Abstract:

In this paper, we revisit the following elliptic equations with critical exponent

where $N\geq 4 $, $ s\in (0,2^*-1) $ with $ 2^*=\frac{2N}{N-2} $, $ \varepsilon>0 $, $ \Omega $ is a smooth bounded domain in $ \mathbb{R}^N $. Under some conditions on $ Q(x) $, Cao and Zhong [Cao D, Zhong X. Nonlin Anal TMA, 1997, 29: 461-483] gave the existence of single-peak solutions for small $\varepsilon$ when $N\geq 4$, $ s\in (1,2^*-1) $. Recently, Duan and Tian [Duan L, Tian S. Discrete Contin Dyn Syst, 2022, 42(8): 4061-4094] proved non-existence of single-peak solutions for small $ \varepsilon $ when $ N\geq 5$, $ s=1 $ and got the existence of single-peak solutions for small $ \varepsilon $ when $ N=4$, $ s=1 $. Here we establish non-existence of single-peak solutions for the case $ N\geq 5 $ and $ s<1 $ (sublinear perturbation) by local Pohozaev identities. Our results show that the concentration of solutions to above problem is delicate and sensitive for the dimension $ N$.

Key words: critical Sobolev exponent, Pohozaev identity, sublinear perturbation, non-existence.