The Solvability of Fractional Elliptic Equation with the Hardy Potential

Abstract

In this paper, we study the existence and nonexistence of solutions to fractional elliptic equations with the Hardy potential where is a bounded Lipschitz domain with , is a fractional Laplace operator, , , is a positive number, , is a parameter depending on , , and is the sharp constant of the Hardy–Sobolev inequality.

1. Introduction

In this paper, we consider the solvability of the following fractional elliptic problem:where is a bounded Lipschitz domain with , , , , , and is the sharp constant of the Hardy–Sobolev inequality; the fractional Laplace operator is defined bywhere P.V. stands for the Cauchy principal value and constant is a constant.

Recently, much attention has been devoted to the study of fractional Laplacian equations. One of the reasons comes from the fact that the fractional Laplacian arises in various areas and different applications, such as phase transitions, finance, stratified materials, flame propagation, ultrarelativistic limits of quantum mechanics, and water waves. For more details, see [1–6] and references therein.

For fractional elliptic problems with the Hardy potential, Abdellaoui et al. [7] obtained the existence and summability of solutions to a class of nonlocal elliptic problem:with and . They mainly considered the summability of solutions to (3) with and the existence and regularity of solutions to (3) with . Mi et al. [8] obtained the combined influence of the Hardy potential and lower order terms on the existence and regularity of solutions to the problem:

Barrios et al. [9] discussed the existence and multiplicity of solutions to the following fractional elliptic equation:where , ,and is a parameter depending on . They shown that problem (5) has at least one solution if and problem (5) has no solution if .

Recently, Shang et al. [10] studied the existence and multiplicity of positive solutions to the following problem:where , , , and . Some other results of fractional elliptic equations with the Hardy potential, see [7, 9, 11–14] and references therein.

The local version of quasilinear problem related to problem (8) has been considered by Boccardo et al. [15]. They analyzed the existence of nontrivial solutions to the following problem:where is a smooth bounded domain, , , is a Carathéodory function, and there exist constants and such that for any .

Motivated by the above works, the aim of this paper is to study the existence of solutions to problem (1) by the method of subsuper solutions and taking into advantage the combined effect of concave and convex nonlinearity.

We make the following assumptions: (F1) where is a parameter depending on . (F2) is Carathéodory function, and there exist constants and , such that, for any , (F3) The function (F4) Define for all , where is the first eigenvalue of in .

Now, we state our main result.

Theorem 1. Suppose hold. Then, there exists a positive constant , such that, for all , problem (1) has at least a nonnegative solution if , where is defined by (12).

Remark 1. In order to prove the above theorem, we study directly to the pseudodifferential operator, without the harmonic extension to an extra dimension by transforming the nonlocal problem into a local problem due to Caffarelli and Silvestre [16].

Remark 2. To establish the upper bound for (see (9)), we consider a radial solution with constant to the problem:We obtainwhereIn order to have homogeneity, we haveThus, we deduce that . Since , we conclude that . Note that the map is decreasing about , see [17, 18]. Hence, there is a unique element such that . Thus, we have , that is,which implies thatTherefore, we can construct a supersolution to problem (1) for , just modifying the found above. Thus, is the threshold for the existence to problem (1).

Now, we consider the nonexistence of solution to problem (1).

Theorem 2. Suppose hold. Then, problem (1) has no solution in if for some , , and , where and are defined by (12) and (13), respectively.

The following two examples also appeared in [15].

Remark 3. An example of function with , which satisfies conditions (10) and (11) for any , such that problem (1) has at least one positive solution. In this condition , by (52), we have . DefineIt is easy to see thatWe have to prove that is smaller than the minimum of . Therefore, we haveMoreover, by (10) and (13), we have for any ,Thus, for any , . Hence, problem (1) has no solution at least . Therefore, the result of the above theorem is more general than [9].

Remark 4. We consider the functionfor . We easily deduce that conditions (10) and (11) are fulfilled and . On the contrary,If , the function is monotone decreasing for . Then,Similarly, in this case, problem (1) has no solution provided

Remark 5. The function is nonincreasing. Hence, if , for some , the results of Theorem 1 will be true for any .

The paper is organized as follows. In Section 2, we present some definitions and preliminary tools, which will be used in the Proof of Theorems 1 and 2. The Proof of Theorems 1 and 2 are given in Section 3 and Section 4, respectively.

2. Preliminaries and Function Setting

In this section, we recall some known results for reader’s convenience.

Denote the spaceequipped with the norm

Let be an open subset of . Given and in the Schwartz class, the distribution in is defined as

We give some useful facts for the fractional Sobolev space.

Definition 1. Let , and define the fractional Sobolev space:We need to consider the space , which is defined aswith the normwhere . The pair yields a Hilibert space (see Lemma 7 in [19]).

We have to use the classical Sobolev theorem.

Theorem 3 (see [20], Theorem 6.5). Let , then there exists a positive constant , such that, for any measurable and compactly supported function , we havewhere is the so-called Sobolev critical exponent.

In this paper, we consider the existences of energy solution to problem (1) with the critical and subcritical cases.

Definition 2. We say that is an energy solution to problem (1), if for any ,

We also need to consider the weak solution to problem (1).

Definition 3. We say that is a weak solution to problem (1), if a.e. in , in ,and for all , ,where in and .

Definition 4. If satisfiesin the weak sense, we say that is a supersolution to problem (1).
If satisfiesin the weak sense, we say that is a subsolution to problem (1).

Now, we recall the comparison lemma.

Lemma 1 (see [9]). Let and be solutions, respectively, toThen, for all if and .

For the supercritical case, we need a prior regularity result, see [9], Lemma 2.2.

Lemma 2. Given , where . There exists a unique weak solution toin the sense thatfor all with in .

Moreover, , for some constant independent of . In addition, if

Then, .

3. The Existence Result

We are now ready to prove Theorem 1 by employing the idea contained in [9, 15], whose proof will be split into several steps.

Proof of Theorem 1.  Step 1: subsolution to problem (1). We first consider the eigenvalue problem: Note that the eigenfunction belongs to . Suppose , where is given in (12), by , for all , taking small enough, we have Therefore, for , Therefore, Thus, is a subsolution to problem (1). Next, we consider supersolution to problem (1) in the subcritical and supercritical case, respectively. Step 2: supersolution for subcritical and critical case: . We look for a supersolution of the form with and as real parameters and verify Since , we obtain By (49), we deduce that for the appropriate choice of . Let . Taking with , which is a suitable constant such that, for small enough, and by (10) we have where appears in (1). By (52), we obtain Thus, we have concluded that is a supersoution to (1) for . Moreover, by (48) and (50), we obtain that Define in is the weak solution to for and . We now check that this definition makes sense and are monotone and satisfy For , there is nothing to prove. Suppose the result is true up to order . Then, So is well-defined by (54) and Lemma 2. By the induction hypothesis, for , and . Then, by Lemma 1, we obtain . Similarly, for any , and . Then, a.e. in . We conclude that (56) holds. We can define in . Moreover, by (9), (54), and (56), Hence, up to a subsequence, we know that in . By monotony, the whole sequence weakly converges. Therefore, we can pass to the limit in (55) and conclude that is a minimal energy solution of (1). Step 3: supersolution for supercritical case: . If , where is given in (9). For constant , there exists a radial function such that Since , then Taking , where the constant is given by (54), we obtain Moreover by (62), satisfies (36). By Lemma 2, define to be the weak solutions of (55). Moreover, similarly, by the induction hypothesis, we can conclude that Note that and hold, for all , define as the solution of Set Then, . By (44), we obtain that We deduce from the comparison principle that in . On the contrary, by (11), the function is nondecreasing. Therefore, By the comparison principle we deduce that in . In particular, for all , is a nondecreasing sequence which is bounded. Therefore, monotone converges in to a weak nonnegative solution to (1) for . Therefore, for small enough, we have built a minimal solution in both subcritical and supercritical case. Let that is, we show that . Step 4: , for . We consider the following eigenvalue problem with the Hardy potential: Since , problem (70) is well defined. Taking as a test function in problem (1), we obtain that Since is a solution to (70), it follows that If . Taking as a test function in (1), where is solution to problem (44), we have Moreover, is also a classical solution (see Remark 2.1 in [21]). From (72), we immediately deduce that Since there exist structural positive constants and such that , for any . From (70) and (73), we obtain that . This implies that for . We complete the Proof of Theorem 1.

4. Nonexistence Result

In this section, we consider the nonexistence of solution to problem (1) in .

Proof of Theorem 2. Suppose that problem (1) has a solution under the conditions of Theorem 2. Then, there exists a constant such that , where is the first eigenfunction of , that is, satisfies (44).
Let , where is a small constant. Denote . Then, we haveFurthermore, for any , ; then, for small enough, we deduce thatThus, according to (75) and (76), we haveHence, it is possible to construct the subsolution and supersolution to the problem:with . However, this is impossible.

Data Availability

Data sharing not applicable to this article as no datasets were generated or analysed during the current study.

Conflicts of Interest

The authors declare that they have no conflicts of interest.

Acknowledgments

This research was partially supported by the National Natural Science Foundation of China (no. 11761059), Program for Yong Talent of State Ethnic Affairs Commission of China (no. XBMU-2019-AB-34), Fundamental Research Funds for the Central Universities (no. 31920200036), and First-Rate Discipline of Northwest Minzu University.