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

Abstract

Abstract 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> \left\{\phantom{\rule[-1.25em]{}{0ex}}\begin{array}{ll}-\Delta u={| u| }^{{2}^{\ast }-2}u+\lambda u+\mu u\log {u}^{2}\hspace{1.0em}&amp; x\in \Omega ,\\ u=0\hspace{1.0em}&amp; x\in \partial \Omega ,\end{array}\right. 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> \Omega \subset {{\mathbb{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> \lambda ,\mu \in {\mathbb{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\ge 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}^{\ast }:= \frac{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>