2022/10/04 by Deng, Yinbin, He, Qihan, Pan, Yiqing +1
#Analysis of PDEs (math.AP) #FOS: Mathematics
paper · doi:10.48550/arxiv.2210.01373
We consider the existence and nonexistence of positive solution for the following Brézis-Nirenberg problem with logarithmic perturbation: \begincases -Δu=|u|^2∗ -2u+λu+μulog u2 amp;x∈ Ω, u=0amp; x∈ ∂ Ω, \endcases where Ω ⊂ \RN is a bounded smooth domain, λ, μ∈ \R, N≥3 and 2∗ :=(2N)/(N-2) is the critical Sobolev exponent for the embedding H10(Ω)\hookrightarrow L2^∗(Ω). The uncertainty of the sign of slog s2 in (0, +∞) 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 and N≥ 4. While the case of μ<0 is thornier. However, for N=3,4 λ∈ (-∞, λ1(Ω)), we can also establish the existence of positive solution under some further suitable assumptions. And a nonexistence result is also obtained for μ<0 and -((N-2)μ)/(2)+((N-2)μ)/(2)log(-((N-2)μ)/(2))+λ-λ1(Ω)≥ 0 if N≥ 3. Comparing with the results in Brézis, H. and Nirenberg, L. (Comm. Pure Appl. Math. 1983), some new interesting phenomenon occurs when the parameter μ on logarithmic perturbation is not zero.