Multiplicity of solutions for Schrödinger–Bopp–Podolsky systems

Abstract

Abstract In this paper, we study the existence and multiplicity of solutions for the Schrödinger–Bopp–Podolsky system { - Δ ⁢ u + V ⁢ ( x ) ⁢ u + ϕ ⁢ u = f ⁢ ( u ) + λ ⁢ | u | 4 ⁢ u in ⁢ ℝ 3 , - Δ ⁢ ϕ + a 2 ⁢ Δ 2 ⁢ ϕ = 4 ⁢ π ⁢ u 2 in ⁢ ℝ 3 , \left\{\begin{aligned} \displaystyle{-}\Delta u+V(x)u+\phi u&\displaystyle=f(u% )+\lambda|u|^{4}u&&\displaystyle\phantom{}\text{in }\mathbb{R}^{3},\\ \displaystyle{-}\Delta\phi+a^{2}\Delta^{2}\phi&\displaystyle=4\pi u^{2}&&% \displaystyle\phantom{}\text{in }\mathbb{R}^{3},\end{aligned}\right. where x ∈ ℝ 3 {x\in\mathbb{R}^{3}} , a > 0 {a>0} , V ⁢ ( x ) ∈ 𝒞 ⁢ ( ℝ 3 , ℝ ) {V(x)\in\mathcal{C}(\mathbb{R}^{3},\mathbb{R})} . Using variational methods and the symmetric mountain pass theorem, we establish the existence of multiple solutions for this system.