Bifurcation and multiplicity results for critical problems involving the p-Grushin operator

In this article, we prove a bifurcation and multiplicity result for a critical problem involving a degenerate nonlinear operator <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:msubsup> <m:mrow> <m:mi>Δ</m:mi> </m:mrow> <m:mrow> <m:mi>γ</m:mi> </m:mrow> <m:mrow> <m:mi>p</m:mi> </m:mrow> </m:msubsup> </m:math> {Δ }γᵖ . We extend to a generic <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mi>p</m:mi> <m:mo>></m:mo> <m:mn>1</m:mn> </m:math> p> 1 a result, which was proved only when <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mi>p</m:mi> <m:mo>=</m:mo> <m:mn>2</m:mn> </m:math> p=2 . When <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mi>p</m:mi> <m:mo>≠</m:mo> <m:mn>2</m:mn> </m:math> p≠ 2 , the nonlinear operator <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mo>−</m:mo> <m:msubsup> <m:mrow> <m:mi>Δ</m:mi> </m:mrow> <m:mrow> <m:mi>γ</m:mi> </m:mrow> <m:mrow> <m:mi>p</m:mi> </m:mrow> </m:msubsup> </m:math> -{Δ }γᵖ has no linear eigenspaces, so our extension is nontrivial and requires an abstract critical theorem, which is not based on linear subspaces. We use an abstract result based on a pseudo-index related to the <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:msub> <m:mrow> <m:mi mathvariant="double-struck">Z</m:mi> </m:mrow> <m:mrow> <m:mn>2</m:mn> </m:mrow> </m:msub> </m:math> {{Z}}₂ -cohomological index that is applicable here. We provide a version of the Lions’ concentration-compactness principle for our operator.