Multiplicity of solutions for elliptic systems via local Mountain Pass method

Abstract

We consider the system$-\varepsilon^{2} \Delta u +W(x)u=Q_{u}(u,v)$ in $\mathbb{R}^N,$$-\varepsilon^{2} \Delta v +V(x)v=Q_{v}(u,v)$ in $\mathbb{R}^N, $$u,v \in H^{1}(\mathbb{R}^N),u(x),v(x)>0$ for each $x \in\mathbb{R}^N,$where $\varepsilon>0$, $W$ and $V$ are positive potentials and $Q$is a homogeneous function with subcritical growth. We relate thenumber of solutions with the topology of the set where $W$ and $V$attain their minimum values. In the proof we applyLjusternik-Schnirelmann theory.