On nonlinear Schrödinger–Poisson equations with general potentials

We study the existence of infinitely many finite energy radial solutions to the nonlinear Schrödinger–Poisson equations { Δ u − u − ϕ ( x ) u + f ( u ) = 0 in  R 3 Δ ϕ + u 2 = 0 , lim | x | → ∞ ϕ ( x ) = 0 in  R 3 (NSPE for short) under some structure conditions on the nonlinearity function f . As consequences of the main result, we can provide examples of f which guarantee the existence of infinitely many finite energy solutions but (i) ) grows faster than t 2 and slower than t p for all p > 2 or ) is the same as | t | t when | t | ≤ t 0 for arbitrarily given t 0 > 0 . ( t ) = | t | p − 1 t , it is known that (NSPE) admits no finite energy nontrivial solutions when p ∈ ( 1 , 2 ] and admits infinitely many finite energy solutions when p ∈ ( 2 , 5 ) so examples (i) and (ii) show some interesting features of (NSPE).