Regularity results for a class of nonlinear fractional Laplacian and singular problems

Abstract

In this article, we investigate the existence, uniqueness, nonexistence, and regularity of weak solutions to the nonlinear fractional elliptic problem of type (P) (see below) involving singular nonlinearity and singular weights in smooth bounded domain. We prove the existence of weak solution in $W_{loc}^{s,p}(\Omega )$ via approximation method. Establishing a new comparison principle of independent interest, we prove the uniqueness of weak solution for $0 \le \delta < 1+s- \frac{1}{p}$ and furthermore the nonexistence of weak solution for $\delta \ge sp.$ Moreover, by virtue of barrier arguments we study the behavior of weak solutions in terms of distance function. Consequently, we prove Hölder regularity up to the boundary and optimal Sobolev regularity for weak solutions.

A Appendix

In this section, we recall the local regularity results for the fractional p-Laplacian. We set for $R>0$ and $y \in {\mathbb {R}}^N$

$$\begin{aligned} Q(u;y,R)= & {} \Vert u\Vert _{L^\infty (B_R(y))} + \left( R^{sp} \int _{(B_R(y))^c} \frac{|u(x)|^{p-1}}{|x-y|^{N+sp}} ~dx\right) ^\frac{1}{p-1}. \end{aligned}$$

Proposition A.1

(Corollary 5.5, [33]) If $u \in {\overline{W}}^{s,p}(B_{2R_0}(y)) \cap L^\infty (B_{2R_0}(y))$ satisfies $|{{(-\Delta )}^{s}_{p}} u| \le K$ weakly in $B_{2R_0}(y)$ for some $R_0>0$, then there exists universal constants $\omega \in (0,1)$ and $C>0$ with the following property:

$$\begin{aligned}&[u]_{C^{\omega }(B_{R_0}(x_0))}:=\sup _{x,y\in B_{R_0}(x_0)}\frac{|u(x)-u(y)|}{|x-y|^{\omega }} \\&\qquad \le C [(K R_0^{sp})^{\frac{1}{p-1}} + Q(u;x_0,2R_0)] R_0^{-\omega }. \end{aligned}$$

Proposition A.2

(Theorem 1.4, [6]) Let $p\in [2,\infty )$ and $u \in W^{s,p}_{loc}(\Omega ) \cap L^\infty _{loc}(\Omega ) \cap L^{p-1}({\mathbb {R}}^N)$ be a local weak solution of ${{(-\Delta )}^{s}_{p}} u = f$ in $\Omega$ with $f \in L^\infty _{loc}(\Omega ).$ Then $u \in C^{\omega }_{loc}(\Omega )$ for every $0< \omega < \min \{\frac{sp}{p-1},1\}.$ More precisely, for every $0< \omega < \min \{\frac{sp}{p-1},1\}$ and every ball $B_{4R}(x_0) \Subset \Omega$, there exists a constant $C= C(N,s,p,\omega )$ such that

$$\begin{aligned}{}[u]_{C^{\omega }(B_{\frac{R}{8}}(x_0))} \le C [(\Vert f\Vert _{L^\infty (B_R(x_0))} R^{sp})^{\frac{1}{p-1}} + Q(u;x_0,R)] R^{-\omega }. \end{aligned}$$

Moreover we recall the following result which is suitable for the acquisition estimates of Theorems 3.2 and 3.3.

Lemma A.1

(Lemma 2.5, [33]) Let $u \in {\overline{W}}^{s,p}_{loc}(\Omega )$. For $\epsilon >0$, let $A_\epsilon \subset {\mathbb {R}}^N \times {\mathbb {R}}^N$ be a neighbourhood of D, the diagonal of ${\mathbb {R}}^N \times {\mathbb {R}}^N$, which satisfies

• (i) $(x,y) \in A_\epsilon$ for all $(y,x) \in A_\epsilon$,

• (ii) $\max \left\{ \sup _{x \in A_\epsilon } {{\,\mathrm{dist}\,}}(x,D), \sup _{y \in D} {{\,\mathrm{dist}\,}}(y,A_\epsilon )\right\} \rightarrow 0$ as $\epsilon \rightarrow 0^+.$

For all $x \in {\mathbb {R}}^N$, we define $A_\epsilon (x)= \{y \in {\mathbb {R}}^N: (x,y) \in A_\epsilon \}$ and

$$\begin{aligned} f_\epsilon (x) = 2\int _{(A_\epsilon (x))^c} \frac{[u(x)-u(y)]^{p-1}}{|x-y|^{N+sp}} ~dy. \end{aligned}$$

Assume that $f_\epsilon \rightarrow f$ in $L^1_{loc}(\Omega )$. Then, u satisifies

$$\begin{aligned} {{(-\Delta )}^{s}_{p}}\, u = f \ \ \ \text {E-weakly in}\ \Omega . \end{aligned}$$