| 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.