POSITIVITY AND NEGATIVITY OF SOLUTIONS TO n × n WEIGHTED SYSTEMS INVOLVING THE LAPLACE OPERATOR ON R N

B Enédicteen´, Enédicte Alziary, Jacqueline Fleckinger, Marie-H, El`ene El`, El`ene Lecureux, N Wei
2012 Electronic Journal of Differential Equations   unpublished
We consider the sign of the solutions of a n × n system defined on the whole space R N , N ≥ 3 and a weight function ρ with a positive part decreasing fast enough, −∆U = λρ(x)M U + F, where F is a vector of functions, M is a n×n matrix with constant coefficients, not necessarily cooperative, and the weight function ρ is allowed to change sign. We prove that the solutions of the n × n system exist and then we prove the local fundamental positivity and local fundamental negativity of the
more » ... when |λσ 1 − λρ| is small enough, where σ 1 is the largest eigenvalue of the constant matrix M and λρ is the "principal" eigenvalue of −∆u = λρ(x)u, lim |x|→∞ u(x) = 0; u(x) > 0, x ∈ R N .
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