LAUSR

In this paper, we study the existence and multiplicity of standing waves with prescribed L2 ‐norm Schrödinger‐Poisson equations with general nonlinearities in R3 : i∂tψ+Δψ−κ(|x|−1*|φ|2)ψ+f(ψ)=0, where κ>0 and f is superlinear and satisfies the monotonicity condition. To this end, we look for critical points of the following functional Eκ(u)=12∫R3|∇u|2+κ4∫R3(|x|−1*u2)u2−∫R3F(u) constrained on the L2 ‐spheres S(c)=u∈H1(R3):||u||22=c,c>0 , where F(s):=∫0sf(t)dt . We consider the case where Eκ is unbounded from below on S(c) and establish the existence of critical points of Eκ on S(c) for c>0 sufficiently small and under some mild assumptions on f . In addition, we show that there are infinitely many radial critical points {unκ} of Eκ on S(c) when f is odd and present a convergence property of unκ as κ↘0 . Our results generalize some recent results in the literature.