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Sarah Rsheed Mohamed Alotaibi,Kamel Saoudi 대한수학회 2020 대한수학회지 Vol.57 No.3
In this work we investigate the nonlocal elliptic equation with critical Hardy-Sobolev exponents as follows, \[({\rm P}) \begin{cases} (-\Delta _p)^su = \lambda |u|^{q-2}u + \frac{|u|^{p^*_s(t)-2}u}{|x|^t} & \textrm{in} \ \Omega ,\\ u=0 & \textrm{in} \ \mathbb{R}^N\setminus\Omega, \end{cases} \] where $\Omega\subset\mathbb{R}^N$ is an open bounded domain with Lipschitz boundary, $0<s<1$, $\lambda >0$ is a parameter, $0<t<sp<N$, $1<q< p< p_s^*$ where $p^*_s = \frac{Np}{N-s p}$, $p^*_s(t) = \frac{p(N-t)}{N-s p}$, are the fractional critical Sobolev and Hardy-Sobolev exponents respectively. The fractional $p$-laplacian $(-\Delta_p)^s u$ with $s\in (0,1)$ is the nonlinear nonlocal operator defined on smooth functions by \[ (-\Delta_p)^s u(x)=2 \underset{\epsilon\searrow 0}{\lim}\int_{\mathbb{R}^{N}\backslash B_\epsilon}\frac{|u(x)-u(y)|^{p-2}(u(x)-u(y))}{|x-y|^{N+ ps}}\,{\rm d}y,\ \ x\in \mathbb{R}^N. \] The main goal of this work is to show how the usual variational methods and some analysis techniques can be extended to deal with nonlocal problems involving Sobolev and Hardy nonlinearities. We also prove that for some $\alpha\in (0,1)$, the weak solution to the problem ({\rm P}) is in $C^{1,\alpha}(\overline{\Omega})$.
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Alotaibi, Sarah Rsheed Mohamed,Saoudi, Kamel Korean Mathematical Society 2020 대한수학회지 Vol.57 No.3
In this work we investigate the nonlocal elliptic equation with critical Hardy-Sobolev exponents as follows, $$(P)\;\{(-{\Delta}_p)^su={\lambda}{\mid}u{\mid}^{q-2}u+{\frac{{\mid}u{\mid}^{p{^*_s}(t)-2}u}{{\mid}x{\mid}^t}}{\hspace{10}}in\;{\Omega},\\u=0{\hspace{217}}in\;{\mathbb{R}}^N{\backslash}{\Omega},$$ where Ω ⊂ ℝ<sup>N</sup> is an open bounded domain with Lipschitz boundary, 0 < s < 1, λ > 0 is a parameter, 0 < t < sp < N, 1 < q < p < p<sup>∗</sup><sub>s</sub> where $p^*_s={\frac{N_p}{N-sp}}$, $p^*_s(t)={\frac{p(N-t)}{N-sp}}$, are the fractional critical Sobolev and Hardy-Sobolev exponents respectively. The fractional p-laplacian (-∆<sub>p</sub>)<sup>s</sup>u with s ∈ (0, 1) is the nonlinear nonlocal operator defined on smooth functions by $\displaystyle(-{\Delta}_p)^su(x)=2{\lim_{{\epsilon}{\searrow}0}}\int{_{{\mathbb{R}}^N{\backslash}{B_{\epsilon}}}}\;\frac{{\mid}u(x)-u(y){\mid}^{p-2}(u(x)-u(y))}{{\mid}x-y{\mid}^{N+ps}}dy$, x ∈ ℝ<sup>N</sup>. The main goal of this work is to show how the usual variational methods and some analysis techniques can be extended to deal with nonlocal problems involving Sobolev and Hardy nonlinearities. We also prove that for some α ∈ (0, 1), the weak solution to the problem (P) is in C<sup>1,α</sup>(${\bar{\Omega}}$).
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