Multiplicity and Hölder regularity of solutions for a nonlocal elliptic PDE involving singularity

Abstract

In this paper, we prove the existence of multiple solutions for a nonlinear nonlocal elliptic PDE involving a singularity which is given as \begin{eqnarray} (-Δ_p)^s u&=& \fracλ{u^γ}+u^q~\text{in}~Ω,\nonumber u&=&0~\text{in}~\mathbb{R}^N\setminusΩ,\nonumber u&>& 0~\text{in}~Ω\nonumber, \end{eqnarray} where $\Omega$ is an open bounded domain in ${\mathbb{R}}^N$ with smooth boundary, $N>ps$, $s\in (0,1)$, $\lambda >0$, $0<\gamma <1$, $1<p<\infty$, $p-1<q\le p_s^{\ast }=\frac{Np}{N-ps}$. We employ variational techniques to show the existence of multiple positive weak solutions of the above problem. We also prove that for some $\beta \in (0,1)$, the weak solution to the problem is in $C^{1,\beta }(\overline{\Omega })$.

42 pages

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• From the long jump random walk to the fractional Laplacian
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