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

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