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 {Δu−u−ϕ(x)u+f(u)=0in R3Δϕ+u2=0,lim|x|→∞ϕ(x)=0in R3 (NSPE for short) under some structure conditions on the nonlinearity function ff. As consequences of the main result, we can provide examples of ff which guarantee the existence of infinitely many finite energy solutions but (i)f(t)f(t) grows faster than t2t2 and slower than tptp for all p>2p>2 or(ii)f(t)f(t) is the same as |t|t|t|t when |t|≤t0|t|≤t0 for arbitrarily given t0>0t0>0. If f(t)=|t|p−1tf(t)=|t|p−1t, it is known that (NSPE) admits no finite energy nontrivial solutions when p∈(1,2]p∈(1,2] and admits infinitely many finite energy solutions when p∈(2,5)p∈(2,5) so examples (i) and (ii) show some interesting features of (NSPE).