Normalized solutions for Sobolev critical fractional Schrödinger equation

In the present study, we investigate the existence of the normalized solutions to Sobolev critical fractional Schrödinger equation: <m:math xmlns:m="http://www.w3.org/1998/Math/MathML" display="block"> <m:mspace width="14em"/> <m:mfenced open="{" close=""> <m:mrow> <m:mtable displaystyle="true"> <m:mtr> <m:mtd columnalign="left"> <m:msup> <m:mrow> <m:mrow> <m:mo>(</m:mo> <m:mrow> <m:mo>−</m:mo> <m:mi>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:mi>λ</m:mi> <m:mi>u</m:mi> <m:mo>=</m:mo> <m:mi>f</m:mi> <m:mrow> <m:mo>(</m:mo> <m:mrow> <m:mi>u</m:mi> </m:mrow> <m:mo>)</m:mo> </m:mrow> <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: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>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:mspace width="0.1em"/> <m:mtext>in</m:mtext> <m:mspace width="0.1em"/> <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:mspace width="12em"/> <m:mpadded> <m:mrow> <m:mo>(</m:mo> <m:mrow> <m:msub> <m:mrow> <m:mi>P</m:mi> </m:mrow> <m:mrow> <m:mi>m</m:mi> </m:mrow> </m:msub> </m:mrow> <m:mo>)</m:mo> </m:mrow> </m:mpadded> </m:mtd> </m:mtr> <m:mtr> <m:mtd columnalign="left"> <m:munder> <m:mrow> <m:mrow> <m:mstyle displaystyle="true"> <m:mo>∫</m:mo> </m:mstyle> </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:msup> <m:mrow> <m:mo>∣</m:mo> <m:mi>u</m:mi> <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:mo>=</m:mo> <m:msup> <m:mrow> <m:mi>m</m:mi> </m:mrow> <m:mrow> <m:mn>2</m:mn> </m:mrow> </m:msup> <m:mo>,</m:mo> <m:mspace width="1.0em"/> </m:mtd> </m:mtr> </m:mtable> </m:mrow> </m:mfenced> </m:math> {14em}\{{array}{ll}{(-Δ )}ˢu+λ u=f(u)+{| u| }^{{2}ₛ*-2}u,{1.0em}& {0.1em}in{0.1em}{0.33em}{{R}}N,{12em}({P}ₘ)\\ { ∫ }_{{{R}}N}{| u| }²{d}x={m}²,{1.0em}{array}. where <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mn>0</m:mn>