The existence of nontrivial solution of a class of Schrödinger–Bopp–Podolsky system with critical growth

Abstract

AbstractWe consider the following Schrödinger–Bopp–Podolsky problem: $$ \textstyle\begin{cases} -\Delta u+V(x) u+\phi u=\lambda f(u)+ \vert u \vert ^{4}u,& \text{in } \mathbb{R}^{3}, \\ -\Delta \phi +\Delta ^{2}\phi = u^{2}, & \text{in } \mathbb{R}^{3}. \end{cases} $$ { − Δ u + V ( x ) u + ϕ u = λ f ( u ) + | u | 4 u , in  R 3 , − Δ ϕ + Δ 2 ϕ = u 2 , in  R 3 . We prove the existence result without any growth and Ambrosetti–Rabinowitz conditions. In the proofs, we apply a cut-off function, the mountain pass theorem, and Moser iteration.