Minimal energy solutions for repulsive nonlinear Schrödinger systems

Abstract

In this paper we establish existence and nonexistence results concerning fully nontrivial minimal energy solutions of the nonlinear Schrödinger system−Δu+u=|u|2q−2u+b|u|q−2u|v|qin Rn,−Δv+ω2v=|v|2q−2v+b|u|q|v|q−2vin Rn. We consider the repulsive case b<0 and assume that the exponent q satisfies 1<q<nn−2 in case n≥3 and 1<q<∞ in case n=1 or n=2. For space dimensions n≥2 and arbitrary b<0 we prove the existence of fully nontrivial nonnegative solutions which converge to a solution of some optimal partition problem as b→−∞. In case n=1 we prove that minimal energy solutions exist provided the coupling parameter b has small absolute value whereas fully nontrivial solutions do not exist if 1<q≤2 and b has large absolute value.