Existence and multiplicity of normalized solutions for a class of Schrödinger‐Poisson equations with general nonlinearities

Abstract

In this paper, we study the existence and multiplicity of standing waves with prescribed L2‐norm Schrödinger‐Poisson equations with general nonlinearities in double-struckR3: 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 Sfalse(cfalse)={}u∈H1false(double-struckR3false):false|false|ufalse|false|22=c,1emc>0, where Ffalse(sfalse):=∫0sffalse(tfalse)dt. We consider the case where Eκ is unbounded from below on Sfalse(cfalse) and establish the existence of critical points of Eκ on Sfalse(cfalse) for c>0 sufficiently small and under some mild assumptions on f. In addition, we show that there are infinitely many radial critical points false{unκfalse} of Eκ on Sfalse(cfalse) when f is odd and present a convergence property of unκ as κ↘0. Our results generalize some recent results in the literature.