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- Title
A fractional Hardy-Sobolev type inequality with applications to nonlinear elliptic equations with critical exponent and Hardy potential.
- Authors
Shen, Yansheng
- Abstract
In this paper we consider the attainability of the optimal constant $ S(N,p,s,\mu) $ associated to the fractional Hardy-Sobolev type embedding which is defined as$\begin{align*} S(N,p,s,\mu): = \inf\limits_{u\in\dot{W}^{s,p}(\mathbb{R}^{N})\backslash\{0\}}\frac{\int_{\mathbb{R}^{N}} \int_{\mathbb{R}^{N}}\frac{|u(x)-u(y)|^{p}}{|x-y|^{N+ps}}dxdy-\mu\int_{\mathbb{R}^{N}}\frac{|u|^{p}} {|x|^{ps}}dx}{\big(\int_{\mathbb{R}^{N}}|u|^{p^{\ast}_{s}}dx\big)^{\frac{p}{p^{\ast}_{s}}}}, \end{align*}$where $ s\in(0,1) $, $ p>1 $ and $ N>ps $, $ 0\leq\mu<\mu_{H} $, the latter being the best constant in the fractional Hardy inequality on $ \mathbb{R}^{N} $, $ p_{s}^{\ast} = \frac{Np}{N-ps} $ is the fractional critical Sobolev exponent. The technique that we use to prove the existence of extremals for $ S(N,p,s,\mu) $ is based on blow-up analysis argument combined with a variational method. Further, as an application of the inequality, we prove an existence result for the critical fractional $ p $-Laplacian equation with Hardy potential and involving continuous nonlinearities having quasicritical growth.
- Subjects
NONLINEAR equations
- Publication
Discrete & Continuous Dynamical Systems: Series A, 2024, Vol 44, Issue 7, p1
- ISSN
1078-0947
- Publication type
Article
- DOI
10.3934/dcds.2024014