The existence of positive solution for an elliptic problem with critical growth and logarithmic perturbation

We consider the existence and nonexistence of the positive solution for the following Brézis-Nirenberg problem with logarithmic perturbation: <m:math xmlns:m="http://www.w3.org/1998/Math/MathML" display="block"> <m:mfenced open="{" close=""> <m:mrow> <m:mspace depth="1.25em"/> <m:mtable displaystyle="true"> <m:mtr> <m:mtd columnalign="left"> <m:mo>−</m:mo> <m:mi mathvariant="normal">Δ</m:mi> <m:mi>u</m:mi> <m:mo>=</m:mo> <m:msup> <m:mrow> <m:mrow> <m:mo stretchy="false">∣</m:mo> <m:mrow> <m:mi>u</m:mi> </m:mrow> <m:mo stretchy="false">∣</m:mo> </m:mrow> </m:mrow> <m:mrow> <m:msup> <m:mrow> <m:mn>2</m:mn> </m:mrow> <m:mrow> <m:mo>∗</m:mo> </m:mrow> </m:msup> <m:mo>−</m:mo> <m:mn>2</m:mn> </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>μ</m:mi> <m:mi>u</m:mi> <m:mi>log</m:mi> <m:msup> <m:mrow> <m:mi>u</m:mi> </m:mrow> <m:mrow> <m:mn>2</m:mn> </m:mrow> </m:msup> <m:mspace width="1.0em"/> </m:mtd> <m:mtd columnalign="left"> <m:mi>x</m:mi> <m:mo>∈</m:mo> <m:mi mathvariant="normal">Ω</m:mi> <m:mo>,</m:mo> </m:mtd> </m:mtr> <m:mtr> <m:mtd columnalign="left"> <m:mi>u</m:mi> <m:mo>=</m:mo> <m:mn>0</m:mn> <m:mspace width="1.0em"/> </m:mtd> <m:mtd columnalign="left"> <m:mi>x</m:mi> <m:mo>∈</m:mo> <m:mo>∂</m:mo> <m:mi mathvariant="normal">Ω</m:mi> <m:mo>,</m:mo> </m:mtd> </m:mtr> </m:mtable> </m:mrow> </m:mfenced> </m:math> \{{[-1.25em]{}{0ex}}{array}{ll}-Δ u={| u| }^{{2}-2}u+λ u+μ ulog {u}²{1.0em}& x∈ Ω ,\\ u=0{1.0em}& x∈ ∂ Ω ,{array}. where <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mi mathvariant="normal">Ω</m:mi> <m:mo>⊂</m:mo> <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:math> Ω ⊂ {{R}}N is a bounded open domain, <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mi>λ</m:mi> <m:mo>,</m:mo> <m:mi>μ</m:mi> <m:mo>∈</m:mo> <m:mi mathvariant="double-struck">R</m:mi> </m:math> λ ,μ ∈ {R} , <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mi>N</m:mi> <m:mo>≥</m:mo> <m:mn>3</m:mn> </m:math> N≥ 3 and <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:msup> <m:mrow> <m:mn>2</m:mn> </m:mrow> <m:mrow> <m:mo>∗</m:mo> </m:mrow> </m:msup> <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:mrow> </m:mfrac> </m:math> {2}:= 2N/N-2 is the critical Sobolev exponent for the embedding <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:msubsup> <m:mrow> <m:mi>H</m:mi> </m:mrow> <m:mrow> <m:mn>0</m:mn> </m:mrow> <m:mrow> <m:mn>1</m:mn> </m:mrow> </m:msubsup> <m:mrow> <m:mo>(</m:mo> <m:mrow> <m:mi mathvariant="normal">Ω</m:mi> </m:mrow> <m:mo>)</m:mo>