Regularity and multiplicity of solutions for a nonlocal problem with critical Sobolev-Hardy nonlinearities

J. Korean Math. Soc. Published online February 10, 2020

Sarah Rsheed Mohamed Alotaibi and kamel saoudi
University of Dammam, University of Sousse

Abstract : In this work we investigate the nonlocal elliptic equation with critical Hardy-Sobolev exponents as follows,
\begin{eqnarray*}({\rm P})
\left\{\begin{array}{ll}
(-\Delta_p)^s u = \lambda |u|^{q-2}u + \frac{|u|^{p^*_s(t)-2}u}{|x|^t}
\;\;\mbox{ in }\,\Omega ,\\

\\
u
= 0 \;\;\mbox{ in }\,\mathbb{R}^N\setminus\Omega,
\end{array}
\right.
\end{eqnarray*}
%-\mu\frac{|u|^{p-2}u}{|x|^p} $0\leq \mu<\overline{\mu}:=\left(\frac{N-p}{p}\right)^p,$
where $\Omega\subset\R^N$ is a bounded domain with Lipschitz boundary, $0<s<1,$ $\lambda >0$ is a parameter, $0<t<sp<N,$ $1<p<\infty$, $0<q-1\leq p-1< p_s^*-1$ 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
\begin{eqnarray*}(-\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,\quad x\in \R^N.
\end{eqnarray*}
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 $C^{1,\alpha}(\overline{\Omega})$.

Keywords : Nonlocal elliptic problems with Sobolev and Hardy nonlinearities, variational methods, multiple positive solutions, regularity of solutions.