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Solutions with multiple spike patterns for an elliptic system

Authors:	Miguel Ramos Hugo Tavares

Institution:	(1) University of Lisbon, CMAF, Faculty of Science, Av. Prof. Gama Pinto 2, 1649-003 Lisboa, Portugal

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.

Keywords:	35J50 58E05
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