Existence and nonexistence of solutions to elliptic equations involving the Hardy potential

JOURNAL OF MATHEMATICAL ANALYSIS AND APPLICATIONS

卷:456

Article

Wang, Ying¹ 

[1] Jiangxi Normal Univ, Dept Math, Nanchang 330022, Jiangxi, Peoples R China

关键词: Hardy potential;    Fractional Laplacian;    Singular solution;    Liouville theorem;   

DOI  :  10.1016/j.jmaa.2017.07.002

来源: Elsevier

【 摘 要 】

The purpose of this paper is to study the nonexistence of nonnegative super solutions to the problem (-Delta)(alpha)u + mu/vertical bar x vertical bar(2 alpha) u >= Qu(p) in R-N \ K, where alpha is an element of (0,1], mu is an element of R, p > 0, IC is a compact set in R-N with N >= 1 and Q is a potential in R-N \ K satisfying that lim inf vertical bar x vertical bar(->+infinity), Q(x)vertical bar x vertical bar(gamma) > 0 for some gamma < 2 alpha.a When alpha = 1, (-Delta)(alpha) is the Laplacian operator, and when alpha is an element of(0,1), it is the fractional Laplacian which is a typical nonlocal operator. In this paper, we find the critical exponent p. > 1 depending on alpha, mu and gamma such that problem (0.1) hag no nontrivial nonnegative super solutions for 0 < p < p*. Furthermore, we also consider the existence and nonexistence of isolated singular solutions to the equation {(-Delta)(alpha)u + mu/vertical bar x vertical bar(2 alpha) u = Qu(v) in R-N \ {0} (vertical bar x vertical bar ->+infinity)lim u(x) = 0 where mu > 0, p > 0 and Q(x)= (1+vertical bar x vertical bar) with gamma is an element of (0, 2 alpha). (C) 2017 Elsevier Inc. All rights reserved.

【 授权许可】

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