Dichotomous concentrating solutions for a Schrödinger–Newton equation.

This paper is concerned with the following Schrödinger–Newton equation - ε 2 Δ u + V (x) u = 1 ε 2 ∫ R 3 u 2 y x - y d y u , x ∈ R 3 , where ε is a positive parameter and V(x) is the potential function. We demonstrate an interesting phenomenon, which we call dichotomy, for concentrating solutions of the above Schrödinger–Newton equation. More specifically, we show the existence of infinitely many concentrating solutions which concentrate both in a bounded domain and near infinity. In addition, the non-degeneracy of the ground state is established for the above Schrödinger–Newton equation with non-constant potentials. [ABSTRACT FROM AUTHOR]