A perturbation approach for the Schrödinger-Born-Infeld system: Solutions in the subcritical and critical case

In this paper, we study the following Schrödinger-Born-Infeld system with a general nonlinearity {−Δu+u+ϕu=f(u)+μ|u|4uinR3,−div([Formula presented])=u2inR3,u(x)→0,ϕ(x)→0,asx→∞, where μ≥0 and f∈C(R,R) satisfies suitable assumptions. This system arises from a suitable coupling of the nonlinear Schrödinger equation and the Born-Infeld theory. We use a new perturbation approach to prove the existence and multiplicity of nontrivial solutions of the above system in the subcritical and critical case. We emphasise that our results cover the case f(u)=|u|p−1u for p∈(2,5/2] and μ=0 which was left in [2] as an open problem.