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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.

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Authors and Affiliations

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

    Miguel Ramos & Hugo Tavares

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  1. Miguel Ramos
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  2. Hugo Tavares
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Correspondence to Miguel Ramos.

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Ramos, M., Tavares, H. Solutions with multiple spike patterns for an elliptic system. Calc. Var. 31, 1–25 (2008). https://doi.org/10.1007/s00526-007-0103-z

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