturbed nonlinear elliptic problems.
The role of the scalar curvature in some singularly perturbed coupled elliptic systems on Riemannian manifolds
Marco Ghimenti Anna Maria Micheletti Angela Pistoia
Discrete & Continuous Dynamical Systems - A 2014, 34(6): 2535-2560 doi: 10.3934/dcds.2014.34.2535
Given a 3-dimensional Riemannian manifold $(M,g)$, we investigate the existence of positive solutions of the Klein-Gordon-Maxwell system $$ \left\{ \begin{array}{cc} -\varepsilon^{2}\Delta_{g}u+au=u^{p-1}+\omega^{2}(qv-1)^{2}u & \text{in }M\\ -\Delta_{g}v+(1+q^{2}u^{2})v=qu^{2} & \text{in }M \end{array}\right. $$ and Schrödinger-Maxwell system $$ \left\{ \begin{array}{cc} -\varepsilon^{2}\Delta_{g}u+u+\omega uv=u^{p-1} & \text{in }M\\ -\Delta_{g}v+v=qu^{2} & \text{in }M \end{array}\right. $$ when $p\in(2,6). $ We prove that if $\varepsilon$ is small enough, any stable critical point $\xi_0$ of the scalar curvature of $g$ generates a positive solution $(u_\varepsilon,v_\varepsilon)$ to both the systems such that $u_\varepsilon$ concentrates at $\xi_0$ as $\varepsilon$ goes to zero.
keywords: scalar curvature Lyapunov-Schmidt reduction. Klein-Gordon-Maxwell systems Scrhödinger-Maxwell systems Riemannian manifolds
On the stability of standing waves of Klein-Gordon equations in a semiclassical regime
Marco Ghimenti Stefan Le Coz Marco Squassina
Discrete & Continuous Dynamical Systems - A 2013, 33(6): 2389-2401 doi: 10.3934/dcds.2013.33.2389
We investigate the orbital stability and instability of standing waves for two classes of Klein-Gordon equations in the semi-classical regime.
keywords: stability semi-classical limit. Klein-Gordon equations

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