On a singular Hamiltonian elliptic systems involving critical growth indimension two

Manassés de Souza

doi:10.3934/cpaa.2012.11.1859

Article Contents

2012, Volume 11, Issue 5: 1859-1874. Doi: 10.3934/cpaa.2012.11.1859

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On a singular Hamiltonian elliptic systems involving critical growth in dimension two

Manassés de Souza^1,

1.
Departamento de Matem

Received: March 2011

Revised: December 2011

Published: September 2012

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Abstract

In this paper we study the existence of nontrivial solutions for the strongly indefinite elliptic system \begin{eqnarray*} -\Delta u + b(x) u = \frac{g(v)}{|x|^\alpha}, v > 0 in R^2, \\ -\Delta v + b(x) v = \frac{f(u)}{|x|^\beta}, u > 0 in R^2, \end{eqnarray*} where $\alpha, \beta \in [0,2)$, $b: \mathbb{R}^2\rightarrow \mathbb{R}$ is a continuous positive potential bounded away from zero and which can be ``large" at the infinity and the functions $f: \mathbb{R}\rightarrow \mathbb{R}$ and $g: \mathbb{R} \rightarrow \mathbb{R}$ behaves like $\exp(\gamma s^2)$ when $|s|\rightarrow+\infty$ for some $\gamma >0$.

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Mathematics Subject Classification: Primary: 35J50, 35J55; Secondary: 35Q55.

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