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: − Δ u = ∣ u ∣ 2 ∗ − 2 u + λ u + μ u log u 2 x ∈ Ω , u = 0 x ∈ ∂ Ω , \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}& x\in \Omega ,\\ u=0\hspace{1.0em}& x\in \partial \Omega ,\end{array}\right. where Ω ⊂ R N \Omega \subset {{\mathbb{R}}}^{N} is a bounded open domain, λ , μ ∈ R \lambda ,\mu \in {\mathbb{R}} , N ≥ 3 N\ge 3 and 2 ∗ ≔ 2 N N − 2 {2}^{\ast }:= \frac{2N}{N-2} is the critical Sobolev exponent for the embedding H 0 1 ( Ω ) ↪ L 2 ∗ ( Ω ) {H}_{0}^{1}\left(\Omega )\hspace{0.33em}\hookrightarrow \hspace{0.33em}{L}^{{2}^{\ast }}\left(\Omega ) . The uncertainty of the sign of s log s 2 s\log {s}^{2} in ( 0 , + ∞ ) \left(0,+\infty ) has some interest in itself. We will show the existence of positive ground state solution, which is of mountain pass type provided λ ∈ R , μ > 0 \lambda \in {\mathbb{R}},\mu \gt 0 and N ≥ 4 N\ge 4 . While the case of μ < 0 \mu \lt 0 is thornier. However, for N = 3 , 4 N=3,4 , λ ∈ ( − ∞ , λ 1 ( Ω ) ) \lambda \in \left(-\infty ,{\lambda }_{1}\left(\Omega )) , we can also establish the existence of positive solution under some further suitable assumptions. A nonexistence result is also obtained for μ < 0 \mu \lt 0 and − ( N − 2 ) μ 2 + ( N − 2 ) μ 2 log − ( N − 2 ) μ 2 + λ − λ 1 ( Ω ) ≥ 0 -\frac{\left(N-2)\mu }{2}+\frac{\left(N-2)\mu }{2}\log \left(-\frac{\left(N-2)\mu }{2}\right)+\lambda -{\lambda }_{1}\left(\Omega )\ge 0 if N ≥ 3 N\ge 3 . Comparing with the results in the study by Brézis and Nirenberg (Positive solutions of nonlinear elliptic equations involving critical Sobolev exponents, Comm. Pure Appl. Math. 36 (1983), 437–477), some new interesting phenomenon occurs when the parameter μ \mu on logarithmic perturbation is not zero.