Existence and convergence of sign-changing solutions for Kirchhoff-type <a:math xmlns:a="http://www.w3.org/1998/Math/MathML"> <a:mi>p</a:mi></a:math>-Laplacian problems involving critical exponent in <b:math xmlns:b="http://www.w3.org/1998/Math/MathM

We investigate the existence of sign-changing solutions for Kirchhoff-type problems with <a:math xmlns:a="http://www.w3.org/1998/Math/MathML"> <a:mi>p</a:mi> </a:math>-Laplacian involving critical exponent: <b:math xmlns:b="http://www.w3.org/1998/Math/MathML" display="block"> <b:mo>−</b:mo> <b:mrow> <b:mo>(</b:mo> <b:mrow> <b:mn>1</b:mn> <b:mo>+</b:mo> <b:mi>b</b:mi> <b:mo fence="false" stretchy="false">|</b:mo> <b:mi mathvariant="normal">∇</b:mi> <b:mi>v</b:mi> <b:msubsup> <b:mo fence="false" stretchy="false">|</b:mo> <b:mi>p</b:mi> <b:mi>p</b:mi> </b:msubsup> </b:mrow> <b:mo>)</b:mo> </b:mrow> <b:msub> <b:mi mathvariant="normal">Δ</b:mi> <b:mi>p</b:mi> </b:msub> <b:mi>v</b:mi> <b:mo>+</b:mo> <b:mi>a</b:mi> <b:mo stretchy="false">(</b:mo> <b:mi>x</b:mi> <b:mo stretchy="false">)</b:mo> <b:mo fence="false" stretchy="false">|</b:mo> <b:mi>v</b:mi> <b:msup> <b:mo fence="false" stretchy="false">|</b:mo> <b:mrow class="MJX-TeXAtom-ORD"> <b:mi>p</b:mi> <b:mo>−</b:mo> <b:mn>2</b:mn> </b:mrow> </b:msup> <b:mi>v</b:mi> <b:mo>=</b:mo> <b:mo fence="false" stretchy="false">|</b:mo> <b:mi>v</b:mi> <b:msup> <b:mo fence="false" stretchy="false">|</b:mo> <b:mrow class="MJX-TeXAtom-ORD"> <b:msup> <b:mi>p</b:mi> <b:mo>∗</b:mo> </b:msup> <b:mo>−</b:mo> <b:mn>2</b:mn> </b:mrow> </b:msup> <b:mi>v</b:mi> <b:mo>+</b:mo> <b:mi>λ</b:mi> <b:mi>f</b:mi> <b:mo stretchy="false">(</b:mo> <b:mi>v</b:mi> <b:mo stretchy="false">)</b:mo> <b:mo>,</b:mo> <b:mspace width="1em"/> <b:mi>x</b:mi> <b:mo>∈</b:mo> <b:msup> <b:mrow class="MJX-TeXAtom-ORD"> <b:mi mathvariant="double-struck">R</b:mi> </b:mrow> <b:mi>N</b:mi> </b:msup> <b:mo>,</b:mo> </b:math> where <ab:math xmlns:ab="http://www.w3.org/1998/Math/MathML"> <ab:mi>b</ab:mi> </ab:math> and <bb:math xmlns:bb="http://www.w3.org/1998/Math/MathML"> <bb:mi>λ</bb:mi> </bb:math> are positive parameters, <cb:math xmlns:cb="http://www.w3.org/1998/Math/MathML"> <cb:msub> <cb:mi mathvariant="normal">Δ</cb:mi> <cb:mi>p</cb:mi> </cb:msub> <cb:mi>v</cb:mi> <cb:mo>=</cb:mo> <cb:mi>div</cb:mi> <cb:mo>⁡</cb:mo> <cb:mo stretchy="false">(</cb:mo> <cb:mo fence="false" stretchy="false">|</cb:mo> <cb:mi mathvariant="normal">∇</cb:mi> <cb:mi>v</cb:mi> <cb:msup> <cb:mo fence="false" stretchy="false">|</cb:mo> <cb:mrow class="MJX-TeXAtom-ORD"> <cb:mi>p</cb:mi> <cb:mo>−</cb:mo> <cb:mn>2</cb:mn> </cb:mrow> </cb:msup> <cb:mi mathvariant="normal">∇</cb:mi> <cb:mi>v</cb:mi> <cb:mo stretchy="false">)</cb:mo> </cb:math> , <nb:math xmlns:nb="http://www.w3.org/1998/Math/MathML"> <nb:msup> <nb:mi>p</nb:mi> <nb:mo>∗</nb:mo> </nb:msup> <nb:mo>=</nb:mo> <nb:mfrac> <nb:mrow> <nb:mi>N</nb:mi> <nb:mi>p</nb:mi> </nb:mrow> <nb:mrow> <nb:mi>N</nb:mi> <nb:mo>−</nb:mo> <nb:mi>p</nb:mi> </nb:mrow> </nb:mfrac> </nb:math> , <ob:math xmlns:ob="http://www.w3.org/1998/Math/MathML"> <ob:mn>1</ob:mn> <ob:mo>&lt;</ob:mo> <ob:mi>p</ob:mi> <ob:mo>&lt;</ob:mo> <ob:mi>N</ob:mi> </ob:math>, and <pb:math xmlns:pb="http://www.w3.org/1998/Math/MathML"> <pb:mo fence="false" stretchy="false">|</pb:mo> <pb:mo>⋅</pb:mo> <pb:msub> <pb:mo fence="false" stretchy="false">|</pb:mo> <pb:mi>p</pb:mi> </pb:msub> </pb:math> is the Lebesgue <ub:math xmlns:ub="http://www.w3.org/1998/Math/MathML"> <ub:msup> <ub:mi>L</ub:mi> <ub:mi>p</ub:mi> </ub:msup> </ub:math>-norm. For sufficiently large <vb:math xmlns:vb="http://www.w3.org/1998/Math/MathML"> <vb:mi>λ</vb:mi> </vb:math>, employing minimization techniques, quantitative deformation lemma and the constrained variational method, we demonstrate the existence of a least-energy sign-changing solution, whose energy is greater than twice that of the ground state solution. Additionally, we show the convergence behavior of the solution as the parameter <wb:math xmlns:wb="http://www.w3.org/1998/Math/MathML"> <wb:mi>b</wb:mi> <wb:mo stretchy="false">↘</wb:mo> <wb:mn>0</wb:mn> </wb:math>. Our findings generalize and extend upon recent results in the literature.