Ground State Solutions for the Nonlinear Schrödinger–Bopp–Podolsky System with Critical Sobolev Exponent
Lin Li, Patrizia Pucci, Xianhua Tang
Abstract
Abstract In this paper, we study the existence of ground state solutions for the nonlinear Schrödinger–Bopp–Podolsky system with critical Sobolev exponent <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mo>{</m:mo> <m:mtable columnspacing="0pt" displaystyle="true" rowspacing="0pt"> <m:mtr> <m:mtd columnalign="right"> <m:mrow> <m:mrow> <m:mo>-</m:mo> <m:mrow> <m:mi mathvariant="normal">Δ</m:mi> <m:mo></m:mo> <m:mi>u</m:mi> </m:mrow> </m:mrow> <m:mo>+</m:mo> <m:mrow> <m:mi>V</m:mi> <m:mo></m:mo> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mi>x</m:mi> <m:mo stretchy="false">)</m:mo> </m:mrow> <m:mo></m:mo> <m:mi>u</m:mi> </m:mrow> <m:mo>+</m:mo> <m:mrow> <m:msup> <m:mi>q</m:mi> <m:mn>2</m:mn> </m:msup> <m:mo></m:mo> <m:mi>ϕ</m:mi> <m:mo></m:mo> <m:mi>u</m:mi> </m:mrow> </m:mrow> </m:mtd> <m:mtd columnalign="left"> <m:mrow> <m:mi/> <m:mo>=</m:mo> <m:mrow> <m:mrow> <m:mi>μ</m:mi> <m:mo></m:mo> <m:msup> <m:mrow> <m:mo stretchy="false">|</m:mo> <m:mi>u</m:mi> <m:mo stretchy="false">|</m:mo> </m:mrow> <m:mrow> <m:mi>p</m:mi> <m:mo>-</m:mo> <m:mn>1</m:mn> </m:mrow> </m:msup> <m:mo></m:mo> <m:mi>u</m:mi> </m:mrow> <m:mo>+</m:mo> <m:mrow> <m:msup> <m:mrow> <m:mo stretchy="false">|</m:mo> <m:mi>u</m:mi> <m:mo stretchy="false">|</m:mo> </m:mrow> <m:mn>4</m:mn> </m:msup> <m:mo></m:mo> <m:mi>u</m:mi> </m:mrow> </m:mrow> </m:mrow> </m:mtd> <m:mtd/> <m:mtd columnalign="right"> <m:mrow> <m:mrow> <m:mtext>in </m:mtext> <m:mo></m:mo> <m:msup> <m:mi>ℝ</m:mi> <m:mn>3</m:mn> </m:msup> </m:mrow> <m:mo>,</m:mo> </m:mrow> </m:mtd> </m:mtr> <m:mtr> <m:mtd columnalign="right"> <m:mrow> <m:mrow> <m:mo>-</m:mo> <m:mrow> <m:mi mathvariant="normal">Δ</m:mi> <m:mo></m:mo> <m:mi>ϕ</m:mi> </m:mrow> </m:mrow> <m:mo>+</m:mo> <m:mrow> <m:msup> <m:mi>a</m:mi> <m:mn>2</m:mn> </m:msup> <m:mo></m:mo> <m:msup> <m:mi mathvariant="normal">Δ</m:mi> <m:mn>2</m:mn> </m:msup> <m:mo></m:mo> <m:mi>ϕ</m:mi> </m:mrow> </m:mrow> </m:mtd> <m:mtd columnalign="left"> <m:mrow> <m:mi/> <m:mo>=</m:mo> <m:mrow> <m:mn>4</m:mn> <m:mo></m:mo> <m:mi>π</m:mi> <m:mo></m:mo> <m:msup> <m:mi>u</m:mi> <m:mn>2</m:mn> </m:msup> </m:mrow> </m:mrow> </m:mtd> <m:mtd/> <m:mtd columnalign="right"> <m:mrow> <m:mrow> <m:mtext>in </m:mtext> <m:mo></m:mo> <m:msup> <m:mi>ℝ</m:mi> <m:mn>3</m:mn> </m:msup> </m:mrow> <m:mo>,</m:mo> </m:mrow> </m:mtd> </m:mtr> </m:mtable> </m:mrow> </m:math> \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 <jats:inline-formula id="j_ans-2020-20