Regularity and multiplicity of solutions for a nonlocal problem with critical Sobolev-Hardy nonlinearities
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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 $C^{1,\alpha}(\overline{\Omega})$.
Abstract
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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