On the singularly perturbation fractional Kirchhoff equations: Critical case
Guangze Gu, Zhipeng Yang
Abstract
Abstract This article deals with the following fractional Kirchhoff problem with critical exponent <m:math xmlns:m="http://www.w3.org/1998/Math/MathML" display="block"> <m:mfenced open="(" close=")"> <m:mrow> <m:mi>a</m:mi> <m:mo>+</m:mo> <m:mi>b</m:mi> <m:munder> <m:mrow> <m:mrow> <m:mo>∫</m:mo> </m:mrow> </m:mrow> <m:mrow> <m:msup> <m:mrow> <m:mi mathvariant="double-struck">R</m:mi> </m:mrow> <m:mrow> <m:mi>N</m:mi> </m:mrow> </m:msup> </m:mrow> </m:munder> <m:mo>∣</m:mo> <m:msup> <m:mrow> <m:mrow> <m:mo>(</m:mo> <m:mrow> <m:mo>−</m:mo> <m:mi mathvariant="normal">Δ</m:mi> </m:mrow> <m:mo>)</m:mo> </m:mrow> </m:mrow> <m:mrow> <m:mstyle displaystyle="false"> <m:mfrac> <m:mrow> <m:mi>s</m:mi> </m:mrow> <m:mrow> <m:mn>2</m:mn> </m:mrow> </m:mfrac> </m:mstyle> </m:mrow> </m:msup> <m:mi>u</m:mi> <m:mspace width="-0.25em"/> <m:msup> <m:mrow> <m:mo>∣</m:mo> </m:mrow> <m:mrow> <m:mn>2</m:mn> </m:mrow> </m:msup> <m:mi mathvariant="normal">d</m:mi> <m:mi>x</m:mi> </m:mrow> </m:mfenced> <m:msup> <m:mrow> <m:mrow> <m:mo>(</m:mo> <m:mrow> <m:mo>−</m:mo> <m:mi mathvariant="normal">Δ</m:mi> </m:mrow> <m:mo>)</m:mo> </m:mrow> </m:mrow> <m:mrow> <m:mi>s</m:mi> </m:mrow> </m:msup> <m:mi>u</m:mi> <m:mo>=</m:mo> <m:mrow> <m:mo>(</m:mo> <m:mrow> <m:mn>1</m:mn> <m:mo>+</m:mo> <m:mi>ε</m:mi> <m:mi>K</m:mi> <m:mrow> <m:mo>(</m:mo> <m:mrow> <m:mi>x</m:mi> </m:mrow> <m:mo>)</m:mo> </m:mrow> </m:mrow> <m:mo>)</m:mo> </m:mrow> <m:msup> <m:mrow> <m:mi>u</m:mi> </m:mrow> <m:mrow> <m:msubsup> <m:mrow> <m:mn>2</m:mn> </m:mrow> <m:mrow> <m:mi>s</m:mi> </m:mrow> <m:mrow> <m:mo>∗</m:mo> </m:mrow> </m:msubsup> <m:mo>−</m:mo> <m:mn>1</m:mn> </m:mrow> </m:msup> <m:mo>,</m:mo> <m:mspace width="1.0em"/> <m:mstyle> <m:mspace width="0.1em"/> <m:mtext>in</m:mtext> <m:mspace width="0.1em"/> </m:mstyle> <m:mspace width="0.33em"/> <m:msup> <m:mrow> <m:mi mathvariant="double-struck">R</m:mi> </m:mrow> <m:mrow> <m:mi>N</m:mi> </m:mrow> </m:msup> <m:mo>,</m:mo> </m:math> \left(a+b\mathop{\int }\limits_{{{\mathbb{R}}}^{N}}| {\left(-\Delta )}^{\tfrac{s}{2}}u\hspace{-0.25em}{| }^{2}{\rm{d}}x\right){\left(-\Delta )}^{s}u=\left(1+\varepsilon K\left(x)){u}^{{2}_{s}^{\ast }-1},\hspace{1.0em}\hspace{0.1em}\text{in}\hspace{0.1em}\hspace{0.33em}{{\mathbb{R}}}^{N}, where <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mi>a</m:mi> <m:mo>,</m:mo> <m:mi>b</m:mi> <m:mo>></m:mo> <m:mn>0</m:mn> </m:math> a,b\gt 0 are given constants, <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mi>ε</m:mi> </m:math> \varepsilon is a small parameter, <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:msubsup> <m:mrow> <m:mn>2</m:mn> </m:mrow> <m:mrow> <m:mi>s</m:mi> </m:mrow> <m:mrow> <m:mo>∗</m:mo> </m:mrow> </m:msubsup> <m:mo>=</m:mo> <m:mfrac> <m:mrow> <m:mn>2</m:mn> <m:mi>N</m:mi> </m:mrow> <m:mrow> <m:mi>N</m:mi> <m:mo>−</m:mo> <m:mn>2</m:mn> <m:mi>s</m:mi> </m:mrow> </m:mfrac> </m:math> {2}_{s}^{\ast }=\frac{2N}{N-2s} with <m:math xmlns:m="http://www.w3.org/1998/Math/MathML