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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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참고문헌 (36)
참고문헌 신청
D. Applebaum / 2009 / Cambridge Stud-ies in Advanced Mathematics, 116 / Cambridge University Press
D. Averna / 2016 / On the existence and multiplicity of solutions for Dirichlet's problem for fractional dierential equations / Fract. Calc. Appl. Anal 19(1):253~266
L. Brasco / 2014 / The fractional Cheeger problem / Interfaces Free Bound 16(3):419~458
L. Brasco / 2016 / The second eigenvalue of the fractional p-Laplacian / Adv. Calc. Var 9(4):323~355
H. Brezis / 1983 / A relation between pointwise convergence of functions and convergence of functionals / Proc. Amer. Math. Soc 88(3):486~490
D. Averna / 2016 / On the existence and multiplicity of solutions for Dirichlet's problem for fractional dierential equations / Fract. Calc. Appl. Anal 19(1):253~266
L. Brasco / 2014 / The fractional Cheeger problem / Interfaces Free Bound 16(3):419~458
L. Brasco / 2016 / The second eigenvalue of the fractional p-Laplacian / Adv. Calc. Var 9(4):323~355
H. Brezis / 1983 / A relation between pointwise convergence of functions and convergence of functionals / Proc. Amer. Math. Soc 88(3):486~490
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