Existence of semiclassical solutions for some critical Schrödinger–Poisson equations with potentials. (September 2020) - Yang, Minbo
Abstract: In this paper we are interested in the existence of semiclassical solutions for the critical Schrödinger–Poisson equations with multiple potentials. Under suitable assumptions on the potentials, we are able to prove the existence of semiclassical solutions ( u ε, ϕ ε ) with maximum points x ε of u ε concentrating at a special set C P characterized by some of the potentials. Moreover, for any sequence x ε → x 0 ∈ C P, by setting v ε ( x ) ≔ u ε ( ε x + x ε ), the ground solutions v ε convergence strongly in H 1 ( R 3 ) to a ground state solution v of − Δ v + V ( x 0 ) v = P ( x 0 ) g ( v ) + Q ( x 0 ) | v | 4 v, in R 3 .