Solutions with multiple spike patterns for an elliptic system

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Abstract

We consider a system of the form \(- \varepsilon^2 \Delta u + V(x)u=g(v)\) , \(-\varepsilon^2 \Delta v + V(x)v=f(u)\) in an open domain \(\Omega\) of \( {\mathbb{R}}^N\) , with Dirichlet conditions at the boundary (if any). We suppose that f and g are power-type non-linearities, having superlinear and subcritical growth at infinity. We prove the existence of positive solutions \(u_{\varepsilon}\) and \(v_{\varepsilon} \) which concentrate, as \(\varepsilon\to 0\) , at a prescribed finite number of local minimum points of V(x), possibly degenerate.