Multiple solutions for a nonhomogeneous Schrödinger–Maxwell system in R3R3

The paper considers the following nonhomogeneous Schrödinger–Maxwell system: equation(SM){−Δu+u+λϕ(x)u=|u|p−1u+g(x),x∈R3,−Δϕ=u2,x∈R3, where λ>0λ>0, p∈(1,5)p∈(1,5), and 0≤g(x)=g(|x|)∈L2(R3)0≤g(x)=g(|x|)∈L2(R3). There seem to be no results on the existence of multiple solutions to problem (SM) for p∈(1,3)p∈(1,3). In this paper, we find that there is a constant Cp>0Cp>0 such that problem (SM) has at least two solutions for all p∈(1,5)p∈(1,5) provided that ‖g‖L2≤Cp‖g‖L2≤Cp, however, for p∈(1,2]p∈(1,2] we need λ>0λ>0 is small. Moreover, Cp=(p−1)2p[(p+1)Sp+12p]1/(p−1), where SS is the Sobolev constant.