Critical nonlocal Schrödinger-Poisson system on the Heisenberg group

In this paper, we are concerned with the following a new critical nonlocal Schrödinger-Poisson system on the Heisenberg group: <m:math xmlns:m="http://www.w3.org/1998/Math/MathML" display="block"> <m:mfenced open="{" close=""> <m:mtable columnalign="left left" rowspacing=".1em" columnspacing="1em"> <m:mtr> <m:mtd> <m:mo>−</m:mo> <m:mfenced open="(" close=")"> <m:mrow> <m:mi>a</m:mi> <m:mo>−</m:mo> <m:mi>b</m:mi> <m:mrow> <m:msub> <m:mo>∫</m:mo> <m:mrow> <m:mi mathvariant="normal">Ω</m:mi> </m:mrow> </m:msub> </m:mrow> <m:mrow> <m:mo>|</m:mo> </m:mrow> <m:msub> <m:mi mathvariant="normal">∇</m:mi> <m:mrow> <m:mi>H</m:mi> </m:mrow> </m:msub> <m:mi>u</m:mi> <m:msup> <m:mrow> <m:mo>|</m:mo> </m:mrow> <m:mrow> <m:mn>2</m:mn> </m:mrow> </m:msup> <m:mi>d</m:mi> <m:mi>ξ</m:mi> </m:mrow> </m:mfenced> <m:msub> <m:mi mathvariant="normal">Δ</m:mi> <m:mrow> <m:mi>H</m:mi> </m:mrow> </m:msub> <m:mi>u</m:mi> <m:mo>+</m:mo> <m:mi>μ</m:mi> <m:mi>ϕ</m:mi> <m:mi>u</m:mi> <m:mo>=</m:mo> <m:mi>λ</m:mi> <m:mrow> <m:mo>|</m:mo> </m:mrow> <m:mi>u</m:mi> <m:msup> <m:mrow> <m:mo>|</m:mo> </m:mrow> <m:mrow> <m:mi>q</m:mi> <m:mo>−</m:mo> <m:mn>2</m:mn> </m:mrow> </m:msup> <m:mi>u</m:mi> <m:mo>+</m:mo> <m:mrow> <m:mo>|</m:mo> </m:mrow> <m:mi>u</m:mi> <m:msup> <m:mrow> <m:mo>|</m:mo> </m:mrow> <m:mrow> <m:mn>2</m:mn> </m:mrow> </m:msup> <m:mi>u</m:mi> <m:mo>,</m:mo> <m:mspace width="1em"/> </m:mtd> <m:mtd> <m:mtext>in</m:mtext> <m:mspace width="thinmathspace"/> <m:mi mathvariant="normal">Ω</m:mi> <m:mo>,</m:mo> </m:mtd> </m:mtr> <m:mtr> <m:mtd> <m:mo>−</m:mo> <m:msub> <m:mi mathvariant="normal">Δ</m:mi> <m:mrow> <m:mi>H</m:mi> </m:mrow> </m:msub> <m:mi>ϕ</m:mi> <m:mo>=</m:mo> <m:msup> <m:mi>u</m:mi> <m:mn>2</m:mn> </m:msup> <m:mo>,</m:mo> <m:mspace width="1em"/> </m:mtd> <m:mtd> <m:mtext>in</m:mtext> <m:mspace width="thinmathspace"/> <m:mi mathvariant="normal">Ω</m:mi> <m:mo>,</m:mo> </m:mtd> </m:mtr> <m:mtr> <m:mtd> <m:mi>u</m:mi> <m:mo>=</m:mo> <m:mi>ϕ</m:mi> <m:mo>=</m:mo> <m:mn>0</m:mn> <m:mo>,</m:mo> <m:mspace width="1em"/> </m:mtd> <m:mtd> <m:mtext>on</m:mtext> <m:mspace width="thinmathspace"/> <m:mi mathvariant="normal">∂</m:mi> <m:mi mathvariant="normal">Ω</m:mi> <m:mo>,</m:mo> </m:mtd> </m:mtr> </m:mtable> </m:mfenced> </m:math> $${equation*}{cases} -(a-b∫Ω|∇Hu|²dξ)ΔHu+μφ u=λ|u|q-2u+|u|²u, &in \, Ω,\\ -ΔHφ=u^2, &in\, Ω,\\ u=φ=0, &on\, ∂Ω, {cases} {equation*}$$ where Δ H is the Kohn-Laplacian on the first Heisenberg group <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:msup> <m:mrow> <m:mi mathvariant="double-struck">H</m:mi> </m:mrow> <m:mn>1</m:mn> </m:msup> </m:math> H¹ , and <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mi mathvariant="normal">Ω</m:mi> <m:mo>⊂</m:mo> <m:msup> <m:mrow> <m:mi mathvariant="double-struck">H</m:mi> </m:mrow> <m:mn>1</m:mn> </m:msup> </m:math> Ω⊂ H¹ is a smooth bounded domain, a , b