Multiple solutions for a class of Schrödinger–Bopp–Podolsky system with indefinite nonlinearities

In this paper, we focus on the existence and multiplicity of solutions for the Schrödinger–Bopp–Podolsky system: {−Δu+ωu+l(x)ϕu=g(x)|u|p−2u+h(x)|u|q−2uin R3,−Δϕ+a2Δ2ϕ=l(x)u2in R3, where 1<q<2<p<4 and ω,a>0 are parameters. The functions l(x) and h(x) satisfy some suitable conditions as well as the nonlinearity g∈C(R3) which is allowed to be sign–changing in R3. By introducing some novel techniques to recover the compactness of the Sobolev embedding H1(R3)↪Lr(R3)(2≤r<6), the Ekeland variational principle and an innovative constraint method of the Nehari manifold are adopted to get three nontrivial solutions for the above problem.