November  2009, 8(6): 1745-1758. doi: 10.3934/cpaa.2009.8.1745

Multiplicity of solutions for elliptic systems via local Mountain Pass method

1. 

Universidade Federal da Campina Grande, Departamento de Matemática, 58109-970, Campina Grande - PB, Brazil

2. 

Universidade Federal do Pará, Departamento de Matemática, 66075-100, Belém - PA, Brazil

3. 

Universidade de Brasília, Departamento de Matemática, 70910-900, Brasília - DF

Received  October 2008 Revised  May 2009 Published  August 2009

We consider the system

$-\varepsilon^{2} \Delta u +W(x)u=Q_{u}(u,v)$ in $\mathbb{R}^N,$

$-\varepsilon^{2} \Delta v +V(x)v=Q_{v}(u,v)$ in $\mathbb{R}^N, $

$u,v \in H^{1}(\mathbb{R}^N),u(x),v(x)>0$ for each $x \in \mathbb{R}^N, $

where $\varepsilon>0$, $W$ and $V$ are positive potentials and $Q$ is a homogeneous function with subcritical growth. We relate the number of solutions with the topology of the set where $W$ and $V$ attain their minimum values. In the proof we apply Ljusternik-Schnirelmann theory.

Citation: Claudianor O. Alves, Giovany M. Figueiredo, Marcelo F. Furtado. Multiplicity of solutions for elliptic systems via local Mountain Pass method. Communications on Pure & Applied Analysis, 2009, 8 (6) : 1745-1758. doi: 10.3934/cpaa.2009.8.1745
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