Extremal functions of generalized critical Hardy inequalities

JOURNAL OF DIFFERENTIAL EQUATIONS

卷:267

Article

Sano, Megumi¹ 

[1] Hiroshima Univ, Grad Sch Engn, Math Lab, Higashihiroshima, Hiroshima 7390046, Japan

关键词: Critical Hardy inequality;    Optimal constant;    Extremal function;    Symmetry breaking;   

DOI  :  10.1016/j.jde.2019.03.024

来源: Elsevier

【 摘 要 】

In this paper, we show the existence and non-existence of minimizers of the following minimization problems which include an open problem mentioned by Horiuchi and Kumlin [20]: G(a ):= inf(u is an element of W01,N(Omega)/{0} )integral(Omega) vertical bar del u vertical bar(N)dx())/(integral(Omega)vertical bar u vertical bar(q) f(a,beta)(x)dx)(N)(/q), Where f(a,beta)(x) := vertical bar x vertical bar(-N)(log aR/vertical bar x vertical bar)(-)(beta). First, we give an answer to the open problem when Omega = B-R(0). Next, we investigate the minimization problems on general bounded domains. In this case, the results depend on the shape of the domain Omega. Finally, symmetry breaking property of the minimizers is proved for sufficiently large beta. (C) 2019 Elsevier Inc. All rights reserved.

【 授权许可】

Free