The Effect of the Domain Topology on the Number of Positive Solutions for Fractional p-laplacian Equation with Critical Growth

作者：Jin, Zhen-Feng; Sun, Hong-Rui^*; Zhang, Weimin

来源：Results in Mathematics, 2023, 78(5): 205.

DOI：10.1007/s00025-023-01968-6

• Fractional p-Laplacian
• critical Sobolev exponent
• multiplicity
• positive solutions

In this paper, we consider positive solutions for the following fractional p-Laplacian problem with critical growth @@@ (-Delta)(p)(s)u = mu u(q-1) + u(p*s-1), u > 0 in Omega, u = 0 in R-n\Omega, @@@ where Omega subset of R-n is a bounded domain with smooth boundary, mu > 0,p >= 2,s is an element of(0,1),n > spandq is an element of [p,p(s)(*)) with ps(*) = np/n-sp* (-Delta)(p)(s) is the fractional p-Laplacian operator. Using minimax methods and Lusternik-Schnirelman theory, we show that the number of positive solutions of the problem with the least number of closed and contractible sets in (Omega) over bar which cover (Omega) over bar, provided that q = p with n >= sp(2) or q is an element of(p,ps(*)) with n > 2s(pq+p-q)/pq+2p-p(2)-q. Our results improve and generalize the result of Alves and Ding (J Math Anal Appl 279:508-521, 2003) and Figueiredo et al. (Calc Var Part Differ Equ 57:1-24, 2018).

• 单位
  兰州大学