On the Schrödinger–Maxwell system involving sublinear terms

In this paper, we study the coupled Schrödinger–Maxwell system equation(SMλSMλ){−Δu+u+eϕu=λα(x)f(u)inR3,−Δϕ=4πeu2inR3, where e>0e>0,α∈L∞(R3)∩L6/(5−q)(R3)α∈L∞(R3)∩L6/(5−q)(R3) for some q∈(0,1)q∈(0,1), and the continuous function f:R→Rf:R→R is superlinear at zero and sublinear at infinity, e.g., f(s)=min(|s|r,|s|p)f(s)=min(|s|r,|s|p) with 00λ>0, we prove a non-existence result for (SMλ)(SMλ), while for λ>0λ>0 large enough, a recent Ricceri-type result guarantees the existence of at least two non-trivial solutions for (SMλ)(SMλ) as well as the ‘stability’ of system (SMλ)(SMλ) with respect to an arbitrary subcritical perturbation of the Schrödinger equation.