Bénédicte Alziary
Jacqueline Fleckinger

Semi-linear cooperative elliptic systems involving Schrödinger operators: Groundstate positivity or negativity

The paper is published: Rostocker Mathematisches Kolloquium, Rostock. Math. Kolloq. 71, 41 - 56 (2018)

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 $U$.

Keywords:   Semilinear elliptic equations, Schrödinger operator

karin.martin@uni-rostock.de
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