Existence of Three Positive Solutions for a Nonlocal Singular Dirichlet Boundary Problem

1 Introduction

In the present paper, we consider the following nonlocal singular problem:

where s∈(0,1), q∈(0,1), λ>0 and Ω is a smooth bounded domain in ℝn. We have the following assumptions on f∈C1⁢([0,∞)):

1. f ⁢ ( 0 ) > 0 .

2. lim u → ∞ ⁡ f ⁢ ( u ) u q + 1 = 0 .

3. u ↦ f ⁢ ( u ) is nondecreasing in ℝ+.

4. There exists a 0<σ1<σ2 such that f⁢(u)uq is nondecreasing on (σ1,σ2).

Note that from (f1) we have limu→0⁡f⁢(u)uq=∞.

The fractional Laplace operator (-Δ)s is defined by

where P.V. denotes the Cauchy principal value and

with Γ being the Gamma function. The fractional Laplacian is the infinitesimal generator of Lévy stable diffusion processes and arises in anomalous diffusion in plasma, population dynamics, geophysical fluid dynamics, flames propagation, chemical reactions in liquids and American options in finance; see [5, 16, 36] for instance. Fractional Sobolev spaces were introduced mainly in the framework of harmonic analysis in the middle part of the last century and are the natural setting to study weak solutions to problems involving the fractional Laplacian. In this regard, the paper of Caffarelli and Silvestre [8] on the harmonic extension problem brought to light the subject of nonlocal equations, and has subsequently motivated many works on equations and systems involving the fractional Laplacian (-Δ)s, s∈(0,1). We refer the reader respectively to [16, 34, 31] for an introduction and surveys about fractional Sobolev spaces and fractional elliptic problems.

In the local case, i.e. s=1, the study of elliptic singular problems starts mainly with the pioneering work of Crandal, Rabinowitz and Tartar [11]. This seminal work inspired a huge list of articles where authors have investigated many different issues (existence/nonexistence, uniqueness/multiplicity, regularity of solutions, etc.) about singular problems in the local and more recently in the nonlocal set up. We cite here some related works with no intent to furnish an exhaustive list. The multiplicity of solutions for singular problems with critical nonlinearity has been studied in [25, 28, 27], while the exponential critical nonlinearity case has been dealt with in [13]. Semilinear elliptic and singular problems with convection term were first studied in [19], whereas [15] brought existence results to elliptic equations involving a singular absorption term. Hölder regularity of weak solutions is discussed in [24]. We refer to the surveys [20, 26] for further details on singular elliptic equations in the local setting. In the nonlocal case, singular problems with critical nonlinearity have been studied in [7, 22, 32]. Recently, Adimurthi, Giacomoni and Santra [2] studied the following nonlocal singular problem:

where δ,λ>0, K:Ω→ℝ+ is a Hölder continuous function in Ω that behaves like dist⁢(x,∂⁡Ω)-β, β∈[0,2⁢s), and f is a positive real-valued C2 function. They established existence, regularity and bifurcation results using the framework of weighted spaces.

Recently, Düzgün and Iannizzotto [17] have established the existence of three non-zero solutions for a Dirichlet-type boundary value problem involving the fractional Laplacian. But the study of three solutions for singular nonlocal problems was completely open till now. Our work brings new results in this regard. We use the method of sub- and supersolutions combined with a fixed point theorem due to Amann to achieve the objective. For the construction of the barrier functions, we have taken some ideas from [29].

The salient feature of this work is the presence of the singular term u-q which is a primary hindrance in making the operator monotone. We slightly transform the problem to a new one and show that the operator associated with it becomes monotone increasing and compact. This idea had been formerly used by Dhanya, Ko and Shivaji [14] in the local case. But here itself we remark that their approach can not be directly applied to problem (Pλ) due to the presence of the nonlocal operator (-Δ)s instead of Δ. Most substantially, [14, Theorem 3.6] can not be adapted here due to the lack of an explicit form of (-Δ)s⁢δs⁢(x), where δ⁢(x)=dist⁢(x,∂⁡Ω) denotes the distance function up to the boundary. To overcome this difficulty, we construct a subsolution v satisfying

where c>0 is constant and Ωr⊂Ω. To obtain this, we separately study, by bifurcation arguments, a nonlocal infinite semipositone problem (Iθ) in Section 6. This leads naturally to a solution of the required problem. In the local setting, we refer the readers to [33, 30, 23] concerning infinite semipositone problems. But we indicate that a “nonlocal” infinite semipositone problem has not been studied in the past. So our results are completely new in this regard. The main result of our paper is accomplished by using a well-known critical point theorem by Amann [3]. Last but not the least, we additionally prove the uniqueness of solutions to (Pλ) when λ becomes sufficiently large under appropriate conditions of f. This result is motivated by the paper [9] of Castro, Eunkyung and Shivaji. Nevertheless, we point out that their approach can not be exactly applied here due to the presence of the nonlocal operator (-Δ)s. We still succeed to obtain the result by an appropriate application of Hardy’s inequality for the fractional Laplacian (see Section 5). Now we state the main results of our paper as follows.

The outline of this paper is as follows: In Section 2, we give some useful preliminaries about the main equation in (Pλ). In Section 3, we construct the sub- and supersolutions used to apply the fixed point theorem of Amann. In Section 4, we prove the main result of our paper: Theorem 1.2. In Section 5, we prove our main uniqueness result: Theorem 1.4. Finally, in Section 6 we investigate the fractional and singular semipositone problem (Iθ) used crucially for proving the strong increasingness of the operator T in Section 4.

2 Preliminaries

We start with defining the function spaces. Given any ϕ∈C0⁢(Ω¯) such that ϕ>0 in Ω, we define

with the usual norm ∥uϕ∥L∞⁢(Ω) and the associated positive cone. We define the following open convex subset of Cϕ⁢(Ω):

In particular, Cϕ+ contains all functions u∈C0⁢(Ω) with k1⁢ϕ≤u≤k2⁢ϕ in Ω for some k1,k2>0. We consider the following fractional Sobolev space:

equipped with the norm

We define the distance function as δ⁢(x):=dist⁢(x,∂⁡Ω), x∈Ω. Let ϕ1,s denote the first positive eigenfunction of (-Δ)s in H~s⁢(Ω) corresponding to its principal eigenvalue λ1,s such that ∥ϕ1,s∥L∞⁢(Ω)=1. We recall that ϕ1,,s∈Cs⁢(ℝn) and also ϕ1,s∈Cδs+⁢(Ω) (see for instance [35, Proposition 1.1 and Theorem 1.2]).

3 Sub- and Supersolutions of (Pλ)

In this section, we show the existence of two pairs of sub-supersolutions (ζ1,ϑ1) and (ζ2,ϑ2) such that ζ1≤ζ2≤ϑ1, ζ1≤ϑ2≤ϑ1 and ζ2⩽̸ϑ2. Moreover, it holds that ζ2,ϑ2 are strict sub- and supersolutions of (Pλ). Let w denote the unique solution of the problem

Then from the proof of [2, Theorem 1.1] we know that w∈H~s⁢(Ω)∩Cϕ1,s+⁢(Ω) and w∈Cs⁢(ℝn). We construct our supersolution ϑ1 first. Since (f2) holds, we get

This implies that if we choose a constant Mλ≫1 sufficiently large such that

then ϑ1=Mλ⁢w∈H~s⁢(Ω)∩Cϕ1,s+⁢(Ω) forms a supersolution of (Pλ). Indeed, using the nondecreasing nature of f, we get

Now since limu→0⁡f⁢(u)uq=∞, we can choose mλ>0 sufficiently small so that

Now we define ζ1=mλ⁢ϕ1,s∈H~s⁢(Ω)∩Cϕ1,s+⁢(Ω), and it is easy to see that

Therefore, ζ1 is a subsolution of (Pλ). It is not hard to see that we always choose mλ small enough so that ζ1≤ϑ1. This completes our construction of the first pair of sub-supersolution.

Our next step is to construct the second pair of sub-supersolution of (Pλ). We first construct our positive supersolution ϑ2 such that ∥ϑ2∥L∞⁢(Ω)=σ1 (see (f4)). Let us define

and assume that

Then using the nondecreasing nature of f, we find that it satisfies

Now we construct our second positive supersolution of (Pλ), which is one of the crucial part of our paper. For this, we let σ∈(0,σ1] be such that f*⁢(σ)=min0<x≤σ⁡f⁢(x)xq, and also define h∈C⁢([0,∞)) such that

so that h is a nondecreasing function on (0,σ1] and h⁢(u)≤f⁢(u)uq for all u≥0. With this definition of h, we consider the nonsingular problem

Let Gs⁢(x,y) denote the Green function associated to (-Δ)s with homogeneous Dirichlet boundary conditions in Ω. Then we have

Let BR^⁢(0) denote the ball (centered at 0, where without loss of generality we assume that 0∈Ω) of largest radius R^ that is inscribed in Ω, and also let R<R^. Suppose K:L2⁢(Ω)→L2⁢(Ω) to be the linear map defined by

Let χR:Ω→ℝ be the characteristic function defined by

Then from [10, Theorem 1.1], for each (x,y)∈Ω×Ω we have that

Therefore, there exists a constant C1>0 depending on R such that

uniformly in x∈Ω. Let M2=(minx∈Ω⁡K⁢(χR)⁢(x))-1>0 and a∈(σ1,σ2]. Also we define v=a⁢χR in Ω. Then if v1 denotes a solution to the problem

then for each x∈Ω we get

where we used the fact that h is nondecreasing and χR≤1 in Ω. Therefore, v1≤σ2 in BR⁢(0) if

Claim: v1≥v in Ω for a certain range of λ. Let x∈Ω∖BR⁢(0). Since Gs⁢(x,y)≥0 for all x,y∈Ω and h⁢(u)>0 for all u≥0, we get v⁢(x)=0≤v1⁢(x). Now let x∈BR⁢(0). Then

So if we assume that λ≥M2⁢ah⁢(a), then by the definition of v we get

Hence we finally get that 0≤v≤v1≤σ2 in Ω if

Since we assumed h to be nondecreasing in (0,σ2) (because of (f4)), we get h⁢(v⁢(x))≤h⁢(v1⁢(x)) for x∈Ω. So v1 weakly satisfies the problem

Since h⁢(v)∈L∞⁢(Ω), using [35, Theorem 1.2], we get that v1∈Cs⁢(ℝn). This gives us that v1 forms a subsolution of (Qλ). Moreover, (3.3) and the strong maximum principle imply that v1>0 in Ω. Therefore, using the fact that h⁢(u)≤f⁢(u)uq for all u≥0 implies that ζ2=v1 forms a positive subsolution of (Pλ).

Therefore, we got that if

then we obtain a positive subsolution ζ2 and a positive supersolution ϑ2 of (Pλ) such that ζ2≰ϑ2. Indeed, ∥ϑ2∥L∞⁢(Ω)=σ1 and ∥ζ2∥L∞⁢(Ω)≥a>σ1.

4 Proof of the Main Result

In this section, we prove our main result after establishing some necessary results. We begin by noticing that our problem (Pλ) can be rewritten as

where f~⁢(u)=λ⁢(f⁢(u)-f⁢(0)uq). We fix that λ∈[λ1,λ2]. Since f∈C1⁢([0,∞)), by the mean value theorem we get f~⁢(u)=λ⁢f′⁢(v)⁢u1-q for some v∈(0,u). Also this implies f~⁢(0)=0 because limt→0⁡|f′⁢(t)|<∞ and q∈(0,1). Therefore, f~ can be considered as a continuous function on [0,∞) such that f~⁢(0)=0. We assume also the following:

1. There exists a constant k~>0 such that f~⁢(t)+k~⁢t is increasing in [0,∞).

We remark that for any φ∈H~s⁢(Ω) and z⁢(x)≥k1⁢δs⁢(x) in Ω, Hardy’s inequality gives that

where k1,C>0 are constants. So now following the arguments of [22, Lemma 3.2], we can prove that a weak solution z∈H~s⁢(Ω) of (P’λ) satisfies (4.1) for every φ∈H~s⁢(Ω) if z≥k1⁢δs⁢(x) in Ω.

We extend the functions f and f~ naturally as f⁢(t)=f⁢(0) and f~⁢(t)=f~⁢(0) for t≤0. Because of the assumption in (h1), without loss of generality we can assume that f~ is increasing in ℝ+. Now we define the map T:C0⁢(Ω¯)→Cϕ1,s⁢(Ω) by T⁢(u)=z if and only if z is a weak solution of

By saying that z∈H~s⁢(Ω) is a weak solution of (Sλ) we mean that it satisfies

for all φ∈Cc∞⁢(Ω). But repeating the arguments from above for (P’λ), we can show that z∈H~s⁢(Ω) of (Sλ) satisfies (4.2) for every φ∈H~s⁢(Ω) if z≥k1⁢δs⁢(x) in Ω.