Acta mathematica scientia,Series B ›› 2022, Vol. 42 ›› Issue (3): 1113-1124.

• Articles •

### A FRACTIONAL CRITICAL PROBLEM WITH SHIFTING SUBCRITICAL PERTURBATION

Qi LI1, Chang-Lin XIANG2

1. 1. College of Science, Wuhan University of Science and Technology, Wuhan, 430065, China;
2. Three Gorges Mathematical Research Center, China Three Gorges University, Yichang, 443002, China
• Received:2020-12-11 Published:2022-06-24
• Contact: Chang-Lin XIANG,E-mail:changlin.xiang@ctgu.edu.cn E-mail:changlin.xiang@ctgu.edu.cn
• Supported by:
Q. Li was supported by the excellent doctorial dissertation cultivation grant (2018YBZZ067) from Central China Normal University. C.-L. Xiang was financially supported by the National Natural Science Foundation of China (11701045) and the Yangtze Youth Fund (2016cqn56).

Abstract: In this paper, we consider a class of fractional problem with subcritical perturbation on a bounded domain as follows: \begin{equation*} (P_{k})\quad \left\{ \begin{array}{ll} \displaystyle (-\Delta)^s u=g(x)[(u-k)^+]^{q-1}+u^{2^{*}_{s}-1},\ \ &x\in \Omega,\\ \displaystyle u>0,\ \ &x\in \Omega,\\ \displaystyle u=0,\ \ &x\in \mathbb{R}^N\backslash \Omega. \end{array} \right. \end{equation*} We prove the existence of nontrivial solutions $u_{k}$ of $(P_{k})$ for each $k\in (0,\infty)$. We also investigate the concentration behavior of the solutions $u_{k}$ as $k\to \infty$.

CLC Number:

• 35J20
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