Global Boundedness of Weak Solutions to Fractional Nonlocal Equations

Abstract

In this paper, we establish the global boundedness of weak solutions to fractional nonlocal equations using the fractional Moser iteration argument and some other ideas. Our results not only extend the boundedness result of Ros-Oton-Serra to general fractional nonlocal equations under a weaker assumption can but also be viewed as a generalization of the boundedness of weak solutions of second-order elliptic equations to nonlocal equations.

1. Introduction

In this paper, we investigate the global boundedness of weak solutions of the following nonlocal equations:

$$\left\{\begin{array}{cc}{\displaystyle L_K^{2s}}u\left(x\right)=g\left(x\right),\hfill & \mathrm{in}\mathrm{\Omega },\hfill \\ u=0,\hfill & \mathrm{in}{\mathbb{R}}^n\setminus \mathrm{\Omega },\hfill \end{array}\right.$$

(1)

and

$$\left\{\begin{array}{cc}{\displaystyle L_K^{2s}}u\left(x\right)={\displaystyle L_b^s}f\left(x\right),\hfill & \mathrm{in}\mathrm{\Omega },\hfill \\ u=0,\hfill & \mathrm{in}{\mathbb{R}}^n\setminus \mathrm{\Omega },\hfill \end{array}\right.$$

(2)

where $0<s<1$, $\mathrm{\Omega }$ is an open bounded domain in ${\mathbb{R}}^n$, $g\in L^2\left(\mathrm{\Omega }\right)$, and the nonlocal operators ${\displaystyle L_K^{2s}}$ and ${\displaystyle L_b^s}$ are defined by

$$\begin{array}{ccc}\hfill {\displaystyle L_K^{2s}}u\left(x\right)& =& C(n,s)P.V.{\int }_{{\mathbb{R}}^n}K(x,y)[u\left(x\right)-u\left(y\right)]dy\hfill \\ & =& C(n,s){\displaystyle \underset{ϵ\to 0}{lim}}{\int }_{{\mathbb{R}}^n\setminus B_{ϵ}\left(x\right)}K(x,y)[u\left(x\right)-u\left(y\right)]dy,\hfill \end{array}$$

and

$$\begin{array}{ccc}\hfill {\displaystyle L_b^s}f\left(x\right)& =& C(n,{\displaystyle \frac{s}{2}})P.V.{\int }_{{\mathbb{R}}^n}b(x,y)[f\left(x\right)-f\left(y\right)]dy\hfill \\ & =& C(n,{\displaystyle \frac{s}{2}}){\displaystyle \underset{ϵ\to 0}{lim}}{\int }_{{\mathbb{R}}^n\setminus B_{ϵ}\left(x\right)}b(x,y)[f\left(x\right)-f\left(y\right)]dy,\hfill \end{array}$$

for any functions u, $f\in \mathcal{S}\left({\mathbb{R}}^n\right)$, where $\mathcal{S}\left({\mathbb{R}}^n\right)$ is the Schwartz space of the rapidly decaying $C^{\infty }$ function in ${\mathbb{R}}^n$. Here, P.V. indicates integration in the sense of a Cauchy principal value and

$$C(n,s)={\left({\int }_{{\mathbb{R}}^n}{\displaystyle \frac{1-cos\left({\zeta }_1\right)}{{\left|\zeta \right|}^{n+2s}}}d\zeta \right)}^{-1}.$$

We always assume that the symmetric kernels $K(x,y)=K(x-y)$ and $b(x,y)=b(x-y)$ satisfy

$${\displaystyle \frac{\lambda }{{|x-y|}^{n+2s}}}\le K(x,y)\le {\displaystyle \frac{\Lambda }{{|x-y|}^{n+2s}}}\mathrm{and}\left|b(x,y)\right|\le {\displaystyle \frac{B}{{|x-y|}^{n+s}}}$$

(3)

for some positive constants $\lambda \le \Lambda$ and B.

Fractional nonlocal equations have very wide applications. For example, they can be used to describe the Lévy process with jumps in stochastic processes. The Lévy process with jumps, as an important branch of modern probability theory, has extensive applications in fields such as statistics, economics, insurance, physics, engineering, and operations research. Many models in fluid mechanics are also nonlocal, for instance, the surface quasi-geotropic equation for simulating sea surface temperature in oceanography [1] and the Benjamin–Ono equation for simulating one-dimensional internal waves in deep water [2].

We say that $u\in {\displaystyle {\mathcal{H}}_0^s}\left(\mathrm{\Omega }\right)$ is a weak solution of (1) if

$${\mathcal{L}}_K(u,\varphi )={\int }_{{\mathbb{R}}^n}{\int }_{{\mathbb{R}}^n}K(x,y)[u\left(x\right)-u\left(y\right)][\varphi \left(x\right)-\varphi \left(y\right)]dydx={\int }_{\mathrm{\Omega }}g\left(x\right)\varphi \left(x\right)dx,$$

(4)

for any $\varphi \in {\displaystyle {\mathcal{H}}_0^s}\left(\mathrm{\Omega }\right)$, where $g\in L^{{\displaystyle \frac{2n}{n+2s}}}\left(\mathrm{\Omega }\right)$, and $u\in {\displaystyle {\mathcal{H}}_0^s}\left(\mathrm{\Omega }\right)$ is a weak solution of (2) if

$${\mathcal{L}}_K(u,\varphi )={\int }_{\mathrm{\Omega }}f\left(x\right){\int }_{{\mathbb{R}}^n}b(x,y)[\varphi \left(x\right)-\varphi \left(y\right)]dydx,\forall \varphi \in {\displaystyle {\mathcal{H}}_0^s}\left(\mathrm{\Omega }\right).$$

(5)

where $f\in {\displaystyle L_0^2}\left(\mathrm{\Omega }\right)$. Here, the assumption that $f=0$ in ${\mathbb{R}}^n\setminus \mathrm{\Omega }$ is very natural. For one thing, it makes the definition of weak solutions for nonlocal Equation (2) and corresponding local equation $div(A\nabla u)=div(\overrightarrow{f})$ consistent; for another, the Moser iteration technique can only be utilized successfully under this condition to obtain the global boundedness of solutions to Equation (2).

Our main results are the following theorems.

Theorem 1. 

Let $0<s<1$ and $n>2s$. Suppose that $g\in L^{{\displaystyle \frac{q}{2s}}}\left(\mathrm{\Omega }\right)$ for some $q>n$. Then, if $u\in {\displaystyle {\mathcal{H}}_0^s}\left(\mathrm{\Omega }\right)$ is a weak solution of (1), then $u\in L^{\infty }\left(\mathrm{\Omega }\right)$ and there exists a positive constant $C=C(n,s,\lambda ,q)$ such that

$${\parallel u\parallel }_{L^{\infty }\left(\mathrm{\Omega }\right)}\le C{\left|\mathrm{\Omega }\right|}^{{\displaystyle \frac{2s}{n}}}{({⨏}_{\mathrm{\Omega }}{\left|g\right|}^{{\displaystyle \frac{q}{2s}}}dx)}^{{\displaystyle \frac{2s}{q}}}.$$

Theorem 2. 

Let $0<s<1$. Suppose that $f\in {\displaystyle L_0^{{\displaystyle \frac{q}{s}}}}\left(\mathrm{\Omega }\right)$ for some $q>n$. Then, if $u\in {\displaystyle {\mathcal{H}}_0^s}\left(\mathrm{\Omega }\right)$ is a weak solution of (2), then $u\in L^{\infty }\left(\mathrm{\Omega }\right)$ and we have

$${\parallel u\parallel }_{L^{\infty }\left(\mathrm{\Omega }\right)}\le C{\left|\mathrm{\Omega }\right|}^{{\displaystyle \frac{s}{n}}}{({⨏}_{\mathrm{\Omega }}{\left|f\right|}^{{\displaystyle \frac{q}{s}}}dx)}^{{\displaystyle \frac{s}{q}}},$$

where $C=C(n,s,\lambda ,B,q)>0$.

Remark 1. 

It is well known that the operator ${\displaystyle L_K^{2s}}$ in Equation (1) becomes the fractional Laplacian when $\lambda =\Lambda =1$ in assumption (3). The global boundedness of weak solutions to the fractional Laplacian equation was investigated in [3] using the De Giorgi iteration method, although the authors did not provide the specific estimate formula. Further more, Ros-Oton and Serra [4] established the boundedness of weak solutions of the fractional Laplacian equation under the assumption that $g\in L^{\infty }\left(\mathrm{\Omega }\right)$ by constructing an appropriate supersolution and applying the Maximum principle; more precisely, they obtained the following estimate:

$${\parallel u\parallel }_{L^{\infty }\left(\mathrm{\Omega }\right)}\le C{\left(diam\mathrm{\Omega }\right)}^{2s}{\parallel g\parallel }_{L^{\infty }\left(\mathrm{\Omega }\right)},$$

for some constant C depending on n and s. From this point of view, our result in Theorem 1 can be seen as an extension of this global boundedness estimate for solutions of the fractional Laplacian equation to general nonlocal equations under a weaker assumption. In addition, a related result for Equation (1) was also studied in [5], where the authors proved the global boundedness of weak solutions to the fractional p-Laplacian equation. In this paper, we provide a simple proof of the global boundedness result for fractional nonlocal equations, which can also be applied to obtain the similar boundedness for the fractional p-Laplacian equation (i.e., Theorem 3). Finally, our results can be viewed as a generalization of the global boundedness of weak solutions of elliptic equations in [6] to fractional nonlocal equations.

Remark 2. 

The proof of the global boundedness of solutions to Equation (2) is a little more difficult than that of Equation (1) due to the nonlocality of the operator ${\displaystyle L_b^s}$. In fact, Equation (2) is a general nonlocal version of local second-order elliptic equations in divergence form of the type

$$div(A\nabla u)=div(\overrightarrow{f}).$$

The De Giorgi–Nash–Moser theory can be used to obtain the boundedness of weak solutions to this equation. Recently, this famous theory was extended to nonlocal operators in [3,7]. More precisely, in an article [3], Servadei and Valdinoci investigated the boundedness of the Dirichlet problem of the fractional Laplacian equation using a fractional version of the classical De Giorgi iteration argument; Castro, Kuusi, and Palatucci [7] established the local boundedness of weak solutions to the fractional p-Laplacian equation using a nonlocal version of the Moser iteration method. For more results about the applications of nonlocal De Giorgi–Nash–Moser arguments, we can refer to [8,9]. However, their arguments cannot be used to obtain the boundedness of weak solutions of Equation (2). This is because we do not have the corresponding chain rule for operator ${\displaystyle L_b^s}$ like that for the gradient operator $\nabla$. In order to overcome this difficulty, we divide our equation into a few simple equations for which the fractional Moser iteration argument can be applied. The downside of our approach is that it deeply depends on the linearity of Equation (2). The global boundedness of weak solutions to Equation (2), with ${\displaystyle L_K^{2s}}$ replaced with a nonlocal nonlinear operator, for example, the fractioanal p-Laplacion, will be studied in our further research.

Using a similar process as Theorem 1, we can establish the global boundedness for weak solutions of the following fractional p-Laplacian equation:

$$\left\{\begin{array}{cc}{\displaystyle {(-\Delta )}_p^s}u\left(x\right)=g\left(x\right),\hfill & \mathrm{in}\mathrm{\Omega },\hfill \\ u=0,\hfill & \mathrm{in}{\mathbb{R}}^n\setminus \mathrm{\Omega },\hfill \end{array}\right.$$

(6)

where $0<s<1$, $1<p<\infty$, and the nonlocal operator ${\displaystyle {(-\Delta )}_p^s}$ is well defined for any $u\in \mathcal{S}\left({\mathbb{R}}^n\right)$ by

$$\begin{array}{ccc}\hfill {\displaystyle {(-\Delta )}_p^s}u\left(x\right)& =& C(n,s)P.V.{\int }_{{\mathbb{R}}^n}{\displaystyle \frac{{|u\left(x\right)-u\left(y\right)|}^{p-2}[u\left(x\right)-u\left(y\right)]}{{|x-y|}^{n+ps}}}dy.\hfill \end{array}$$

Here, we say $u\in {\displaystyle {\mathcal{W}}_0^{s,p}}\left(\mathrm{\Omega }\right)$ is a weak solution of (6) if

$${\int }_{{\mathbb{R}}^n}{\int }_{{\mathbb{R}}^n}{\displaystyle \frac{{|u\left(x\right)-u\left(y\right)|}^{p-2}[u\left(x\right)-u\left(y\right)][\varphi \left(x\right)-\varphi \left(y\right)]}{{|x-y|}^{n+ps}}}dydx={\int }_{\mathrm{\Omega }}g\left(x\right)\varphi \left(x\right)dx,$$

(7)

for any $\varphi \in {\displaystyle {\mathcal{W}}_0^{s,p}}\left(\mathrm{\Omega }\right)$, where $g\in L^{{\displaystyle \frac{np}{n(p-1)+ps}}}\left(\mathrm{\Omega }\right)$.

Theorem 3. 

Let $0<s<1$ and $1<p<\infty$ such that $n>ps$. Suppose that $g\in L^{{\displaystyle \frac{q}{ps}}}\left(\mathrm{\Omega }\right)$ for some $q>n$. Then, if $u\in {\displaystyle {\mathcal{W}}_0^{s,p}}\left(\mathrm{\Omega }\right)$ is a weak solution of (6), then $u\in L^{\infty }\left(\mathrm{\Omega }\right)$ and there exists a positive constant $C=(n,s,p,q)$ such that

$${\parallel u\parallel }_{L^{\infty }\left(\mathrm{\Omega }\right)}\le C{[{\left|\mathrm{\Omega }\right|}^{{\displaystyle \frac{ps}{n}}}{({⨏}_{\mathrm{\Omega }}{\left|g\right|}^{{\displaystyle \frac{q}{ps}}}dx)}^{{\displaystyle \frac{ps}{q}}}]}^{{\displaystyle \frac{1}{p-1}}}.$$

Remark 3. 

The proof of Theorem 3 is a combination of the process of the proof of Theorem 1 and the convexity of function $f\left(t\right)=t^p$ for $p>1$. The later implies that $u_{\pm }\in W^{s,p}\left({\mathbb{R}}^n\right)$ if $u\in W^{s,p}\left({\mathbb{R}}^n\right)$ (see Section 4). Thus, we can take a test function $\tilde{h}\left(w\right)\in {\displaystyle {\mathcal{W}}_0^{s,p}}\left(\mathrm{\Omega }\right)$ similar to that in Section 3.

Remark 4. 

It is well known that in order to obtain the global boundedness of weak solutions to elliptic partial differential equations

$$div(A\nabla u)=div\left(\overrightarrow{f}\right)+g,\mathit{in}\mathrm{\Omega },$$

we need $f^i\in L^q\left(\mathrm{\Omega }\right)$ and $g\in L^{{\displaystyle \frac{q}{2}}}\left(\mathrm{\Omega }\right)$ for some $q>n$. Naturally, for the fractional nonlocal equations in this paper, the similar condition is imposed on f and g in Theorems 1–3. In addition, the boundedness results in this paper can be viewed as the first step to obtain more regularity results for weak solutions of fractional nonlocal equations.

This paper is organized as follows. In Section 2, notations for related function spaces and some basic results are concluded. In Section 3, we show the global boundedness of weak solutions to fractional nonlocal linear equations. The boundedness of weak solutions of the fractional p-Laplacian equation is considered in Section 4.

2. Notations and Preliminaries

2.1. Notations for Function Spaces

(1) $L^p\left(\mathrm{\Omega }\right)=\left\{u\right|{\parallel u\parallel }_{L^p\left(\mathrm{\Omega }\right)}<\infty \}$, where ${\parallel u\parallel }_{L^p\left(\mathrm{\Omega }\right)}=({\int }_{\mathrm{\Omega }}{\left|u\right|}^p{dx)}^{1/p}$ for $1\le p<\infty$.

(2) ${\displaystyle L_0^p}\left(\mathrm{\Omega }\right)=\{u\in L^p\left({\mathbb{R}}^n\right)|u=0\mathrm{in}{\mathbb{R}}^n\setminus \mathrm{\Omega }\}$ for $1\le p<\infty$.

(3) Let $\mathrm{\Omega }$ be an open set in ${\mathbb{R}}^n$. For any $0<s<1$ and for any $1\le p<\infty$, the fractional Sobolev space $W^{s,p}\left(\mathrm{\Omega }\right)$ is defined as follows:

$$W^{s,p}\left(\mathrm{\Omega }\right)=\left\{u\in L^p\left(\mathrm{\Omega }\right)|{\displaystyle \frac{\left|u\right(x)-u(y\left)\right|}{{|x-y|}^{\frac{n}{p}+s}}}\in L^p(\mathrm{\Omega }\times \mathrm{\Omega })\right\},$$

which is an intermediary Banach space between $L^p\left(\mathrm{\Omega }\right)$ and $W^{1,p}\left(\mathrm{\Omega }\right)$, endowed with the norm

$${\parallel u\parallel }_{W^{s,p}\left(\mathrm{\Omega }\right)}={\left({\displaystyle {\parallel u\parallel }_{L^p\left(\mathrm{\Omega }\right)}^p}+{\displaystyle {\left[u\right]}_{W^{s,p}\left(\mathrm{\Omega }\right)}^p}\right)}^{{\displaystyle \frac{1}{p}}},$$

where

$${\left[u\right]}_{W^{s,p}\left(\mathrm{\Omega }\right)}={\left({\int }_{\mathrm{\Omega }}{\int }_{\mathrm{\Omega }}{\displaystyle \frac{{|u\left(x\right)-u\left(y\right)|}^p}{{|x-y|}^{n+ps}}}dxdy\right)}^{{\displaystyle \frac{1}{p}}},$$

is the Gagliardo seminorm of u. Let ${\displaystyle W_0^{s,p}}\left(\mathrm{\Omega }\right)$ be the closure of ${\displaystyle C_0^{\infty }}\left(\mathrm{\Omega }\right)$ in the norm ${\parallel ·\parallel }_{W^{s,p}\left(\mathrm{\Omega }\right)}$; then, ${\displaystyle W_0^{s,p}}\left({\mathbb{R}}^n\right)={\displaystyle W_0^{s,p}}\left({\mathbb{R}}^n\right)$. For every open and bounded domain $\mathrm{\Omega }\subset {\mathbb{R}}^n$, define ${\displaystyle {\mathcal{W}}_0^{s,p}}\left(\mathrm{\Omega }\right)$ as the closure of ${\displaystyle C_0^{\infty }}\left(\mathrm{\Omega }\right)$ with respect to the norm

$${\parallel u\parallel }_{{\mathcal{W}}^{s,p}\left(\mathrm{\Omega }\right)}={\left({\int }_{{\mathbb{R}}^n}{\int }_{{\mathbb{R}}^n}{\displaystyle \frac{{|u\left(x\right)-u\left(y\right)|}^p}{{|x-y|}^{n+ps}}}dxdy\right)}^{{\displaystyle \frac{1}{p}}}.$$

In fact, the space ${\displaystyle {\mathcal{W}}_0^{s,p}}\left(\mathrm{\Omega }\right)$ can be equivalently defined as the closure of ${\displaystyle C_0^{\infty }}\left(\mathrm{\Omega }\right)$ with respect to the norm

$${\left({\int }_{\mathrm{\Omega }}{\left|u\right|}^{p}dx\right)}^{{\displaystyle \frac{1}{p}}}+{\left({\int }_{{\mathbb{R}}^n}{\int }_{{\mathbb{R}}^n}{\displaystyle \frac{{|u\left(x\right)-u\left(y\right)|}^p}{{|x-y|}^{n+ps}}}dxdy\right)}^{{\displaystyle \frac{1}{p}}}.$$

If $sp\ne 1$ and $\mathrm{\Omega }$ is an open bounded Lipschitz set in ${\mathbb{R}}^n$, the space ${\displaystyle {\mathcal{W}}_0^{s,p}}\left(\mathrm{\Omega }\right)$ coincides with ${\displaystyle W_0^{s,p}}\left(\mathrm{\Omega }\right)$; see [10]. For the case $sp=1$, the strict inclusion holds, i.e., ${\displaystyle {\mathcal{W}}_0^{s,p}}\left(\mathrm{\Omega }\right)⫋{\displaystyle W_0^{s,p}}\left(\mathrm{\Omega }\right)$; see [5].

When $p=2$, for any $0<s<1$, the fractional Sobolev space $W^{s,2}\left({\mathbb{R}}^n\right)$ turns out to be a Hilbert space, usually denoted by $H^s\left({\mathbb{R}}^n\right)$. Actually, we can also define $H^s\left({\mathbb{R}}^n\right)$ by the Fourier transform

$$H^s\left({\mathbb{R}}^n\right)=\left\{u\in L^2\left({\mathbb{R}}^n\right)|{\int }_{{\mathbb{R}}^n}{(1+|\xi |}^{2s}\left)\right|\mathcal{F}u\left(\xi \right){|}^{2}d\xi <\infty \right\},$$

where $\mathcal{F}$ denotes the Fourier transform. For any open bounded domain $\mathrm{\Omega }\subset {\mathbb{R}}^n$, we denote ${\displaystyle {\mathcal{W}}_0^{s,2}}\left(\mathrm{\Omega }\right)={\displaystyle {\mathcal{H}}_0^s}\left(\mathrm{\Omega }\right)$, which is a Hilbert space with the scalar product

$$<u,v{>}_{{\displaystyle {\mathcal{H}}_0^s}\left(\mathrm{\Omega }\right)}={\int }_{{\mathbb{R}}^n}{\int }_{{\mathbb{R}}^n}{\displaystyle \frac{\left[u\right(x)-u(y\left)\right]\left[v\right(x)-v(y\left)\right]}{{|x-y|}^{n+2s}}}dxdy,$$

and corresponding norm

$${\parallel u\parallel }_{{\displaystyle {\mathcal{H}}_0^s}\left(\mathrm{\Omega }\right)}={\left[u\right]}_{H^s\left({\mathbb{R}}^n\right)}={\left({\int }_{{\mathbb{R}}^n}{\int }_{{\mathbb{R}}^n}{\displaystyle \frac{{[u\left(x\right)-u\left(y\right)]}^2}{{|x-y|}^{n+2s}}}dxdy\right)}^{{\displaystyle \frac{1}{2}}}.$$

We recall that ${\displaystyle {\mathcal{H}}_0^s}\left(\mathrm{\Omega }\right)=\left\{u\in H^s\left({\mathbb{R}}^n\right)|u=0\mathrm{in}{\mathbb{R}}^n\setminus \mathrm{\Omega }\right\}$ for any bounded domain with a continuous boundary; see [11].

For more information about fractional Sobolev spaces, we can also refer to [12,13,14,15,16].

2.2. Preliminaries

Lemma 1. 

Let $s\in (0,1)$ and $\omega \in H^s\left({\mathbb{R}}^n\right)$. Assume the kernel of operator ${\displaystyle L_b^s}$ satisfies (3); then

$${\displaystyle {\parallel {\displaystyle L_b^s}\omega \parallel }_{L^2\left({\mathbb{R}}^n\right)}^2}\le {\displaystyle \frac{B^{2}C(n,s)}{2}}{\displaystyle {\left[\omega \right]}_{H^s\left({\mathbb{R}}^n\right)}^2}.$$

Proof. 

It can be proved using a process similar to that in [14] (Section 3). More precisely, using the standard changing variable formula and a second-order Taylor expansion, we have

$${\displaystyle L_b^s}\omega \left(x\right)=-{\displaystyle \frac{C(n,\frac{s}{2})}{2}}{\int }_{{\mathbb{R}}^n}b\left(y\right)[\omega (x+y)+\omega (x-y)-2\omega \left(x\right)]dy,$$

for any $\omega \in \mathcal{S}\left({\mathbb{R}}^n\right)$. By applying the Fourier transform on variable x, we get

$$\begin{array}{ccc}\hfill \mathcal{F}\left({\displaystyle L_b^s}\omega \right)& =& -{\displaystyle \frac{C(n,\frac{s}{2})}{2}}{\int }_{{\mathbb{R}}^n}b\left(y\right)\mathcal{F}[\omega (x+y)+\omega (x-y)-2\omega \left(x\right)]dy\hfill \\ & =& -{\displaystyle \frac{C(n,\frac{s}{2})}{2}}{\int }_{{\mathbb{R}}^n}b\left(y\right)[e^{ix·\xi }+e^{-ix·\xi }-2]dy\left(\mathcal{F}\omega \right)\left(\xi \right)\hfill \\ & =& C(n,{\displaystyle \frac{s}{2}}){\int }_{{\mathbb{R}}^n}b\left(y\right)[1-cos(\xi ·y)]dy\left(\mathcal{F}\omega \right)\left(\xi \right).\hfill \end{array}$$

Then, by combining assumption (3) and the fact that

$${\int }_{{\mathbb{R}}^n}{\displaystyle \frac{1-cos(\xi ·y)}{{\left|y\right|}^{n+s}}}dy=C{(n,{\displaystyle \frac{s}{2}})}^{-1}{\left|\xi \right|}^s,$$

in [14], we have

$$\left|\mathcal{F}\right({\displaystyle L_b^s}\omega \left)\right|\le {B\left|\xi \right|}^s\left|\mathcal{F}\omega \right|.$$

This, together with Proposition 3.4 in [14], leads to

$${\displaystyle {\parallel {\displaystyle L_b^s}\omega \parallel }_{L^2\left({\mathbb{R}}^n\right)}^2}={\displaystyle {\parallel \mathcal{F}\left({\displaystyle L_b^s}\omega \right)\parallel }_{L^2\left({\mathbb{R}}^n\right)}^2}\le B^2{\displaystyle {\parallel {\left|\xi \right|}^s\mathcal{F}\omega \parallel }_{L^2\left({\mathbb{R}}^n\right)}^2}={\displaystyle \frac{B^{2}C(n,s)}{2}}{\displaystyle {\left[\omega \right]}_{H^s\left({\mathbb{R}}^n\right)}^2}.$$

Then, the finial estimate follows for any $\omega \in H^s\left({\mathbb{R}}^n\right)$ by the fact that the space $\mathcal{S}\left({\mathbb{R}}^n\right)$ is dense in $H^s\left({\mathbb{R}}^n\right)$; see [17]. □

Lemma 2. 

Let the kernels of operators ${\displaystyle L_K^{2s}}$ and ${\displaystyle L_b^s}$ satisfy condition (3). Then, for $f\in {\displaystyle L_0^2}\left(\mathrm{\Omega }\right)$, Dirichlet problem (2) is uniquely solvable.

Proof. 

Let $\mathcal{H}={\displaystyle {\mathcal{H}}_0^s}\left(\mathrm{\Omega }\right)$ and $F\left(v\right)={\int }_{\mathrm{\Omega }}f\left(x\right){\displaystyle L_b^s}v\left(x\right)dx$ for $v\in \mathcal{H}$. Then, the bilinear form ${\mathcal{L}}_K$ defined by (4) is bounded and coercive on $\mathcal{H}$, and $F\in {\mathcal{H}}^{*}$, since

$$\lambda {\displaystyle {\left[u\right]}_{H^s\left({\mathbb{R}}^n\right)}^2}\le {\mathcal{L}}_K(u,u)\le \Lambda {\displaystyle {\left[u\right]}_{H^s\left({\mathbb{R}}^n\right)}^2},$$

and

$$\left|F\left(v\right)\right|\le {\parallel f\parallel }_{L^2\left(\mathrm{\Omega }\right)}\parallel {\displaystyle L_b^s}{v\parallel }_{L^2\left({\mathbb{R}}^n\right)}\le B{\left({\displaystyle \frac{C(n,s)}{2}}\right)}^{{\displaystyle \frac{1}{2}}}{\parallel f\parallel }_{L^2\left(\mathrm{\Omega }\right)}{\left[v\right]}_{H^s\left({\mathbb{R}}^n\right)}.$$

Thus, we can conclude the unique solvability of Dirichlet problem (2) from the Lax–Milgram Theorem, i.e., Theorem 5.8 in [6]. □

3. Proof of Theorem 1 and Theorem 2

In this section, we provide the proof of the global boundedness of weak solutions for fractional nonlocal Equations (1) and (2) using the Moser iteration method in [6] and some ideas in [5].

Proof of Theorem 1. 

For any $\beta \ge 1$ and $N>k>0$, let

$$H\left(t\right)=\left\{\begin{array}{cc}{t}^{\beta }-k^{\beta },\hfill & \mathrm{if}t\in [k,N],\hfill \\ \beta N^{\beta -1}(t-N)+N^{\beta }-k^{\beta },\hfill & \mathrm{if}t>N,\hfill \end{array}\right.$$

and $h\left(t\right)={\displaystyle {\int }_k^t}{\left(H^{\prime }\left(s\right)\right)}^{2}ds.$ Let $w=u^{+}+k$; then, we have $h\left(w\right)$, $H\left(w\right)\in {\displaystyle {\mathcal{H}}_0^s}\left(\mathrm{\Omega }\right)$. Taking the test function $\varphi =h\left(w\right(x\left)\right)$ in (4), we have

$$\begin{array}{c}\hfill I={\int }_{{\mathbb{R}}^n}{\int }_{{\mathbb{R}}^n}K(x,y)[u\left(x\right)-u\left(y\right)][h\left(w\left(x\right)\right)-h\left(w\left(y\right)\right)]dydx={\int }_{\mathrm{\Omega }}g\left(x\right)h\left(w\left(x\right)\right)dx=II.\end{array}$$

For the left-hand side, notice that

$$\begin{array}{ccc}\hfill I& =& {\int }_{{\mathbb{R}}^n\cap \{u>0\}}{\int }_{{\mathbb{R}}^n\cap \{u>0\}}K(x,y)[w\left(x\right)-w\left(y\right)][h\left(w\left(x\right)\right)-h\left(w\left(y\right)\right)]dydx\hfill \\ & & +2{\int }_{{\mathbb{R}}^n\cap \{u>0\}}{\int }_{{\mathbb{R}}^n\cap \{u\le 0\}}K(x,y)[u\left(x\right)-u\left(y\right)]h\left(w\left(x\right)\right)dydx\hfill \\ & \ge & {\int }_{{\mathbb{R}}^n}{\int }_{{\mathbb{R}}^n}K(x,y)[w\left(x\right)-w\left(y\right)][h\left(w\left(x\right)\right)-h\left(w\left(y\right)\right)]dydx,\hfill \end{array}$$

where we used the fact that

$$u\left(x\right)-u\left(y\right)\ge u\left(x\right)=(u^{+}\left(x\right)+k)-(u^{+}\left(y\right)+k)=w\left(x\right)-w\left(y\right)$$

whenever $u\left(x\right)>0\ge u\left(y\right)$. Then, by assumption (3) and the general inequality

$$(a-b)(h\left(a\right)-h\left(b\right))=(a-b){\displaystyle {\int }_b^a}{\left(H^{\prime }\left(t\right)\right)}^{2}dt\ge {[H\left(a\right)-H\left(b\right)]}^2\mathrm{for}\mathrm{any}a,b\ge k,$$

we have

$$I\ge \lambda {\int }_{{\mathbb{R}}^n}{\int }_{{\mathbb{R}}^n}{\displaystyle \frac{{[H\left(w\left(x\right)\right)-H\left(w\left(y\right)\right)]}^2}{{|x-y|}^{n+2s}}}dydx.$$

For the right-hand side, since $h\left(w\right)\le w{\left[H^{\prime }\left(w\right)\right]}^2$, by the Hölder inequality, we get

$$II\le {\int }_{\mathrm{\Omega }}\left|g\right|w{\left[H^{\prime }\left(w\right)\right]}^2\le {\int }_{\mathrm{\Omega }}{\displaystyle \frac{\left|g\right|}{k}}{\left[w{H}^{\prime }\left(w\right)\right]}^{2}dx\le {\displaystyle \frac{1}{k}}{\parallel g\parallel }_{L^{{\displaystyle \frac{q}{2s}}}\left(\mathrm{\Omega }\right)}{\displaystyle {\parallel w{H}^{\prime }\left(w\right)\parallel }_{L^{{\displaystyle \frac{2q}{q-2s}}}\left(\mathrm{\Omega }\right)}^2}.$$

Combining the above inequalities, and then applying the fractional Sobolev inequality, i.e, Theorem 6.5 in [14], we have

$$\begin{array}{ccc}\hfill {\parallel H\left(w\right)\parallel }_{L^{{\displaystyle \frac{2n}{n-2s}}}\left(\mathrm{\Omega }\right)}& \le & c(n,s){\left({\int }_{{\mathbb{R}}^n}{\int }_{{\mathbb{R}}^n}{\displaystyle \frac{{[H\left(w\left(x\right)\right)-H\left(w\left(y\right)\right)]}^2}{{|x-y|}^{n+2s}}}dydx\right)}^{{\displaystyle \frac{1}{2}}}\hfill \\ & \le & c(n,s,\lambda ){\left({\displaystyle \frac{{\parallel g\parallel }_{L^{\frac{q}{2s}}\left(\mathrm{\Omega }\right)}}{k}}\right)}^{{\displaystyle \frac{1}{2}}}{\parallel w{H}^{\prime }\left(w\right)\parallel }_{L^{{\displaystyle \frac{2q}{q-2s}}}\left(\mathrm{\Omega }\right)}.\hfill \end{array}$$

Let N go to infinity. By the triangle inequality, we get

$$\begin{array}{ccc}\hfill {\parallel w^{\beta }\parallel }_{L^{{\displaystyle \frac{2n}{n-2s}}}\left(\mathrm{\Omega }\right)}& \le & {\parallel w^{\beta }-k^{\beta }\parallel }_{L^{{\displaystyle \frac{2n}{n-2s}}}\left(\mathrm{\Omega }\right)}+k^{\beta }{\left|\mathrm{\Omega }\right|}^{{\displaystyle \frac{n-2s}{2n}}}\hfill \\ & \le & c(n,s,\lambda ){\left({\displaystyle \frac{{\parallel g\parallel }_{L^{\frac{q}{2s}}\left(\mathrm{\Omega }\right)}}{k}}\right)}^{{\displaystyle \frac{1}{2}}}\beta \parallel w^{\beta }{\parallel }_{L^{{\displaystyle \frac{2q}{q-2s}}}\left(\mathrm{\Omega }\right)}+{\left|\mathrm{\Omega }\right|}^{{\displaystyle \frac{s}{q}}-{\displaystyle \frac{s}{n}}}{\parallel w^{\beta }\parallel }_{L^{{\displaystyle \frac{2q}{q-2s}}}\left(\mathrm{\Omega }\right)}\hfill \\ & =& \left[c(n,s,\lambda ){\left({\displaystyle \frac{{\parallel g\parallel }_{L^{\frac{q}{2s}}\left(\mathrm{\Omega }\right)}}{k}}\right)}^{{\displaystyle \frac{1}{2}}}+{\left|\mathrm{\Omega }\right|}^{{\displaystyle \frac{s}{q}}-{\displaystyle \frac{s}{n}}}\right]\beta {\parallel w^{\beta }\parallel }_{L^{{\displaystyle \frac{2q}{q-2s}}}\left(\mathrm{\Omega }\right)},\hfill \end{array}$$

where the last inequality is due to the fact that

$${\parallel w^{\beta }\parallel }_{L^{{\displaystyle \frac{2q}{q-2s}}}\left(\mathrm{\Omega }\right)}\ge k^{\beta }{\left|\mathrm{\Omega }\right|}^{{\displaystyle \frac{q-2s}{2q}}}.$$

Taking $k={\parallel g\parallel }_{L^{{\displaystyle \frac{q}{2s}}}\left(\mathrm{\Omega }\right)}{\left|\mathrm{\Omega }\right|}^{{\displaystyle \frac{2s}{n}}-{\displaystyle \frac{2s}{q}}}$, we have

$${\parallel w^{\beta }\parallel }_{L^{{\displaystyle \frac{2n}{n-2s}}}\left(\mathrm{\Omega }\right)}\le {c(n,s,\lambda )\beta \left|\mathrm{\Omega }\right|}^{{\displaystyle \frac{s}{q}}-{\displaystyle \frac{s}{n}}}{\parallel w^{\beta }\parallel }_{L^{{\displaystyle \frac{2q}{q-2s}}}\left(\mathrm{\Omega }\right)}.$$

Let $q^{*}={\displaystyle \frac{2q}{q-2s}}$ and $\chi ={\displaystyle \frac{n(q-2s)}{q(n-2s)}}>1$. We get

$${\parallel w\parallel }_{L^{\beta q^{*}\chi }\left(\mathrm{\Omega }\right)}\le {\left({c(n,s,\lambda )\left|\mathrm{\Omega }\right|}^{{\displaystyle \frac{s}{q}}-{\displaystyle \frac{s}{n}}}\right)}^{{\displaystyle \frac{1}{\beta }}}{\beta }^{{\displaystyle \frac{1}{\beta }}}{\parallel w\parallel }_{L^{\beta q^{*}}\left(\mathrm{\Omega }\right)}.$$

Taking $\beta ={\chi }^i$, $i=0$, 1, 2, ⋯, and then by iteration, for any positive integer m,

$$\begin{array}{ccc}\hfill {\parallel w\parallel }_{L^{q^{*}{\chi }^m}\left(\mathrm{\Omega }\right)}& \le & {\left({c(n,s,\lambda )\left|\mathrm{\Omega }\right|}^{{\displaystyle \frac{s}{q}}-{\displaystyle \frac{s}{n}}}\right)}^{{\displaystyle {\sum }_{i=0}^{m-1}}{\chi }^{-i}}{\chi }^{{\displaystyle {\sum }_{i=0}^{m-1}}i{\chi }^{-i}}{\parallel w\parallel }_{L^{q^{*}}\left(\mathrm{\Omega }\right)}\hfill \\ & \le & {C(n,s,\lambda ,q)\left|\mathrm{\Omega }\right|}^{-{\displaystyle \frac{1}{q^{*}}}}{\parallel w\parallel }_{L^{q^{*}}\left(\mathrm{\Omega }\right)}.\hfill \end{array}$$

Letting $m\to \infty$, we obtain

$${\parallel w\parallel }_{L^{\infty }\left(\mathrm{\Omega }\right)}\le {C(n,s,\lambda ,q)\left|\mathrm{\Omega }\right|}^{-{\displaystyle \frac{1}{q^{*}}}}{\parallel w\parallel }_{L^{q^{*}}\left(\mathrm{\Omega }\right)},$$

which together with the interpolation inequality that

$${\parallel w\parallel }_{L^{q^{*}}\left(\mathrm{\Omega }\right)}\le {ϵ|\parallel w\parallel }_{\infty }+{ϵ}^{-\mu }{\parallel w\parallel }_{L^2\left(\mathrm{\Omega }\right)}\mathrm{for}\mathrm{any}ϵ>0,\mathrm{where}\mu ={\displaystyle \frac{2s}{q-2s}},$$

imply that

$${\parallel w\parallel }_{L^{\infty }\left(\mathrm{\Omega }\right)}\le {C(n,s,\lambda ,q)\left|\mathrm{\Omega }\right|}^{-{\displaystyle \frac{1}{2}}}{\parallel w\parallel }_{L^2\left(\mathrm{\Omega }\right)}.$$

Consequently,

$$\begin{array}{ccc}\hfill \parallel u^{+}{\parallel }_{L^{\infty }\left(\mathrm{\Omega }\right)}& \le & C(n,s,\lambda ,q)\left[\right|\mathrm{\Omega }{|}^{-{\displaystyle \frac{1}{2}}}\parallel u^{+}{\parallel }_{L^2\left(\mathrm{\Omega }\right)}+k]\hfill \\ & \le & C(n,s,\lambda ,q)\left[{\left({⨏}_{\mathrm{\Omega }}{\left|u\right|}^{2}dx\right)}^{{\displaystyle \frac{1}{2}}}+{\left|\mathrm{\Omega }\right|}^{{\displaystyle \frac{2s}{n}}}{\left({⨏}_{\mathrm{\Omega }}{\left|g\right|}^{{\displaystyle \frac{q}{2s}}}dx\right)}^{{\displaystyle \frac{2s}{q}}}\right].\hfill \end{array}$$

Then, by repeating the above process to $-u$, we get

$${\parallel u\parallel }_{L^{\infty }\left(\mathrm{\Omega }\right)}\le C(n,s,\lambda ,q)\left[{\left({⨏}_{\mathrm{\Omega }}{\left|u\right|}^{2}dx\right)}^{{\displaystyle \frac{1}{2}}}+{\left|\mathrm{\Omega }\right|}^{{\displaystyle \frac{2s}{n}}}{\left({⨏}_{\mathrm{\Omega }}{\left|g\right|}^{{\displaystyle \frac{q}{2s}}}dx\right)}^{{\displaystyle \frac{2s}{q}}}\right].$$

(8)

Taking $\varphi =u$ in (4), then by assumption (3), the Hölder inequality, and the fractional Sobolev inequality, we get

$$\begin{array}{ccc}\hfill \lambda {\displaystyle {\left[u\right]}_{H^s\left({\mathbb{R}}^n\right)}^2}& \le & {\int }_{{\mathbb{R}}^n}{\int }_{{\mathbb{R}}^n}K(x,y){[u\left(x\right)-u\left(y\right)]}^{2}dydx\hfill \\ & =& {\int }_{\mathrm{\Omega }}g\left(x\right)u\left(x\right)dx\hfill \\ & \le & {\parallel g\parallel }_{L^{{\displaystyle \frac{q}{2s}}}\left(\mathrm{\Omega }\right)}{\parallel u\parallel }_{L^{{\displaystyle \frac{q}{q-2s}}}\left(\mathrm{\Omega }\right)}\hfill \\ & \le & {\left|\mathrm{\Omega }\right|}^{{\displaystyle \frac{1}{2}}+{\displaystyle \frac{s}{n}}-{\displaystyle \frac{2s}{q}}}{\parallel g\parallel }_{L^{{\displaystyle \frac{q}{2s}}}\left(\mathrm{\Omega }\right)}{\parallel u\parallel }_{L^{{\displaystyle \frac{2n}{n-2s}}}\left(\mathrm{\Omega }\right)}\hfill \\ & \le & {C(n,s)\left|\mathrm{\Omega }\right|}^{{\displaystyle \frac{1}{2}}+{\displaystyle \frac{s}{n}}-{\displaystyle \frac{2s}{q}}}{\parallel g\parallel }_{L^{{\displaystyle \frac{q}{2s}}}\left(\mathrm{\Omega }\right)}{\left[u\right]}_{H^s\left({\mathbb{R}}^n\right)},\hfill \end{array}$$

which together with the Hölder inequality imply that

$$\begin{array}{c}\hfill {\left({⨏}_{\mathrm{\Omega }}{\left|u\right|}^2\right)}^{{\displaystyle \frac{1}{2}}}\le {\left|\mathrm{\Omega }\right|}^{{\displaystyle \frac{s}{n}}-{\displaystyle \frac{1}{2}}}{\parallel u\parallel }_{L^{{\displaystyle \frac{2n}{n-2s}}}}\le {C(n,s)\left|\mathrm{\Omega }\right|}^{{\displaystyle \frac{s}{n}}-{\displaystyle \frac{1}{2}}}{\left[u\right]}_{H^s\left({\mathbb{R}}^n\right)}\le C(n,s,\lambda ){\left|\mathrm{\Omega }\right|}^{{\displaystyle \frac{2s}{n}}}{\left({⨏}_{\mathrm{\Omega }}{\left|g\right|}^{{\displaystyle \frac{q}{2s}}}\right)}^{{\displaystyle \frac{2s}{q}}}.\end{array}$$

Then, the final estimate for u follows by this estimate and inequality (8). Thus, we finish the proof of Theorem 1. □

Proof of Theorem 2. 

(i) First, we suppose that $f\ge 0$ and $b(x,y)\ge 0$. Assume $\beta$, k, and functions w, $h\left(t\right)$, and $H\left(t\right)$ are all as in the proof of Theorem 1. Taking the test function $\varphi =h\left(w\right(x\left)\right)$ in (5), we have

$$\begin{array}{cc}& {\int }_{{\mathbb{R}}^n}{\int }_{{\mathbb{R}}^n}K(x,y)[u\left(x\right)-u\left(y\right)][h\left(w\left(x\right)\right)-h\left(w\left(\left(y\right)\right)\right]dydx\hfill \\ =& {\int }_{\mathrm{\Omega }}f\left(x\right){\int }_{{\mathbb{R}}^n}b(x,y)[h(w\left(\left(x\right)\right)-h\left(w\left(y\right)\right)]dydx.\hfill \end{array}$$

(9)

For the left-hand side, we still have

$$\begin{array}{cc}& {\int }_{{\mathbb{R}}^n}{\int }_{{\mathbb{R}}^n}K(x,y)[u\left(x\right)-u\left(y\right)][h\left(w\left(x\right)\right)-h\left(w\left(\left(y\right)\right)\right]dydx\hfill \\ \ge & \lambda {\int }_{{\mathbb{R}}^n}{\int }_{{\mathbb{R}}^n}{\displaystyle \frac{{[H\left(w\left(x\right)\right)-H\left(w\left(y\right)\right)]}^2}{{|x-y|}^{n+2s}}}dydx.\hfill \end{array}$$

(10)

For the right-hand side, we can find

$$\begin{array}{cc}& {\int }_{\mathrm{\Omega }}f\left(x\right){\int }_{{\mathbb{R}}^n}b(x,y)[h(w\left(\left(x\right)\right)-h\left(w\left(y\right)\right)]dydx\hfill \\ \le & C(n,s,B){\int }_{\mathrm{\Omega }}{f}^2\left(x\right){\left[H^{\prime }\left(w\left(x\right)\right)\right]}^{2}dx+{\displaystyle \frac{1}{2}}{\int }_{{\mathbb{R}}^n}{\int }_{{\mathbb{R}}^n}{\displaystyle \frac{{[H\left(w\left(x\right)\right)-H\left(w\left(y\right)\right)]}^2}{{|x-y|}^{n+2s}}}dydx.\hfill \end{array}$$

(11)

Indeed, since $f\ge 0$, $b\ge 0$, and

$$h\left(a\right)-h\left(b\right)={\displaystyle {\int }_b^a}{h}^{\prime }\left(t\right)dt\le H^{\prime }\left(a\right)[H\left(a\right)-H\left(b\right)]\mathrm{for}\mathrm{any}a,b\ge k,$$

then by assumption (3), the Hölder inequality, and the Cauchy inequality, for any $ϵ\in (0,1)$, we have

$$\begin{array}{cc}& {\int }_{\mathrm{\Omega }}f\left(x\right){\int }_{{\mathbb{R}}^n}b(x,y)[h(w\left(\left(x\right)\right)-h\left(w\left(y\right)\right)]dydx\hfill \\ \le & {\int }_{\mathrm{\Omega }}f\left(x\right)H^{\prime }\left(w\left(x\right)\right){\int }_{{\mathbb{R}}^n}b(x,y)[H(w\left(\left(x\right)\right)-H\left(w\left(y\right)\right)]dydx\hfill \\ \le & {\left({\int }_{\mathrm{\Omega }}{f}^2\left(x\right){\left[H^{\prime }\left(w\left(x\right)\right)\right]}^{2}dx\right)}^{{\displaystyle \frac{1}{2}}}{\left[{\int }_{\mathrm{\Omega }}{\left({\int }_{{\mathbb{R}}^n}b(x,y)[H(w\left(\left(x\right)\right)-H\left(w\left(y\right)\right)]dy\right)}^{2}dx\right]}^{{\displaystyle \frac{1}{2}}}\hfill \\ \le & {\displaystyle \frac{1}{ϵ}}{\int }_{\mathrm{\Omega }}{f}^2\left(x\right){\left[H^{\prime }\left(w\left(x\right)\right)\right]}^{2}dx+ϵ{\int }_{\mathrm{\Omega }}{\left({\int }_{{\mathbb{R}}^n}b(x,y)[H(w\left(\left(x\right)\right)-H\left(w\left(y\right)\right)]dy\right)}^{2}dx.\hfill \\ \le & {\displaystyle \frac{1}{ϵ}}{\int }_{\mathrm{\Omega }}{f}^2\left(x\right){\left[H^{\prime }\left(w\left(x\right)\right)\right]}^{2}dx+{\displaystyle \frac{B^{2}C(n,s)ϵ}{2}}{\int }_{{\mathbb{R}}^n}{\int }_{{\mathbb{R}}^n}{\displaystyle \frac{[H{(w\left(\left(x\right)\right)-H\left(w\left(y\right)\right)]}^2}{{|x-y|}^{n+2s}}}dydx.\hfill \end{array}$$

(12)

where the last inequality is due to Lemma 1. Taking $ϵ={\displaystyle \frac{1}{B^{2}C(n,s)}}$ in (12), we obtain (11).

Combining (9)–(11) and applying the fractional Sobolev inequality, we get

$$\begin{array}{ccc}\hfill {\parallel H\left(w\right)\parallel }_{L^{{\displaystyle \frac{2n}{n-2s}}}\left(\mathrm{\Omega }\right)}& \le & C(n,s){\left[{\int }_{{\mathbb{R}}^n}{\int }_{{\mathbb{R}}^n}{\displaystyle \frac{{[H\left(w\left(x\right)\right)-H\left(w\left(y\right)\right)]}^2}{{|x-y|}^{n+2s}}}dydx\right]}^{{\displaystyle \frac{1}{2}}}\hfill \\ & \le & C(n,s,\lambda ,B)({\int }_{\mathrm{\Omega }}{f}^2{\left[H^{\prime }\left(w\left(x\right)\right)\right]}^2{dx)}^{{\displaystyle \frac{1}{2}}}\hfill \\ & \le & {\displaystyle \frac{C(n,s,\lambda ,B)}{k}}{\left({\int }_{\mathrm{\Omega }}{f}^2{\left[w{H}^{\prime }\left(w\left(x\right)\right)\right]}^{2}dx\right)}^{{\displaystyle \frac{1}{2}}}\hfill \\ & \le & {\displaystyle \frac{C(n,s,\lambda ,B)}{k}}{\parallel f\parallel }_{L^{{\displaystyle \frac{q}{s}}}\left(\mathrm{\Omega }\right)}{\parallel w{H}^{\prime }\left(w\right)\parallel }_{L^{{\displaystyle \frac{2q}{q-2s}}}\left(\mathrm{\Omega }\right)}.\hfill \end{array}$$

Let N go to infinity and take $k={\parallel f\parallel }_{L^{{\displaystyle \frac{q}{s}}}\left(\mathrm{\Omega }\right)}{\left|\mathrm{\Omega }\right|}^{{\displaystyle \frac{s}{n}}-{\displaystyle \frac{s}{q}}}$. Then, using the same process as that in Theorem 1, we get

$${\parallel w\parallel }_{L^{\infty }\left(\mathrm{\Omega }\right)}\le {C(n,s,\lambda ,B,q)\left|\mathrm{\Omega }\right|}^{-{\displaystyle \frac{1}{2}}}{\parallel w\parallel }_{L^2\left(\mathrm{\Omega }\right)}.$$

which implies

$$\begin{array}{ccc}\hfill \parallel u^{+}{\parallel }_{L^{\infty }\left(\mathrm{\Omega }\right)}& \le & C(n,s,\lambda ,B,q)\left[\right|\mathrm{\Omega }{|}^{-{\displaystyle \frac{1}{2}}}\parallel u^{+}{\parallel }_{L^2\left(\mathrm{\Omega }\right)}+k]\hfill \\ & \le & C(n,s,\lambda ,B,q)\left[{({⨏}_{\mathrm{\Omega }}{\left|u\right|}^{2}dx)}^{{\displaystyle \frac{1}{2}}}+{\left|\mathrm{\Omega }\right|}^{{\displaystyle \frac{s}{n}}}{({⨏}_{\mathrm{\Omega }}{\left|f\right|}^{{\displaystyle \frac{q}{s}}}dx)}^{{\displaystyle \frac{s}{q}}}\right].\hfill \end{array}$$

The final estimate for the case when f and b both are non-negative follows by repeating the above process to $-u$.

(ii) Generally, let $f=f_{+}-f_{-}$ and $b=b_{+}-b_{-}$. By Lemma 2, there exists the unique $u_1\in {\displaystyle {\mathcal{H}}_0^s}\left(\mathrm{\Omega }\right)$ satisfying

$$\left\{\begin{array}{cc}{\displaystyle L_K^{2s}}{u}_1\left(x\right)={\displaystyle L_{b_{+}}^s}{f}_{+}\left(x\right),\hfill & \mathrm{in}\mathrm{\Omega },\hfill \\ u_1=0,\hfill & \mathrm{in}{\mathbb{R}}^n\setminus \mathrm{\Omega }.\hfill \end{array}\right.$$

(13)

Similarly, there exist $u_2$ and $u_3\in {\displaystyle {\mathcal{H}}_0^s}\left(\mathrm{\Omega }\right)$ satisfying

$$\left\{\begin{array}{cc}{\displaystyle L_K^{2s}}{u}_2\left(x\right)={\displaystyle L_{b_{-}}^s}{f}_{+}\left(x\right),\hfill & \mathrm{in}\mathrm{\Omega },\hfill \\ u_2=0,\hfill & \mathrm{in}{\mathbb{R}}^n\setminus \mathrm{\Omega },\hfill \end{array}\right.$$

and

$$\left\{\begin{array}{cc}{\displaystyle L_K^{2s}}{u}_3\left(x\right)={\displaystyle L_{b_{+}}^s}{f}_{-}\left(x\right),\hfill & \mathrm{in}\mathrm{\Omega },\hfill \\ u_3=0,\hfill & \mathrm{in}{\mathbb{R}}^n\setminus \mathrm{\Omega },\hfill \end{array}\right.$$

respectively. Let $u_4=u-u_1+u_2+u_3$; we can find that $u_4\in {\displaystyle {\mathcal{H}}_0^s}\left(\mathrm{\Omega }\right)$ is a weak solution of

$$\left\{\begin{array}{cc}{\displaystyle L_K^{2s}}{u}_4\left(x\right)={\displaystyle L_{b_{-}}^s}{f}_{-}\left(x\right),\hfill & \mathrm{in}\mathrm{\Omega },\hfill \\ u_4=0,\hfill & \mathrm{in}{\mathbb{R}}^n\setminus \mathrm{\Omega }.\hfill \end{array}\right.$$

By (i), we have

$${\parallel u_i\parallel }_{L^{\infty }\left(\mathrm{\Omega }\right)}\le C(n,s,\lambda ,B,q)\left[{\left({⨏}_{\mathrm{\Omega }}{\left|u_i\right|}^{2}dx\right)}^{{\displaystyle \frac{1}{2}}}+{\left|\mathrm{\Omega }\right|}^{{\displaystyle \frac{s}{n}}}{({⨏}_{\mathrm{\Omega }}{\left|f_{+}\right|}^{{\displaystyle \frac{q}{s}}}dx)}^{{\displaystyle \frac{s}{q}}}\right],i=1,2,$$

(14)

and

$${\parallel u_i\parallel }_{L^{\infty }\left(\mathrm{\Omega }\right)}\le C(n,s,\lambda ,B,q)\left[{\left({⨏}_{\mathrm{\Omega }}{\left|u_i\right|}^{2}dx\right)}^{{\displaystyle \frac{1}{2}}}+{\left|\mathrm{\Omega }\right|}^{{\displaystyle \frac{s}{n}}}{({⨏}_{\mathrm{\Omega }}{\left|f_{-}\right|}^{{\displaystyle \frac{q}{s}}}dx)}^{{\displaystyle \frac{s}{q}}}\right],i=3,4.$$

(15)

Multiply (13) by $u_1$ and integrate over ${\mathbb{R}}^n$; then, by assumption (3), the Hölder inequality, and the Lemma 1, we get

$$\begin{array}{ccc}\hfill \lambda {\displaystyle {\left[u_1\right]}_{H^s\left({\mathbb{R}}^n\right)}^2}& \le & {\int }_{{\mathbb{R}}^n}{\int }_{{\mathbb{R}}^n}K(x,y){[u_1\left(x\right)-u_1\left(y\right)]}^{2}dydx\hfill \\ & =& {\int }_{\mathrm{\Omega }}{f}_{+}{\displaystyle L_{b_{+}}^s}{u}_{1}dx\hfill \\ & \le & {\parallel f_{+}\parallel }_{L^2\left(\mathrm{\Omega }\right)}{\parallel {\displaystyle L_{b_{+}}^s}{u}_1\parallel }_{L^2\left(\mathrm{\Omega }\right)}\hfill \\ & \le & {C(n,s,B)\left|\mathrm{\Omega }\right|}^{{\displaystyle \frac{1}{2}}-{\displaystyle \frac{s}{q}}}{\parallel f_{+}\parallel }_{L^{{\displaystyle \frac{q}{s}}}\left(\mathrm{\Omega }\right)}{\left[u_1\right]}_{H^s\left({\mathbb{R}}^n\right)}\hfill \end{array}$$

which together with the Hölder inequality and the fractional Sobolev inequality imply that

$$\begin{array}{ccc}\hfill {\left({⨏}_{\mathrm{\Omega }}{\left|u_1\right|}^2\right)}^{{\displaystyle \frac{1}{2}}}& \le & {\left|\mathrm{\Omega }\right|}^{{\displaystyle \frac{s}{n}}-{\displaystyle \frac{1}{2}}}{\parallel u_1\parallel }_{L^{{\displaystyle \frac{2n}{n-2s}}}}\hfill \\ & \le & {C(n,s)\left|\mathrm{\Omega }\right|}^{{\displaystyle \frac{s}{n}}-{\displaystyle \frac{1}{2}}}{\left[u_1\right]}_{H^s\left({\mathbb{R}}^n\right)}\hfill \\ & \le & {C(n,s,\lambda ,B)\left|\mathrm{\Omega }\right|}^{{\displaystyle \frac{s}{n}}}{\left({⨏}_{\mathrm{\Omega }}{\left|f_{+}\right|}^{{\displaystyle \frac{q}{s}}}\right)}^{{\displaystyle \frac{s}{q}}}.\hfill \end{array}$$

A similar inequality holds for $u_2$, $u_3$, and $u_4$. Combining these inequalities and (14) and (15), we obtain

$$\begin{array}{ccc}\hfill {\parallel u\parallel }_{L^{\infty }\left(\mathrm{\Omega }\right)}\le {\displaystyle \sum _{i=1}^4}{\parallel u_i\parallel }_{L^{\infty }\left(\mathrm{\Omega }\right)}& \le & {C(n,s,\lambda ,B,q)\left|\mathrm{\Omega }\right|}^{{\displaystyle \frac{s}{n}}}\left[{\left({⨏}_{\mathrm{\Omega }}{\left|f_{+}\right|}^{{\displaystyle \frac{q}{s}}}\right)}^{{\displaystyle \frac{s}{q}}}+{\left({⨏}_{\mathrm{\Omega }}{\left|f_{-}\right|}^{{\displaystyle \frac{q}{s}}}\right)}^{{\displaystyle \frac{s}{q}}}\right]\hfill \\ & \le & {C(n,s,\lambda ,B,q)\left|\mathrm{\Omega }\right|}^{{\displaystyle \frac{s}{n}}}{\left({⨏}_{\mathrm{\Omega }}{\left|f\right|}^{{\displaystyle \frac{q}{s}}}\right)}^{{\displaystyle \frac{s}{q}}}.\hfill \end{array}$$

This completes the proof of Theorem 2. □

4. Proof of Theorem 3

In this section, we mainly establish the global boundedness of weak solutions to fractional p-Laplacian Equation (6).

Proof of Theorem 3. 

For any $\beta \ge 1$ and $N>k>0$, let $H\left(t\right)$ be the same as that in Section 3 and $\tilde{h}\left(t\right)={\displaystyle {\int }_k^t}{\left[H^{\prime }\left(s\right)\right]}^{p}ds.$ Let $w=u^{+}+k$; then we have $\tilde{h}\left(w\right)$, $H\left(w\right)\in {\displaystyle {\mathcal{W}}_0^{s,p}}\left(\mathrm{\Omega }\right)$. In fact, we only need to show $u_{\pm }\in W^{s,p}\left({\mathbb{R}}^n\right)$ from $u\in W^{s,p}\left({\mathbb{R}}^n\right)$. We write

$${\displaystyle {\left[u\right]}_{W^{s,p}\left({\mathbb{R}}^n\right)}^p}-{\displaystyle {\left[u_{+}\right]}_{W^{s,p}\left({\mathbb{R}}^n\right)}^p}-{\displaystyle {\left[u_{-}\right]}_{W^{s,p}\left({\mathbb{R}}^n\right)}^p}={\int }_{{\mathbb{R}}^n}{\int }_{{\mathbb{R}}^n}{\displaystyle \frac{I(x,y)}{{|x-y|}^{n+ps}}}dydx,$$

where $I(x,y)={|u\left(x\right)-u\left(y\right)|}^p-|u_{+}\left(x\right)-u_{+}{\left(y\right)|}^p-{|u_{-}\left(x\right)-u_{-}\left(y\right)|}^p$ for any $(x,y)\in {\mathbb{R}}^n\times {\mathbb{R}}^n$. We can find that $I(x,y)\ge 0$ in ${\mathbb{R}}^n\times {\mathbb{R}}^n$ if we notice

$${\mathbb{R}}^n\times {\mathbb{R}}^n=({\mathrm{\Omega }}_{+}\times {\mathrm{\Omega }}_{+})\cup ({\mathrm{\Omega }}_{+}\times {\mathrm{\Omega }}_{-})\cup ({\mathrm{\Omega }}_{-}\times {\mathrm{\Omega }}_{+})\cup ({\mathrm{\Omega }}_{-}\times {\mathrm{\Omega }}_{-})$$

where ${\mathrm{\Omega }}_{+}=\{u\ge 0\}$ and ${\mathrm{\Omega }}_{-}=\{u<0\}$. Thus, we have $u_{\pm }\in W^{s,p}\left({\mathbb{R}}^n\right)$. Taking the test function $\varphi =\tilde{h}\left(w\left(x\right)\right)$ in (7), we have

$$\begin{array}{ccc}\hfill I& =& {\int }_{{\mathbb{R}}^n}{\int }_{{\mathbb{R}}^n}{\displaystyle \frac{{|u\left(x\right)-u\left(y\right)|}^{p-2}[u\left(x\right)-u\left(y\right)][\tilde{h}\left(w\left(x\right)\right)-\tilde{h}\left(w\left(y\right)\right)]}{{|x-y|}^{n+ps}}}dydx\hfill \\ & =& {\int }_{\mathrm{\Omega }}g\left(x\right)\tilde{h}\left(w\left(x\right)\right)dx=II.\hfill \end{array}$$

For the left-hand side, we have

$$\begin{array}{ccc}\hfill I& =& {\int }_{{\mathbb{R}}^n\cap \{u>0\}}{\int }_{{\mathbb{R}}^n\cap \{u>0\}}{\displaystyle \frac{{|w\left(x\right)-w\left(y\right)|}^{p-2}[w\left(x\right)-w\left(y\right)][\tilde{h}\left(w\left(x\right)\right)-\tilde{h}\left(w\left(y\right)\right)]}{{|x-y|}^{n+ps}}}dydx\hfill \\ & & +2{\int }_{{\mathbb{R}}^n\cap \{u>0\}}{\int }_{{\mathbb{R}}^n\cap \{u\le 0\}}{\displaystyle \frac{{|u\left(x\right)-u\left(y\right)|}^{p-2}[u\left(x\right)-u\left(y\right)]\tilde{h}\left(w\left(x\right)\right)}{{|x-y|}^{n+ps}}}dydx\hfill \\ & \ge & {\int }_{{\mathbb{R}}^n}{\int }_{{\mathbb{R}}^n}{\displaystyle \frac{{|w\left(x\right)-w\left(y\right)|}^{p-2}[w\left(x\right)-w\left(y\right)][\tilde{h}\left(w\left(x\right)\right)-\tilde{h}\left(w\left(y\right)\right)]}{{|x-y|}^{n+ps}}}dydx,\hfill \end{array}$$

where we used the fact that

$${|u\left(x\right)-u\left(y\right)|}^{p-2}[u\left(x\right)-u\left(y\right)]\ge {\left[u\left(x\right)\right]}^{p-1}={|w\left(x\right)-w\left(y\right)|}^{p-2}[w\left(x\right)-w\left(y\right)],$$

whenever $u\left(x\right)>0\ge u\left(y\right)$. Then, by the inequality

$${|a-b|}^{p-2}(a-b)[\tilde{h}\left(a\right)-\tilde{h}\left(b\right)]={|a-b|}^{p-1}\left|{\displaystyle {\int }_b^a}{\left[H^{\prime }\left(t\right)\right]}^{p}dt\right|\ge {|H\left(a\right)-H\left(b\right)|}^p,$$

for any a, $b\ge k$, which is due to the Hölder inequality, we have

$$I\ge {\int }_{{\mathbb{R}}^n}{\int }_{{\mathbb{R}}^n}{\displaystyle \frac{{|H\left(w\left(x\right)\right)-H\left(w\left(y\right)\right)|}^p}{{|x-y|}^{n+ps}}}dydx.$$

For the right-hand side, since $\tilde{h}\left(w\right)\le w{\left[H^{\prime }\left(w\right)\right]}^p$, then by the Hölder inequality, we get

$$II\le {\int }_{\mathrm{\Omega }}\left|g\right|w{\left[H^{\prime }\left(w\right)\right]}^p\le {\int }_{\mathrm{\Omega }}{\displaystyle \frac{\left|g\right|}{k^{p-1}}}{\left[w{H}^{\prime }\left(w\right)\right]}^{p}dx\le {\displaystyle \frac{1}{k^{p-1}}}{\parallel g\parallel }_{L^{{\displaystyle \frac{q}{ps}}}\left(\mathrm{\Omega }\right)}{\displaystyle {\parallel w{H}^{\prime }\left(w\right)\parallel }_{L^{{\displaystyle \frac{pq}{q-ps}}}\left(\mathrm{\Omega }\right)}^p}.$$

Combining the above inequalities, we have

$${\int }_{{\mathbb{R}}^n}{\int }_{{\mathbb{R}}^n}{\displaystyle \frac{{|H\left(w\left(x\right)\right)-H\left(w\left(y\right)\right)|}^p}{{|x-y|}^{n+ps}}}dydx\le {\displaystyle \frac{1}{k^{p-1}}}{\parallel g\parallel }_{L^{{\displaystyle \frac{q}{ps}}}\left(\mathrm{\Omega }\right)}{\displaystyle {\parallel w{H}^{\prime }\left(w\right)\parallel }_{L^{{\displaystyle \frac{pq}{q-ps}}}\left(\mathrm{\Omega }\right)}^p}.$$

Then, by applying fractional the Sobolev inequality, we get

$$\begin{array}{ccc}\hfill {\parallel H\left(w\right)\parallel }_{L^{{\displaystyle \frac{np}{n-sp}}}\left(\mathrm{\Omega }\right)}& \le & c(n,s,p){\left({\int }_{{\mathbb{R}}^n}{\int }_{{\mathbb{R}}^n}{\displaystyle \frac{{|H\left(w\left(x\right)\right)-H\left(w\left(y\right)\right)|}^p}{{|x-y|}^{n+ps}}}dydx\right)}^{{\displaystyle \frac{1}{p}}}\hfill \\ & \le & c(n,s,p){\left({\displaystyle \frac{{\parallel g\parallel }_{L^{\frac{q}{ps}}\left(\mathrm{\Omega }\right)}}{k^{p-1}}}\right)}^{{\displaystyle \frac{1}{p}}}{\parallel w{H}^{\prime }\left(w\right)\parallel }_{L^{{\displaystyle \frac{pq}{q-ps}}}\left(\mathrm{\Omega }\right)}.\hfill \end{array}$$

Let N go to infinity; then, by the triangle inequality and the fact that

$${\parallel w^{\beta }\parallel }_{L^{{\displaystyle \frac{pq}{q-ps}}}\left(\mathrm{\Omega }\right)}\ge k^{\beta }{\left|\mathrm{\Omega }\right|}^{{\displaystyle \frac{q-ps}{pq}}},$$

we have

$$\begin{array}{ccc}\hfill \parallel w^{\beta }{\parallel }_{L^{{\displaystyle \frac{np}{n-sp}}}\left(\mathrm{\Omega }\right)}& \le & {\parallel w^{\beta }-k^{\beta }\parallel }_{L^{{\displaystyle \frac{np}{n-sp}}}\left(\mathrm{\Omega }\right)}+k^{\beta }{\left|\mathrm{\Omega }\right|}^{{\displaystyle \frac{n-sp}{np}}}\hfill \\ & \le & c(n,s,p){\left({\displaystyle \frac{{\parallel g\parallel }_{L^{\frac{q}{ps}}\left(\mathrm{\Omega }\right)}}{k^{p-1}}}\right)}^{{\displaystyle \frac{1}{p}}}\beta \parallel w^{\beta }{\parallel }_{L^{{\displaystyle \frac{pq}{q-ps}}}\left(\mathrm{\Omega }\right)}+{\left|\mathrm{\Omega }\right|}^{{\displaystyle \frac{s}{q}}-{\displaystyle \frac{s}{n}}}{\parallel w^{\beta }\parallel }_{L^{{\displaystyle \frac{2q}{q-2s}}}\left(\mathrm{\Omega }\right)}.\hfill \end{array}$$

Taking $k={\displaystyle {\parallel g\parallel }_{L^{{\displaystyle \frac{q}{ps}}}\left(\mathrm{\Omega }\right)}^{{\displaystyle \frac{1}{p-1}}}}{\left|\mathrm{\Omega }\right|}^{({\displaystyle \frac{s}{n}}-{\displaystyle \frac{s}{q}}){\displaystyle \frac{p}{p-1}}}$, we have

$${\parallel w^{\beta }\parallel }_{L^{{\displaystyle \frac{pn}{n-sp}}}\left(\mathrm{\Omega }\right)}\le {c(n,s,p)\beta \left|\mathrm{\Omega }\right|}^{{\displaystyle \frac{s}{q}}-{\displaystyle \frac{s}{n}}}{\parallel w^{\beta }\parallel }_{L^{{\displaystyle \frac{pq}{q-ps}}}\left(\mathrm{\Omega }\right)}.$$

Setting ${\tilde{q}}^{*}={\displaystyle \frac{pq}{q-ps}}$ and $\tilde{\chi }={\displaystyle \frac{n(q-ps)}{q(n-ps)}}>1$ due to $q>n$, we get

$${\parallel w\parallel }_{L^{\beta {\tilde{q}}^{*}\tilde{\chi }}\left(\mathrm{\Omega }\right)}\le {\left({c(n,s,p)\left|\mathrm{\Omega }\right|}^{{\displaystyle \frac{s}{q}}-{\displaystyle \frac{s}{n}}}\right)}^{{\displaystyle \frac{1}{\beta }}}{\beta }^{{\displaystyle \frac{1}{\beta }}}{\parallel w\parallel }_{L^{\beta {\tilde{q}}^{*}}\left(\mathrm{\Omega }\right)}.$$

By iteration, for any positive integer m, we have

$${\parallel w\parallel }_{L^{{\tilde{q}}^{*}{\tilde{\chi }}^m}\left(\mathrm{\Omega }\right)}\le {\left({c(n,s,p)\left|\mathrm{\Omega }\right|}^{{\displaystyle \frac{s}{q}}-{\displaystyle \frac{s}{n}}}\right)}^{{\displaystyle {\sum }_{i=0}^{m-1}}{\tilde{\chi }}^{-i}}{\tilde{\chi }}^{{\displaystyle {\sum }_{i=0}^{m-1}}i{\tilde{\chi }}^{-i}}{\parallel w\parallel }_{L^{{\tilde{q}}^{*}}\left(\mathrm{\Omega }\right)}.$$

Letting $m\to \infty$, we obtain

$${\parallel w\parallel }_{L^{\infty }\left(\mathrm{\Omega }\right)}\le {C(n,s,p,q)\left|\mathrm{\Omega }\right|}^{-{\displaystyle \frac{1}{{\tilde{q}}^{*}}}}{\parallel w\parallel }_{L^{{\tilde{q}}^{*}}\left(\mathrm{\Omega }\right)}.$$

Consequently,

$$\begin{array}{ccc}\hfill {\parallel u^{+}\parallel }_{L^{\infty }\left(\mathrm{\Omega }\right)}& \le & C(n,s,p,q)\left[\right|\mathrm{\Omega }{|}^{-{\displaystyle \frac{1}{{\tilde{q}}^{*}}}}\parallel u^{+}{\parallel }_{L^{{\tilde{q}}^{*}}\left(\mathrm{\Omega }\right)}+k]\hfill \\ & \le & C(n,s,p,q)\left[{\left|\mathrm{\Omega }\right|}^{-{\displaystyle \frac{1}{{\tilde{q}}^{*}}}}{\parallel u\parallel }_{L^{{\tilde{q}}^{*}}\left(\mathrm{\Omega }\right)}+{\left({\left|\mathrm{\Omega }\right|}^{{\displaystyle \frac{ps}{n}}}{\left({⨏}_{\mathrm{\Omega }}{\left|g\right|}^{{\displaystyle \frac{q}{ps}}}dx\right)}^{{\displaystyle \frac{ps}{q}}}\right)}^{{\displaystyle \frac{1}{p-1}}}\right].\hfill \end{array}$$

(16)

By taking $\varphi =u$ in (7), then by the Hölder inequality and fractional Sobolev inequality, we get

$$\begin{array}{ccc}\hfill {\displaystyle {\left[u\right]}_{W^{s,p}\left({\mathbb{R}}^n\right)}^p}& =& {\int }_{{\mathbb{R}}^n}{\int }_{{\mathbb{R}}^n}{\displaystyle \frac{{|u\left(x\right)-u\left(y\right)|}^p}{{|x-y|}^{n+ps}}}dydx\hfill \\ & =& {\int }_{\mathrm{\Omega }}g\left(x\right)u\left(x\right)dx\hfill \\ & \le & {\left|\mathrm{\Omega }\right|}^{{\displaystyle \frac{p-1}{p}}+{\displaystyle \frac{s}{n}}-{\displaystyle \frac{ps}{q}}}{\parallel g\parallel }_{L^{{\displaystyle \frac{q}{ps}}}\left(\mathrm{\Omega }\right)}{\parallel u\parallel }_{L^{{\displaystyle \frac{np}{n-sp}}}\left(\mathrm{\Omega }\right)}\hfill \\ & \le & {C(n,s,p)\left|\mathrm{\Omega }\right|}^{{\displaystyle \frac{p-1}{p}}+{\displaystyle \frac{s}{n}}-{\displaystyle \frac{ps}{q}}}{\parallel g\parallel }_{L^{{\displaystyle \frac{q}{ps}}}\left(\mathrm{\Omega }\right)}{\left[u\right]}_{W^{s,p}\left({\mathbb{R}}^n\right)},\hfill \end{array}$$

which together with the Hölder inequality imply that

$$\begin{array}{ccc}\hfill {\left|\mathrm{\Omega }\right|}^{-{\displaystyle \frac{1}{{\tilde{q}}^{*}}}}{\parallel u\parallel }_{L^{{\tilde{q}}^{*}}\left(\mathrm{\Omega }\right)}& \le & {\left|\mathrm{\Omega }\right|}^{{\displaystyle \frac{s}{n}}-{\displaystyle \frac{1}{p}}}{\parallel u\parallel }_{L^{{\displaystyle \frac{np}{n-sp}}}}\hfill \\ & \le & {C(n,s,p)\left|\mathrm{\Omega }\right|}^{{\displaystyle \frac{s}{n}}-{\displaystyle \frac{1}{p}}}{\left[u\right]}_{W^{s,p}\left({\mathbb{R}}^n\right)}\hfill \\ & \le & C(n,s,p){\left[{\mathrm{\Omega }|}^{{\displaystyle \frac{ps}{n}}}{\left({⨏}_{\mathrm{\Omega }}{\left|g\right|}^{{\displaystyle \frac{q}{ps}}}\right)}^{{\displaystyle \frac{ps}{q}}}\right]}^{{\displaystyle \frac{1}{p-1}}}.\hfill \end{array}$$

Combining this with (16) leads to

$${\parallel u_{+}\parallel }_{L^{\infty }\left(\mathrm{\Omega }\right)}\le C(n,s,p,q){\left[{\mathrm{\Omega }|}^{{\displaystyle \frac{ps}{n}}}{\left({⨏}_{\mathrm{\Omega }}{\left|g\right|}^{{\displaystyle \frac{q}{ps}}}\right)}^{{\displaystyle \frac{ps}{q}}}\right]}^{{\displaystyle \frac{1}{p-1}}}.$$

Then, by repeating the above process to $-u$, we obtain the final estimate for u. This completes the proof of Theorem 3. □

5. Conclusions

In this paper, we establish the global boundedness of weak solutions to fractional nonlocal equations using the fractional Moser iteration argument and some other ideas. By choosing the test function appropriately, we obtain a reversed Hölder inequatlity, i.e.,

$${\parallel u\parallel }_{L^{p_1}\left(\mathrm{\Omega }\right)}\le C{\parallel u\parallel }_{L^{p_2}\left(\mathrm{\Omega }\right)},$$

for $p_1>p_2$. Then, by iteration, we obtain the global boundedness results of weak solutions for the fractional linear nonlocal equation with the right-hand side g and ${\displaystyle L_b^s}f$ in Theorem 1 and Theorem 2, respectively. The proof of Theorem 2 is a little more difficult than that of Theorem 1 due to the nonlocality of the operator ${\displaystyle L_b^s}$. Finally, we extend the boundedness result in Theorem 1 to the fractional p-Laplace equation, i.e., Theorem 3. However, the method used in the proof of Theorem 2 is difficult to generalize to a fractional p-Laplace equation because it depends deeply on the linearity of the equation. This will be studied in our next paper.