Ground State Solutions for the Nonlinear Schrödinger–Bopp–Podolsky System with Critical Sobolev Exponent

By Lin Li, Patrizia Pucci, Xianhua Tang. Advanced Nonlinear Studies, 2020

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Abstract In this paper, we study the existence of ground state solutions for the nonlinear Schrödinger–Bopp–Podolsky system with critical Sobolev exponent { - Δ ⁢ u + V ⁢ ( x ) ⁢ u + q 2 ⁢ ϕ ⁢ u = μ ⁢ | u | p - 1 ⁢ u + | 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+q^{2}\phi u&% \displaystyle=\mu|u|^{p-1}u+|u|^{4}u&&\displaystyle\phantom{}\mbox{in }\mathbb% {R}^{3},\\ \displaystyle{-}\Delta\phi+a^{2}\Delta^{2}\phi&\displaystyle=4\pi u^{2}&&% \displaystyle\phantom{}\mbox{in }\mathbb{R}^{3},\end{aligned}\right. where μ>0{\mu>0} is a parameter and 2<p<5{2<p<5}. Under certain assumptions on V, we prove the existence of a nontrivial ground state solution, using the method of the Pohozaev–Nehari manifold, the arguments of Brézis–Nirenberg, the monotonicity trick and a global compactness lemma.

Subfields: Applied Mathematics, Mathematical Physics

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