Nodal solutions for the Schrodinger-Poisson equations with convolution terms

This paper deals with the following nonlinear Schrodinger-Poisson system with convolution terms: {-Delta u + V(vertical bar x vertical bar)u + b phi u = (I-alpha * vertical bar u vertical bar(p)) vertical bar u vertical bar(p-2)u in R-3, (SPC) -Delta phi = u(2) in R-3, where b > 0 is a parameter, V is an element of C([0,infinity), R+), alpha is an element of (0, 3), I-alpha : R-3 -> R is the Riesz potential and p is an element of (3+alpha/3, 3 + alpha). The presence of nonlocal terms phi u and (I-alpha * vertical bar u vertical bar(p)) vertical bar u vertical bar(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 -> infinity, there is a subsequence, still denoted by {b(n)}, such that u(k)(bn) converges to u(k)(0) in H-1(R-3) as n -> infinity, where u(k)(0) also changes sign exactly k times and is a sign-changing radial solution of the Choquard equation -Delta u + V(vertical bar x vertical bar)u = (I-alpha * vertical bar u vertical bar(p)) vertical bar u vertical bar(p-2)u in R-3. Our result generalizes the existing ones for the Schrodinger-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. (C) 2020 Elsevier Ltd. All rights reserved.