Marie-Hélène Lécureux

Comparison with ground state for solutions of non cooperative systems of Schrödinger operators on $\mathbb{R}^N$

The paper is published: Rostocker Mathematisches Kolloquium, Rostock. Math. Kolloq. 65, 51 - 69 (2010)

MSC: 35B51   Comparison principles

Abstract:   We study the sign of solutions of a system $\mathcal L U=\lambda U+MU+F$, on the whole space $\RR^N$, more precisely, we compare the components of $U$ with the ground state solution. Here $\mathcal L$ is a diagonal matrix of Schrödinger operators of the form $Lu\;:=\;-\Delta u+q u$, $F$ is a vector of functions in $L^2(\RR^N)$, and $M$ is a matrix, not necessarily cooperative. When $M$ is a constant matrix, we prove the existence of a real $\Lambda$ playing the role of principal eigenvalue: if $\left|\lambda-\Lambda\right|$ is sufficiently small, $U$ exists and the sign of each entry is fixed. The sign of each entry changes as $\lambda$ grows and get over $\Lambda$. We study the case of a variable $M$ for a $2\times2$ system.

Keywords:   Comparison principles
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