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Positive solutions for a class of elliptic systems with singular potentials

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Abstract

We deal with the existence of positive solutions for the following class of elliptic system

$$\left\{\begin{array}{lll}-\varepsilon^{2}\Delta u+V_{1}(x)u = K(x)Q_{u}(u, v) & {\rm in} \, \mathbb{R}^{N},\\ -\varepsilon^{2}\Delta v+V_{2}(x)v = K(x)Q_{v}(u, v) & {\rm in}\,\mathbb{R}^{N},\\ u,v \in W^{1,2}(\mathbb{R}^{N}),\quad u,v > 0 & {\rm in}\,\mathbb{R}^{N},\\ {\rm \lim}_{|x|\rightarrow \infty}u(x) = {\rm \lim}_{|x|\rightarrow \infty}v(x)=0,\end{array}\right.\quad\quad\quad{(S)}$$

where \({\varepsilon}\) is a small positive parameter, V 1, V 2, K are nonnegative potentials, Q is a (p + 1)-homogeneous function and p is subcritical; that is, 1 < p < 2* − 1, where 2* = 2N/(N − 2) is the critical Sobolev exponent for \({N\geq 3}\).

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Authors and Affiliations

  1. Departamento de Matemática, Universidade Federal de Minas Gerais, Belo Horizonte, MG, 31270-010, Brazil

    P. C. Carrião

  2. Departamento de Ciências Exatas, Universidade Estadual de Montes Claros, Montes Claros, MG, 39401-089, Brazil

    N. H. Lisboa

  3. Departamento de Matemática, Universidade Federal de Juiz de Fora, Juiz de Fora, MG, 36036-330, Brazil

    O. H. Miyagaki

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  1. P. C. Carrião
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  2. N. H. Lisboa
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  3. O. H. Miyagaki
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Correspondence to O. H. Miyagaki.

Additional information

N. H. Lisboa partially supported by Fapemig/Brazil.

O. H. Miyagaki partially supported by CNPq/Brazil and INCTMat/Brazil.

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Carrião, P.C., Lisboa, N.H. & Miyagaki, O.H. Positive solutions for a class of elliptic systems with singular potentials. Z. Angew. Math. Phys. 66, 317–339 (2015). https://doi.org/10.1007/s00033-014-0402-0

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