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- W2951794352 abstract "We consider the problem of attainability of the best constant in the following critical fractional Hardy-Sobolev inequality: begin{equation*} mu_{gamma,s}(R^n):= inflimits_{u in H^{frac{alpha}{2}} (R^n)setminus {0}} frac{ int_{R^n} |({-}{ Delta})^{frac{alpha}{4}}u|^2 dx - gamma int_{R^n} frac{|u|^2}{|x|^{alpha}}dx }{(int_{R^n} frac{|u|^{2_{alpha}^*(s)}}{|x|^{s}}dx)^frac{2}{2_{alpha}^*(s)}}, end{equation*} where $0leq s alpha$, ${2_{alpha}^*(s)}:=frac{2(n-s)}{n-{alpha}},$ and $gamma in mathbb{R}$. This allows us to establish the existence of nontrivial weak solutions for the following doubly critical problem on $R^n$, begin{equation*} left{begin{array}{lll} ({-}{ Delta})^{frac{alpha}{2}}u- gamma frac{u}{|x|^{alpha}}&= |u|^{2_{alpha}^*-2} u + {frac{|u|^{2_{alpha}^*(s)-2}u}{|x|^s}} & text{in } {R^n} hfill u&>0 & text{in } R^n, end{array}right. end{equation*} where $2_{alpha}^*:=frac{2 n}{n-{alpha}}$ is the critical $alpha$-fractional Sobolev exponent, and $gamma < gamma_H:=2^alpha frac{Gamma^2(frac{n+alpha}{4})}{Gamma^2(frac{n-alpha}{4})}$, the latter being the best fractional Hardy constant on $R^n$." @default.
- W2951794352 created "2019-06-27" @default.
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- W2951794352 date "2015-03-27" @default.
- W2951794352 modified "2023-09-27" @default.
- W2951794352 title "Borderline variational problems involving fractional Laplacians and critical singularities" @default.
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