Multiplicity of solutions for coupled elliptic systems on Riemannian manifolds

Abstract:

Let (M,g) be a smooth compact 3-dimensional Riemannian manifold. Given real numbers a>0, q>0, -\sqrt{a}<w<\sqrt{a}, we consider the following Klein Gordon Maxwell (KGM) system:

- eps^2 \Delta_g  u +au =u^{p-1} +w^2(qv-1)^2      on M
- \Delta_g  v+(1+q^2u^2)v=qu^2                           on M
u>0,v>0

Given real numbers q>0, w>0 we consider the following Schroedinger Maxwell (SM) systems

- eps^2 \Delta_g  u +u+wuv =u^{p-1}            on M
- \Delta_g  v+v=qu^2                                   on M
u>0,v>0

We show that the topology on the manifold (M,g) has an effect on the number of solutions of KGM and SM systems. In particular we consider the Lusternik Schnirelmann category cat(M) of M in itself. Also the geometry of the manifold (M,g) influences the number of solutions. We show that the scalar curvature S_g relative to the metric g is the geometric property which influences the number of solutions.