Extremal functions of generalized critical Hardy inequalities

2018/08/02 by Sano, Megumi
#35A23 #35J20 #46B50 #Analysis of PDEs (math.AP) #FOS: Mathematics

paper · doi:10.48550/arxiv.1808.00695

Abstract

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 in 2012: G_a := inf_u ∈ W₀^1,N(Ω) ∖ \ 0\ \dfrac∫_Ω |∇ u |^N dx( ∫_Ω |u|^q f_a,β(x) dx )^(N)/(q), where f_{a, β}(x):=|x|^-N( log (aR)/(|x|) )^-β. First, we give an answer to the open problem when Ω=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 Ω. Finally, symmetry breaking property of the minimizers is proved for sufficiently large β.

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