Large solutions to elliptic equations involving fractional Laplacian
Large solutions to elliptic equations involving fractional Laplacian
复制标题
DOI：
10.1016/j.anihpc.2014.08.001
复制
发表时间：
2013-11
期刊：
影响因子：
--
通讯作者：
Huyuan Chen;P. Felmer;A. Quaas
中科院分区：
文献类型：
--
作者：
Huyuan Chen;P. Felmer;A. Quaas
本文研究形式为(0.1){(−Δ)αu(X)+|u|p−1 u(X)=f(X)，x∈Ω，u(X)=0，x∈Ω‘c，lim x∈Ω，x→∂Ωu(X)=+∞；1，Ω是R-N，N-≥-2的开有界C2区域，算子(−Δ)α，α∈(0，1)是分数次拉普拉斯函数，f：Ω→-R是满足一定条件的连续函数.当1+2−<；p<；1−2ατ0(α)时，问题(0.1)允许d(X)τ2αpτ1(α)的解，并且当{1−2ατ0+τ0(α)+1τ0，1}<；p<；1−2ατ0时，问题(0.1)有无限多个类似d(X)Backup 0(Max)的边值行为的解.此外，我们还得到了问题(0.1)的一些唯一性和不存在性结果。
The purpose of this paper is to study boundary blow up solutions for semi-linear fractional elliptic equations of the form (0.1){(− Δ) α u (x)+| u| p− 1 u (x)= f (x), x∈ Ω, u (x)= 0, x∈ Ω¯ c, lim x∈ Ω, x→∂ Ω u (x)=+∞, where p> 1, Ω is an open bounded C 2 domain of R N, N≥ 2, the operator (− Δ) α with α∈(0, 1) is the fractional Laplacian and f: Ω→ R is a continuous function which satisfies some appropriate conditions. We obtain that problem (0.1) admits a solution with boundary behavior like d (x)− 2 α p− 1, when 1+ 2 α< p< 1− 2 α τ 0 (α), for some τ 0 (α)∈(− 1, 0), and has infinitely many solutions with boundary behavior like d (x) τ 0 (α), when max{1− 2 α τ 0+ τ 0 (α)+ 1 τ 0, 1}< p< 1− 2 α τ 0. Moreover, we also obtained some uniqueness and non-existence results for problem (0.1).