MSC: | 35J61 | Semilinear elliptic equations |
35J10 | Schrödinger operator |
Abstract:
We study here the behavior of the solutions to a $2\times 2$ semi-linear cooperative system involving Schr\" odinger operators (considered in its variational form):
$$LU:=(-\Delta + q(x))U = AU+\mu U + F(x,U)\; \mbox{ in }\; \R^N$$
$$U(x)_{|x|\rightarrow \infty} \rightarrow 0 $$
where $q$ is a continuous positive potential tending to $+\infty$ at infinity; $\mu$ is a real parameter varying near the principal eigenvalue of the system; $U$ is a column vector with components $u_1$ and $u_2$ and $A$ is a
square cooperative matrix with constant coefficient.
$F$ is a column vector with components $f_1$ and $f_2$ depending eventually on $U$.