Fractional Hardy-Sobolev equations with nonhomogeneous terms

This paper deals with existence and multiplicity of positive solutions to the following class of nonlocal equations with critical nonlinearity: <m:math xmlns:m="http://www.w3.org/1998/Math/MathML" display="block"> <m:mtable rowspacing="4pt" columnspacing="1em"> <m:mtr> <m:mtd> <m:mstyle displaystyle="true"> <m:mfenced open="{" close=""> <m:mtable columnalign="left left" rowspacing=".1em" columnspacing="1em"> <m:mtr> <m:mtd> <m:mo stretchy="false">(</m:mo> <m:mo>−</m:mo> <m:mrow class="MJX-TeXAtom-ORD"> <m:mi class="MJX-tex-mathit" mathvariant="italic">Δ</m:mi> </m:mrow> <m:msup> <m:mo stretchy="false">)</m:mo> <m:mi>s</m:mi> </m:msup> <m:mi>u</m:mi> <m:mo>−</m:mo> <m:mi>γ</m:mi> <m:mstyle> <m:mfrac> <m:mi>u</m:mi> <m:mrow> <m:mrow class="MJX-TeXAtom-ORD"> <m:mo stretchy="false">|</m:mo> </m:mrow> <m:mi>x</m:mi> <m:msup> <m:mrow class="MJX-TeXAtom-ORD"> <m:mo stretchy="false">|</m:mo> </m:mrow> <m:mrow class="MJX-TeXAtom-ORD"> <m:mn>2</m:mn> <m:mi>s</m:mi> </m:mrow> </m:msup> </m:mrow> </m:mfrac> </m:mstyle> <m:mo>=</m:mo> <m:mi>K</m:mi> <m:mo stretchy="false">(</m:mo> <m:mi>x</m:mi> <m:mo stretchy="false">)</m:mo> <m:mstyle> <m:mfrac> <m:mrow> <m:mrow class="MJX-TeXAtom-ORD"> <m:mo stretchy="false">|</m:mo> </m:mrow> <m:mi>u</m:mi> <m:msup> <m:mrow class="MJX-TeXAtom-ORD"> <m:mo stretchy="false">|</m:mo> </m:mrow> <m:mrow class="MJX-TeXAtom-ORD"> <m:msubsup> <m:mn>2</m:mn> <m:mi>s</m:mi> <m:mo>∗</m:mo> </m:msubsup> <m:mo stretchy="false">(</m:mo> <m:mi>t</m:mi> <m:mo stretchy="false">)</m:mo> <m:mo>−</m:mo> <m:mn>2</m:mn> </m:mrow> </m:msup> <m:mi>u</m:mi> </m:mrow> <m:mrow> <m:mrow class="MJX-TeXAtom-ORD"> <m:mo stretchy="false">|</m:mo> </m:mrow> <m:mi>x</m:mi> <m:msup> <m:mrow class="MJX-TeXAtom-ORD"> <m:mo stretchy="false">|</m:mo> </m:mrow> <m:mi>t</m:mi> </m:msup> </m:mrow> </m:mfrac> </m:mstyle> <m:mo>+</m:mo> <m:mi>f</m:mi> <m:mo stretchy="false">(</m:mo> <m:mi>x</m:mi> <m:mo stretchy="false">)</m:mo> <m:mspace width="1em"/> <m:mtext>in</m:mtext> <m:mspace width="1em"/> <m:msup> <m:mrow class="MJX-TeXAtom-ORD"> <m:mi mathvariant="double-struck">R</m:mi> </m:mrow> <m:mi>N</m:mi> </m:msup> <m:mo>,</m:mo> </m:mtd> </m:mtr> <m:mtr> <m:mtd> <m:mspace width="2em"/> <m:mspace width="2em"/> <m:mspace width="2em"/> <m:mspace width="1em"/> <m:mi>u</m:mi> <m:mo>∈</m:mo> <m:msup> <m:mrow class="MJX-TeXAtom-ORD"> <m:mover> <m:mi>H</m:mi> <m:mo>˙</m:mo> </m:mover> </m:mrow> <m:mi>s</m:mi> </m:msup> <m:mo stretchy="false">(</m:mo> <m:msup> <m:mrow class="MJX-TeXAtom-ORD"> <m:mi mathvariant="double-struck">R</m:mi> </m:mrow> <m:mi>N</m:mi> </m:msup> <m:mo stretchy="false">)</m:mo> <m:mo>,</m:mo> </m:mtd> </m:mtr> </m:mtable> </m:mfenced> </m:mstyle> </m:mtd> </m:mtr> </m:mtable> </m:math> $${array}{} {cases} (-{Δ})^s u -γ{u}{|x|²ˢ}=K(x){|u|2^*_s(t)-2u}{|x|^t}+f(x) R^N,\\ u∈ Ḣ^s( R^N), {cases} {array}$$ where N > 2 s , s ∈ (0, 1), 0 ≤ t < 2 s < <jats: