Investigates ground state solutions in nonlinear Schrödinger–Bopp–Podolsky system, suggesting new insights into critical Sobolev exponent behavior.
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> \{{aligned} {}{-}Δ u+V(x)u+q²φ u&% =μ|u|ᵖ⁻¹u+|u|⁴u&&{}in % {R}³,\\ {-}Δφ+a²Δ²φ&=4π u²&&% {}in R³,{aligned}. where <jats:inline-formula id="j_ans-2020-20
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Li et al. (2020) studied this question.
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