A note on non-radial sign-changing solutions for the Schrödinger-Poisson problem

Abstract. We study the following system of equations known as Schrödinger-Poisson problem −2∆v + v + φv = f(v) in RN −∆φ = aNv2 in RN φ → 0 as |x | → +∞ where > 0 is a small parameter, f: R → R is given, N ≥ 3, aN is the surface measure of the unit sphere in RN and the unknowns are v, φ: RN → R. We prove that the set of sign-changing solutions has a rich structure in the semiclassical limit: we construct non-radial multi-peak solutions with an arbitrary large number of positive and negative peaks which are displaced in suitable symmetric configurations and which collapse to the same point as → 0. The proof is based on the Lyapunov-Schmidt reduction. 1