Laure Cardoulis
Quoc Anh Ngo
Hoang Quoc Toan

Existence of non-negative Solutions for cooperative elliptic Systems involving Schr\"{o}dinger Operators in the whole Space

The paper is published: Rostocker Mathematisches Kolloquium, Rostock. Math. Kolloq. 63, 63 - 77 (2008)

MSC: 35J60   Nonlinear PDE of elliptic type
  35J65   Nonlinear boundary value problems for linear elliptic PDE; boundary value problems for nonlinear elliptic PDE
ZDM: -  
CR: -  
PACS: -  

Abstract:   In this paper, we obtain some new results on the existence of non-negative solutions for systems of the form \[ (-\Delta +q_i)u_i = \mu_i m_iu_i+\sum_{j=1;j\neq i}^n a_{ij} u_j +f_i(x,u_1,...,u_n) \mbox{ in } \mathbb{R}^N,\; i=1,...,n, \] where each of the $q_i$ are positive potentials satisfying $\lim_{|x|\rightarrow +\infty} q_i(x)=+\infty$, each of the $m_i$ are bounded positive weights, each of the $a_{ij}$, $i \neq j$, are bounded non-negative weights and each of the $\mu_i$ are real parameters. Depending upon the hypotheses on $f_i$, we obtain some new results by using sub- and super-solution methods and the Schauder Fixed Point Theorem.

Keywords:   Nonlinear PDE of elliptic type, boundary value problems for nonlinear elliptic PDE
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