On some Hardy-Sobolev’s type variable exponent inequality and its application


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Mamedov F. I. , Mammadli S. M. , Zeren Y.

Transactions Issue Mathematics, Azerbaijan National Academy of Sciences, vol.37, pp.102-110, 2017 (Refereed Journals of Other Institutions) identifier

  • Publication Type: Article / Article
  • Volume: 37
  • Publication Date: 2017
  • Title of Journal : Transactions Issue Mathematics, Azerbaijan National Academy of Sciences
  • Page Numbers: pp.102-110

Abstract

In this paper, it has been proved a Sobolev’s type variable exponent inequality u(x) x(l − x) p(x);(0,l) ≤ C l u 0 (x) p(x);(0,l) , ∀u ∈ W' 1 p(.) (0, l) where the exponent function p : (0, l) → (1, ∞), is a monotone increasing near little neighborhood of origin and monotone decreasing near l satisfying the conditions: Z l a t − 1 p0(t) dt t ≤ C2a − 1 p0(a) , and Z l a t − 1 p0(l−t) dt t ≤ C1a − 1 p0(l−a) , for 0 < a < l. Applying this inequality and Browder-Minty theory methods, it has been proved an existence result of solution for some variable exponent equation.