Multiplicity of solutions for elliptic systems via local Mountain Pass method

Pages: 1745 - 1758, Volume 8, Issue 6, November 2009

doi:10.3934/cpaa.2009.8.1745       Abstract        Full Text (204.6K)       Related Articles

Claudianor O. Alves - Universidade Federal da Campina Grande, Departamento de Matemática, 58109-970, Campina Grande - PB, Brazil (email)
Giovany M. Figueiredo - Universidade Federal do Pará, Departamento de Matemática, 66075-100, Belém - PA, Brazil (email)
Marcelo F. Furtado - Universidade de Brasília, Departamento de Matemática, 70910-900, Brasília - DF, Brazil (email)

Abstract: 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.

Keywords:  Elliptic systems, Schrödinger equation, Ljusternik-Schnirelmann theory, positive solutions.
Mathematics Subject Classification:  Primary: 35J20, 35J50; Secondary: 58E05.

Received: October 2008;      Revised: May 2009;      Published: August 2009.