Nodal solutions for the Schrödinger–Poisson equations with convolution terms. (July 2020) - Guo, Hui; Wu, Dan
Abstract: This paper deals with the following nonlinear Schrödinger–Poisson system with convolution terms: ( S P C ) − Δ u + V ( | x | ) u + b ϕ u = I α ∗ | u | p | u | p − 2 u in R 3, − Δ ϕ = u 2 in R 3, where b > 0 is a parameter, V ∈ C ( [ 0, ∞ ), R + ), α ∈ ( 0, 3 ), I α : R 3 → R is the Riesz potential and p ∈ ( 3 + α 3, 3 + α ) . The presence of nonlocal terms ϕ u and I α ∗ | u | p | u | p − 2 u makes the variational functional of (SPC) totally different from the case of b = 0 or the case with pure power nonlinearity. Taking advantage of the results from the matrix theory and the Brouwer degree theory, we introduce some new analytic techniques to prove that for any given integer k ≥ 1, (SPC) admits a sign changing radial solution u k b for p > 4, which changes sign exactly k times. Furthermore, for any sequence { b n } with b n → 0 + as n → ∞, there is a subsequence, still denoted by { b n }, such that u k b n converges to u k 0 in H 1 ( R 3 ) as n → ∞, where u k 0 also changes sign exactly k times and is a sign-changing radial solution of the Choquard equation − Δ u + V ( | x | ) u = I α ∗ | u | p | u | p − 2 u in R 3 . Our result generalizes the existing ones for the Schrödinger–Poisson equations and Choquard equations, and seems to be the first result of such radial solutions for an equation with two competing convolution terms. Besides, we show that the degeneracy for this existence result happens for p < 2 .