Abstract
ABSTRACT In this paper, we study the following nonlinear Schrödinger–Newton system { Δ u − V ( x ) u + Ψ u = 0 , x ∈ R 3 , Δ Ψ + 1 2 u 2 = 0 , x ∈ R 3 , which is a nonlinear system obtained by coupling the linear Schrödinger equation of quantum mechanics with the gravitation law of Newtonian mechanics. Assuming that V ( y ) is radial and satisfies some algebraic decay at the infinity, we construct infinitely many non-radial positive solutions which have polygonal symmetry with respect to y 1 and y 2 and are even in y 2 for the system above by the Lyapunov-Schmidt reduction method. Moreover, we have overcame some new difficulties caused by the non-local term.
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