Existence results for critical growth Kohn-Laplace equations with jumping nonlinearities

Abstract This article is concerned with the existence of nontrivial solutions to critical growth Kohn-Laplace equations with jumping nonlinearities. Or, more specifically, we consider the following Kohn-Laplace problem: <m:math xmlns:m="http://www.w3.org/1998/Math/MathML" display="block"> <m:mfenced open="{" close=""> <m:mrow> <m:mtable displaystyle="true"> <m:mtr> <m:mtd columnalign="left"> <m:mo>−</m:mo> <m:msub> <m:mrow> <m:mi>Δ</m:mi> </m:mrow> <m:mrow> <m:mi>H</m:mi> </m:mrow> </m:msub> <m:mi>u</m:mi> <m:mo>=</m:mo> <m:mi>b</m:mi> <m:msup> <m:mrow> <m:mi>u</m:mi> </m:mrow> <m:mrow> <m:mo>+</m:mo> </m:mrow> </m:msup> <m:mo>−</m:mo> <m:mi>a</m:mi> <m:msup> <m:mrow> <m:mi>u</m:mi> </m:mrow> <m:mrow> <m:mo>−</m:mo> </m:mrow> </m:msup> <m:mo>+</m:mo> <m:msup> <m:mrow> <m:mo>∣</m:mo> <m:mi>u</m:mi> <m:mo>∣</m:mo> </m:mrow> <m:mrow> <m:msup> <m:mrow> <m:mi>Q</m:mi> </m:mrow> <m:mrow> <m:mo>∗</m:mo> </m:mrow> </m:msup> <m:mo>−</m:mo> <m:mn>2</m:mn> </m:mrow> </m:msup> <m:mi>u</m:mi> <m:mo>,</m:mo> <m:mspace width="1.0em"/> </m:mtd> <m:mtd columnalign="left"> <m:mi mathvariant="normal">in</m:mi> <m:mspace width="0.33em"/> <m:mi mathvariant="normal">Ω</m:mi> <m:mo>,</m:mo> </m:mtd> </m:mtr> <m:mtr> <m:mtd columnalign="left"> <m:mi>u</m:mi> <m:mo>=</m:mo> <m:mn>0</m:mn> <m:mo>,</m:mo> <m:mspace width="1.0em"/> </m:mtd> <m:mtd columnalign="left"> <m:mi mathvariant="normal">on</m:mi> <m:mspace width="0.33em"/> <m:mo>∂</m:mo> <m:mi mathvariant="normal">Ω</m:mi> <m:mo>,</m:mo> </m:mtd> </m:mtr> </m:mtable> </m:mrow> </m:mfenced> </m:math> \left\{\begin{array}{ll}-{\Delta }_{H}u=b{u}^{+}-a{u}^{-}+{| u| }^{{Q}^{\ast }-2}u,\hspace{1.0em}&amp; {\rm{in}}\hspace{0.33em}\Omega ,\\ u=0,\hspace{1.0em}&amp; {\rm{on}}\hspace{0.33em}\partial \Omega ,\end{array}\right. where <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:msub> <m:mrow> <m:mi>Δ</m:mi> </m:mrow> <m:mrow> <m:mi>H</m:mi> </m:mrow> </m:msub> </m:math> {\Delta }_{H} is the Kohn-Laplacian on the 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:mrow> <m:mi>n</m:mi> </m:mrow> </m:msup> </m:math> {{\mathbb{H}}}^{n} with <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mi>n</m:mi> <m:mo>&gt;</m:mo> <m:mn>1</m:mn> </m:math> n\gt 1 , <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:msup> <m:mrow> <m:mi>Q</m:mi> </m:mrow> <m:mrow> <m:mo>∗</m:mo> </m:mrow> </m:msup> <m:mo>=</m:mo> <m:mn>2</m:mn> <m:mo>+</m:mo> <m:mfrac> <m:mrow> <m:mn>2</m:mn> </m:mrow> <m:mrow> <m:mi>n</m:mi> </m:mrow> </m:mfrac> </m:math> {Q}^{\ast }=2+\frac{2}{n} is the critical Sobolev exponent for <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:msub> <m:mrow> <m:mi>Δ</m:mi> </m:mrow> <m:mrow> <m:mi>H</m:mi> </m:mrow> </m:msub> </m:math> {\Delta }_{H} ,