Abstract

The aim of this paper is to establish a weak comparison principle for a class fractional -Laplacian equation with weight. The nonlinear term is a Carathéodory function which is possibly unbounded both at the origin and at infinity and such that decreases with respect to for a.e. .

1. Introduction and Main Results

In this paper, we study a weak comparison principle for the following fractional -Laplacian problem

where is a smooth bounded domain of containing the origin, , , , and is a general Carathéodory function, which is possibly unbounded both at the origin and at infinity and such that decreases with respect to for a.e. . The weighted fractional -Laplacian is the pseudodifferential operator defined as here P.V. denotes the principal value of the integral.

The simplest model is where and , and and are nonnegative functions.

The interest on the nonlocal operators continues to grow in recent years since such problems arise in various fields. The fractional -Laplacian , on one hand, is an extension of the local operator . Note that, for this type of operator, the Caffarelli-Kohn-Nirenberg inequality plays an important role, see [1–10]. On the other hand,, which appears in a natural way when dealing with the fractional Laplace problem with Hardy potential. More precisely, let be a solution to the following problem where , is the Hardy constant, and

Then, according to ground state representation [11, 12], satisfies where

Fractional Laplace operator can be defined using Fourier analysis, functional calculus, singular integrals, or Lévy processes. Thus, rich mathematical concepts allow in general rich properties. For some abstract definitions and tools of fractional Laplace operator, see [13]. For more recent results of fractional Laplace elliptic problem, see [14–16] and the reference therein.

There are many works on the study of fractional -Laplacian equations. Canino et al. [17] investigated the existence and uniqueness of solutions to

When , Mukherjee and Sreenadh [18] used variational methods to show the existence and multiplicity of positive solution problem (1) with critical growth and singular nonlinearity. Abdellaoui et al. [19] prove the existence of a weak solution to (1) under some hypotheses on .

The main idea of this paper comes from the seminal papers [20, 21]. In [21], Brezis and Oswald have shown the existence and uniqueness of a solution to a Laplace elliptic problem. Recently, Durastanti and Oliva [20] obtained the existence and uniqueness of positive solutions of an elliptic boundary value problem modeled by

The main result of this paper is the following weak comparison principle.

Theorem 1. Assume that are nonnegative functions such that either or is decreasing with respect to , and for almost every , for almost every and for all . Suppose that and are weak solutions to the problem Then, almost everywhere in .

Consequently, we obtain the uniqueness of the solution to problem (1).

Corollary 2. Assume that is a nonnegative function such that is decreasing with respect to for almost every . Then, there exists at most one weak solution to problem (1).

The paper is organized as follows. In Section 2, we present some definitions and preliminary tools, which will be used in the proof of Theorem 1. The proof of Theorem 1 and Corollary 2 be given in Section 3.

2. Preparations

First of all, we give the definition of truncation function.

Definition 3. For every and , define Let , the weighted fractional Sobolev space is defined by endowed with the norm where The space is the completion of with respect to the above norm.

Definition 4. A positive function is a weak solution to problem (1) if and for any , where It is worth pointing out that the formulation (16) can be extended for -test functions by standard argument.

The following fractional Picone inequality appears in Proposition 2.15 of [22] with . A slight change in this proof actually shows that the fractional Picone inequality also holds for fractional -Laplacian .

Lemma 5. Consider with . Assume that is a positive bounded Radon measure in . Then

Proof. Take and , where is a constant. It is easy to show that . By similar arguments as in the proof of Proposition 2.10 of [22], we can easily prove the following equation where we use discrete Picone inequality. For more details, see Proposition 4.2 of [23]. Letting and in (19) leads to (18).

3. Proof of Main Theorem

Proof. Suppose that is decreasing with respect to for almost every . A slight change will be needed provided is decreasing with respect to for almost every but no essential difference.
For fixed and , define where is defined by (12) and is the positive part of the function .
In the following proof, I show that , which leads to almost everywhere in .
Choosing and as test functions in equations of and , respectively, we find Subtracting the two equations (21) and (22), we obtain Decompose as , where Therefore, here .
By the definition of and , we get Now, rewrite as where For simplicity of notation, denote Obviously, Now consider . By (26), we get where Note that and for . Thus, . By the monotonicity of the function , we find , which implies that For , we have since for .
For , we find where Recalling that since and for . This fact, together with for , yields . Consequently Now, we consider . here we use the fractional Picone inequality; see Lemma 5.
For , we have Now, we consider the first term of the right-hand side of (39). Suppose that , Suppose that , the above inequation holds also since for .
For the second term of the right-hand side of (39), by a similar argument, we get Thus, taking into account (40) and (41), we derive that For , it is easily seen that For , repeating the previous argument of leads to For , we have where satisfies .
For the right-hand side of (23), according to (10), we have Using the monotonicity of , we know that since for , where appears in (45). Thus, taking into account (45)–(47), we obtain, for small enough , This fact, together with (23), (30), (33), (34), (37), (38), (42)–(44), leads to which implies that for , that is for any . This completes the proof of Theorem 1.

The proof of Corollary 2 is immediate, which is omitted.

Data Availability

No data were used in this study.

Conflicts of Interest

The authors declare no conflict of interest.