Existence and Concentration of Semiclassical Bound States for a Quasilinear Schrödinger-Poisson System

In the paper we consider the following quasilinear Schrödinger–Poisson system in the whole space R 3 - ε 2 Δ u + ( V + ϕ ) u = u u p - 1 - Δ ϕ - β Δ 4 ϕ = u 2 , where 1 < p < 5 , β > 0 , V : R 3 → ] 0 , ∞ [ , and look for solutions u , ϕ : R 3 → R in the semiclassical regime, namely when ε → 0 . By means of the Lyapunov–Schmidt method we estimate the number of solutions by the cup-length of the critical manifold of the external potential V .