Electronic Journal of Differential Equations, Vol. 2020 (2020), No. 81, pp. 1–31.

ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu

EXISTENCE OF POSITIVE SOLUTIONS FOR FRACTIONAL

LAPLACIAN EQUATIONS: THEORY AND NUMERICAL

EXPERIMENTS

MAYA CHHETRI, PETR GIRG, ELLIOTT HOLLIFIELD

Abstract. We consider a class of nonlinear fractional Laplacian problems
satisfying the homogeneous Dirichlet condition on the exterior of a bounded

domain. We prove the existence of positive weak solution for classes of sublin-

ear nonlinearities including logistic type. A method of sub- and supersolution,
without monotone iteration, is established to prove our existence results. We

also provide numerical bifurcation diagrams and the profile of positive solu-

tions, corresponding to the theoretical results using the finite element method
in one dimension.

1. Introduction

We investigate the existence of positive solutions for a class of nonlocal problems
of the form

(−∆)su = λf(x, u) in Ω;

u = 0 in RN \ Ω ,
(1.1)

where λ > 0 is a bifurcation parameter and Ω ⊂ RN is a bounded domain with
C1,1 boundary ∂Ω for N ≥ 2 (or bounded open interval if N = 1). For a fixed
s ∈ (0, 1), (−∆)s is the fractional Laplacian operator defined by

(−∆)su(x) := CN,s P.V.

∫
RN

u(x)− u(y)

|x− y|N+2s
dy ,

where CN,s := s22sπ−
N
2 Γ(N+2s

2 )/Γ(1 − s) is a positive normalizing constant with
Γ as the usual gamma function, and P.V. stands for the Cauchy principal value of
the singular integral. The nonlinearity f : Ω × R → R is a Carathéodory function
(that is, f(·, t) is measurable for each t ∈ R and f(x, ·) is continuous for a.e. x ∈ Ω).

We seek to establish the existence of a weak solution (to be defined) of the nonlo-
cal problem (1.1) under suitable conditions on the nonlinearity f using the method
of sub- and supersolution. In [1] and [16], sub- and supersolution methods for
fractional Laplacian equations were established for L1-very weak solutions, which
required a rather complicated structure of the space of test functions. Therefore,
we first present a sub- and supersolution result in a framework that is analogous

2010 Mathematics Subject Classification. 35J60, 35J61, 35R11.
Key words and phrases. Fractional Laplacian; sub- and supersolution; sublinear;

logistic equation; finite element method.
c©2020 Texas State University.

Submitted June 28, 2019. Published July 28, 2020.

1



2 M. CHHETRI, P. GIRG, E. HOLLIFIELD EJDE-2020/81

to the weak solution framework for the Laplacian case. An advantage of our ap-
proach is in the possibility of employing principal eigenfunction corresponding to
the variational principal eigenvalue of (−∆)s in the construction of positive sub-
and supersolutions. See also [6], where a sub- and supersolution method was proved
for the fractional p(x)-Laplacian equation for weak solutions, but with the addi-
tional requirement that the nonlinearity is monotone with respect to the solution
variable.

Next, using the method of sub- and supersolutions we establish the existence of
positive weak solutions to (1.1) for classes of nonlinearities: sublinear at infinity,
weighted logistic problems, and logistic problems with constant yield harvesting.

We also discuss the existence and properties of a principal eigenvalue and the
corresponding eigenfunction for a weighted fractional eigenvalue problem. This
eigenfunction plays a crucial role in the construction of some positive sub- and
supersolutions.

Finally, using a finite element method, we present numerical bifurcation diagrams
and illustrate the typical profile of positive solutions for examples of nonlinearity f
satisfying the hypotheses of our theoretical results. Using these numerical experi-
ments, we formulate some relevant conjectures.

The fractional Laplacian operator (−∆)s, is associated with superdiffusion driven
by Lévy flights and appears to be of interest from the application point of view,
see [10, 23, 24, 27, 31] and references therein. More specifically, such operators are
associated with the efficient foraging strategy of living organisms, see e.g. [3, 21]
and references therein.

In Section 2, we discuss function spaces, define terminologies, and state our
main results. In particular, we first state a general sub- and supersolution result,
Theorem 2.3. Then, we state existence results for classes of nonlinearity f that
are sublinear at infinity. In particular, Theorem 2.5 deals with the positone case
f(0) > 0 and Theorem 2.6 deals with the case f(0) = 0. Next, we state an existence
result for a weighted logistic problem in Theorem 2.8, and a logistic problem with
constant yield harvesting (semipositone, that is, f(0) < 0) in Theorem 2.9. In Sec-
tion 3, we discuss a weighted fractional eigenvalue problem and a fractional linear
problem necessary in the construction of sub- and supersolutions in later sections.
In Section 4, we prove Theorem 2.3 using the Schauder fixed point theorem. In
Section 5, we prove Theorem 2.5 and Theorem 2.6. In Section 6, we prove Theo-
rem 2.8 and Theorem 2.9. In Section 7, we present numerical bifurcation diagrams
and profiles of positive solutions obtained using the finite element method in one
dimension. In the Appendix, we show that the norms generated by inner products
(2.1) and (2.2) are equivalent in the fractional Sobolev space Hs

0(Ω) (defined below)
in one dimension as well.

2. Preliminaries and statement of results

We first discuss some properties of fractional Sobolev spaces, see [15, 24] for
more details. Let

Hs(RN ) :=
{
w ∈ L2(RN ) : ‖w‖Hs(RN ) < +∞

}
,

where ‖w‖Hs(RN ) :=
(
‖w‖2L2(RN ) + [w]2Hs(RN )

)1/2
and

[w]Hs(RN ) :=
(∫

RN

∫
RN

|w(x)− w(y)|2

|x− y|N+2s
dxdy

)1/2



EJDE-2020/81 FRACTIONAL LAPLACIAN EQUATIONS 3

is the Gagliardo seminorm of w. Then, the fractional Sobolev space Hs(RN ) is a
Hilbert space with respect to the inner product

〈v, w〉Hs(RN ) :=

∫
RN

vw dx+

∫
RN

∫
RN

[v(x)− v(y)][w(x)− w(y)]

|x− y|N+2s
dxdy . (2.1)

Further, the fractional Sobolev space

Hs
0(Ω) :=

{
w ∈ Hs(RN ) : w ≡ 0 a.e. RN \ Ω

}
is also a Hilbert space with respect to the inner product

〈v, w〉Hs0 (Ω) :=

∫
RN

∫
RN

[v(x)− v(y)][w(x)− w(y)]

|x− y|N+2s
dx dy . (2.2)

The norms generated by (2.1) and (2.2) are equivalent in Hs
0(Ω). This fact follows

from [24, Lemmas 1.28 and 1.29] for N ≥ 2 and Lemma 8.1 for N = 1 with
Ω = (0, 1) ⊂ R. We utilize this important fact in our analysis.

To simplify notation, for ψ, φ ∈ Hs(RN ), we define

E(ψ, φ) :=
CN,s

2

∫
RN

∫
RN

[ψ(x)− ψ(y)][φ(x)− φ(y)]

|x− y|N+2s
dxdy . (2.3)

Definition 2.1. We say that a function u ∈ Hs
0(Ω) is a weak solution of (1.1) if

for all φ ∈ Hs
0(Ω), it holds

E(u, φ) = λ

∫
Ω

f(x, u)φ(x) dx . (2.4)

Definition 2.2. A function u ∈ Hs(RN ) is called a weak supersolution of (1.1) if,
for all φ ∈ Hs

0(Ω) such that φ ≥ 0 a.e. Ω, the following inequality holds

E(u, φ) ≥ λ
∫

Ω

f(x, u(x))φ(x) dx , (2.5)

u ≥ 0 a.e. in RN \ Ω . (2.6)

A function u ∈ Hs(RN ) is called a weak subsolution of (1.1) if the inequalities are
reversed in (2.5) and (2.6).

Now we state a general sub- and supersolution result without monotonicity as-
sumption for the problem

(−∆)su = g(x, u) in Ω;

u = 0 in RN \ Ω,
(2.7)

where g : Ω × R → R is a Carathéodory function satisfying the following assump-
tions:

(H1) for all r > 0, there is ar ∈ L∞(Ω) such that |g(x, t)| ≤ ar(x) for all |t| ≤ r
a.e. x ∈ Ω;

(H2) for all r > 0, there is a continuous nondecreasing function br with br(0) = 0
such that |g(x, t1)− g(x, t2)| ≤ br(|t1 − t2|) for all |t1|, |t2| ≤ r a.e. x ∈ Ω.

Theorem 2.3. Assume (H1) and (H2) hold. Let u and u ∈ Hs(RN ) ∩ L∞(Ω) be
weak subsolution and weak supersolution, respectively of (2.7) satisfying u ≤ u a.e.
in Ω. Then, there exists a weak solution u to (2.7) satisfying u ≤ u ≤ u a.e. in Ω.



4 M. CHHETRI, P. GIRG, E. HOLLIFIELD EJDE-2020/81

Remark 2.4. The hypotheses of Theorem 2.3 are satisfied by a function of the form
g(x, t) = k(x)g̃(t), where k ∈ L∞(Ω) and g̃ : R → R is Hölder continuous. Indeed,
clearly g(x, t) is a Carathéodory function. For any r > 0 and for all |t| ≤ r, we
have |g(x, t)| ≤ ‖k‖L∞(Ω) max|t|≤r |g̃(t)| and hence (H1) is satisfied. By the Hölder
continuity of g̃, |g(x, t1)−g(x, t2)| ≤ A‖k‖L∞(Ω)|t1−t2|η for all |t1|, |t2| ≤ r for some
η ∈ (0, 1) and A > 0. Then, (H2) is satisfied with br(|t1−t2|) := A‖k‖L∞(Ω)|t1−t2|η.

We employ Theorem 2.3 to discuss the existence of positive weak solutions of
(1.1) for classes of nonlinearities f . First, we consider the case f(x, t) = f(t)
satisfying sublinear condition at infinity

lim
t→+∞

f(t)

t
= 0 . (2.8)

Our first result deals with the case when f > 0.

Theorem 2.5. Suppose f : [0,∞)→ (0,+∞) is a Hölder continuous function and
(2.8) is satisfied. Then, (1.1) has a positive weak solution for each λ > 0.

To state our second result, let λ1 be the principal eigenvalue of the eigenvalue
problem (3.3), with q ≡ 1, corresponding to the fractional Laplacian operator
(−∆)s. Then, we prove the following existence result for the case f(0) = 0.

Theorem 2.6. Suppose f : [0,+∞)→ [0,+∞) is a C1 function such that f(0) = 0,
f ′(0) > 0 with f(t) > 0 for all t > 0, and (2.8) is satisfied. Then, (1.1) has a
positive weak solution for any λ > λ1

f ′(0) .

Remark 2.7. An example satisfying the hypotheses of Theorem 2.5 is the reaction

term f(t) = e
κt
κ+t for t ≥ 0 with κ > 0, referred in the literature as perturbed Gelfand

problem when considered with Laplacian operator, see [7, Chap. 2]. In Section 7,
Figures 1-3 give numerical bifurcation diagrams corresponding to this nonlinearity
f depending on the value of κ, illustrating the result in Theorem 2.5. A simple
example satisfying the hypotheses of Theorem 2.6 is f(t) = 3(1 + t)1/3 − 3 for
t ≥ 0. In Section 7, Figure 4 gives the numerical bifurcation diagrams illustrating
the result in Theorem 2.6.

Finally, we discuss existence of positive weak solution for two logistic type prob-
lems. First, let q ∈ L∞(Ω) be such that 0 ≤ q ≤ 1 a.e. in Ω and q(x) > 1

2 on a set
of positive measure. By λ1,q, we denote the principal eigenvalue of the weighted
eigenvalue problem (3.3) with weight q. Then, we prove the following result.

Theorem 2.8. The fractional logistic problem

(−∆)su = λu(q(x)− u) in Ω;

u = 0 in RN \ Ω,
(2.9)

has a positive weak solution for any λ > λ1,q.

Next, let λ1 denote the first eigenvalue of (3.3) with q ≡ 1. Then, we prove the
following existence result for the logistic problem with constant yield harvesting.

Theorem 2.9. For any λ > λ1, there exists a∗ = a∗(λ) > 0 such that the logistic
problem with constant yield harvesting

(−∆)su = λ[u(1− u)− a] in Ω;

u = 0 in RN \ Ω,
(2.10)

has a positive weak solution for a ∈ (0, a∗).



EJDE-2020/81 FRACTIONAL LAPLACIAN EQUATIONS 5

Remark 2.10. For derivation of the time dependent fractional logistic model ut+
(−∆)su = λu(1 − u) with u = u(x, t) and (x, t) ∈ R2, for a simple two particle
reaction scheme, see [10]. Theorem 2.8 complements the existence result obtained
in [12], where existence of nonnegative solution for logistic growth problem was
established using energy minimization. Theorem 2.9 is analogous to the existence
result obtained in [26] for the Laplacian case.

In Section 7, the numerical bifurcation diagrams given in Figures 7-9, illustrate
the result of Theorem 2.8 and demonstrate the influence of q(x) and s ∈ (0, 1)
on the shape of positive solution. In Figure 6, we give the numerical bifurcation
diagrams illustrating the result of Theorem 2.9 for a = 0.05. In this case, we observe
numerically that the bifurcation diagrams contain positive as well as sign-changing
solutions.

In the proof of Theorem 2.8, the difficulty is due to the presence of the weight
function q(x). In the proof Theorem 2.9, the harvesting term “a” poses a challenge
in the investigation of positive solutions. Combining the methods used in the proofs
of Theorem 2.8 and Theorem 2.9, one can prove an existence result if f(x, u) =
λ[u(q(x)− u)− a] with q as in Theorem 2.8.

3. Auxiliary problems

To prove Theorems 2.5-2.9, we utilize the positive weak solutions of the following
auxiliary problems in the construction of weak sub- and supersolutions.

First, consider the linear problem

(−∆)se = 1 in Ω;

e = 0 in RN \ Ω .
(3.1)

Then, there exists a unique weak solution e ∈ Hs
0(Ω) ⊂ Hs(RN ) of (3.1) such that

e > 0 a.e. in Ω, see [20, Thm. 12] for N ≥ 2 > 2s, and for N = 1 the explicit
formula of the solution is given in [29, eqn. (1.4)]. Moreover, it follows from [28,
Lem. 7.3] and [29, Thm. 1.2] that there exist c1, c2 > 0 such that

c1δ
s(x) ≤ e(x) ≤ c2δs(x) a.e. in Ω , (3.2)

where δ(x) is the distance function to the boundary ∂Ω. The function e plays a
crucial role in the construction of both sub- and supersolutions in Section 5.

Second, we consider the following weighted fractional eigenvalue problem

(−∆)sϕ = λq(x)ϕ in Ω;

ϕ = 0 in RN \ Ω ,
(3.3)

where q ∈ L∞(Ω) is such that q ≥ 0 a.e. in Ω and positive on a set of positive
measure. The following results hold true by using arguments similar to the case
q ≡ 1, cf. [24, Prop 3.1 & Cor. 4.8]. We outline the proof for completeness.

Proposition 3.1. Let s ∈ (0, 1) be fixed and Ω ⊂ RN be an open, bounded set.
Then, the following holds.

(a) There exists a principle eigenvalue λ1,q > 0 of (3.3) that can be character-
ized as

λ1,q = inf
φ∈Hs0 (Ω)\{0}

E(ϕ1,q, ϕ1,q)∫
Ω
q(x)|φ(x)|2 dx

. (3.4)



6 M. CHHETRI, P. GIRG, E. HOLLIFIELD EJDE-2020/81

(b) There exists a nonnegative eigenfunction ϕ1,q ∈ Hs
0(Ω) corresponding to

λ1,q, attaining the minimum in (3.4), that is,

λ1,q =
E(ϕ1,q, ϕ1,q)∫

Ω
q(x)|ϕ1,q(x)|2 dx

. (3.5)

Moreover, ϕ1,q satisfies

E(ϕ1,q, φ) = λ1,q

∫
Ω

q(x)ϕ1,q(x)φ(x) dx (3.6)

for every φ ∈ Hs
0(Ω).

(c) λ1,q is simple, that is, if ψ ∈ Hs
0(Ω) is a solution of the equation

E(ψ, φ) = λ1,q

∫
Ω

q(x)ψ(x)φ(x) dx

for every φ ∈ Hs
0(Ω), then ψ = kϕ1,q for some k ∈ R.

(d) If Ω is C1,1 for N ≥ 2 (or bounded open interval if N = 1), then there exist
positive constants c̃1(q), c̃2(q) such that

0 < c̃1(q)δs(x) ≤ ϕ1,q(x) ≤ c̃2(q)δs(x) a.e. in Ω . (3.7)

(e) If Ω is C1,1 for N ≥ 2 (or bounded open interval if N = 1), then

λ1,q = inf
φ∈Hs0(Ω)

φ≥δs a.e. in Ω

E(φ, φ)∫
Ω
q(x)|φ(x)|2 dx

. (3.8)

Proof. For N ≥ 2, parts (a)–(c) can be obtained by repeating the argument of
[24, Prop 3.1] with L2(Ω) norm replaced with weighted L2 norm

∫
Ω
q(x)|φ(x)|2 dx

in constructing the principal eigenvalue λ1,q as Rayleigh quotient given by (3.4).
For N = 1, these follow from the fact that our definition of Hs

0(Ω), via Hs(RN ),
allows us to prove the compact embedding Hs

0(Ω) ↪→ L2(Ω) without considering
an extension domain (cf. proof of [15, Thm. 7.1]). Then, q ∈ L∞(Ω) gives con-
tinuous embedding L2(Ω) ↪→ L2((Ω); q), and hence the principal eigenvalue can be
constructed as the Rayleigh quotient given in (3.4).

For part (d), the C1,1 smoothness assumption on ∂Ω is used to establish the
inequalities of (3.7). In particular, the arguments used in establishing the left
inequality in [28, Lem 7.3] and the right inequality in [29, Thm. 1.2] apply in this
case as well, which are independent of dimension N .

For part (e), clearly, ≤ holds in (3.8). Using the definition of the infimum and
using the fact that ϕ1,q ≥ δs a.e. in Ω (after suitable scaling of ϕ1,q) due to (3.7),
we find

inf
φ∈Hs0(Ω)

φ≥δs a.e. in Ω

E(φ, φ)∫
Ω
q(x)|φ(x)|2 dx

≤ E(ϕ1,q, ϕ1,q)∫
Ω
q(x)|ϕ1,q(x)|2 dx

= λ1,q ,

which establishes ≥ in (3.8), completing part (e). This completes the proof of
Proposition 3.1. �

Next, for k = 2, 3, . . ., we consider the weighted fractional eigenvalue problem

(−∆)sϕ = λγk(x)ϕ in Ω;

ϕ = 0 in RN \ Ω ,
(3.9)



EJDE-2020/81 FRACTIONAL LAPLACIAN EQUATIONS 7

where

γk(x) :=

{
0 if 0 ≤ q(x) < 1/k

q(x) if q(x) ≥ 1/k ,
(3.10)

for q ∈ L∞(Ω) with 0 ≤ q ≤ 1 a.e. in Ω and q(x) > 1/2 on a set of positive measure.
Then, for each k = 2, 3, . . ., the weighted fractional eigenvalue problem (3.9) has a
principal eigenvalue λ1,γk and a corresponding eigenfunction ϕ1,γk satisfying (a)-(d)
of Proposition 3.1.

Finally, we establish the following useful relationship between λ1,q and λ1,γk .

Proposition 3.2. Let q ∈ L∞(Ω) with 0 ≤ q ≤ 1 a.e. in Ω and q(x) > 1/2 on a
set of positive measure, and γk be given by (3.10). Then λ1,γk ↘ λ1,q as k → +∞.

Proof. The properties of q and γk imply that the inequalities∫
Ω

q(x)|φ(x)|2 dx ≥
∫

Ω

γk+1(x)|φ(x)|2 dx ≥
∫

Ω

γk(x)|φ(x)|2 dx > 0 (3.11)

hold for every k ≥ 2 for all φ ∈ Hs
0(Ω) with φ ≥ δs a.e. in Ω. First, we show

λ1,q ≤ λ1,γk+1
≤ λ1,γk for each k ≥ 2. Indeed, it follows from (3.11) that the

inequalities

E(φ, φ)∫
Ω
q(x)|φ(x)|2 dx

≤ E(φ, φ)∫
Ω
γk+1(x)|φ(x)|2 dx

≤ E(φ, φ)∫
Ω
γk(x)|φ(x)|2 dx

(3.12)

hold for all φ ∈ Hs
0(Ω) with φ ≥ δs a.e. in Ω. By taking the infimum over all such

φ, inequalities (3.12) imply λ1,q ≤ λ1,γk+1
≤ λ1,γk , using (3.8), as desired.

Now we show λ1,γk → λ1,q as k → +∞. By (3.8) with k ≥ 2, we see

λ1,γk = inf
φ∈Hs0(Ω)

φ≥δs a.e. in Ω

E(φ, φ)∫
Ω
γk(x)|φ(x)|2 dx

.

Let ϕ1,q be the principal eigenfunction scaled such that ϕ1,q ≥ δs a.e. in Ω. Then,
using the same argument as in the proof of Proposition 3.1 (e), we obtain

λ1,q = inf
φ∈Hs0 (Ω)

{ E(φ, φ)∫
Ω
q(x)|φ(x)|2 dx

: φ ≥ δs a.e. in Ω , ‖φ‖Hs0 (Ω) ≤ ‖ϕ1,q‖Hs0 (Ω)

}
.

By the definition of the infimum, for each k ∈ N, we can find φk ∈ Hs
0(Ω), φk ≥ δs

a.e. in Ω and ‖φk‖Hs0 (Ω) ≤ ‖ϕ1,q‖Hs0 (Ω) such that

λ1,q ≥
E(φk, φk)∫

Ω
q(x)|φk(x)|2 dx

− 2−k .

Thus, for k ≥ 2, we have

λ1,q ≥
E(φk, φk)∫

Ω
q(x)|φk(x)|2 dx

− 2−k

=
E(φk, φk)∫

Ω
γk(x)|φk(x)|2 dx

·
∫

Ω
γk(x)|φk(x)|2 dx∫

Ω
q(x)|φk(x)|2 dx

− 2−k

≥
(

inf
φ∈Hs0(Ω)

φ≥δs a.e. in Ω

E(φ, φ)∫
Ω
γk(x)|φ(x)|2 dx

)∫
Ω
γk(x)|φk(x)|2 dx∫

Ω
q(x)|φk(x)|2 dx

− 2−k

≥ λ1,γk

∫
Ω
γk(x)|φk(x)|2 dx∫

Ω
q(x)|φk(x)|2 dx

− 2−k .



8 M. CHHETRI, P. GIRG, E. HOLLIFIELD EJDE-2020/81

This yields

λ1,q ≤ λ1,γk ≤ (λ1,q + 2−k)

∫
Ω
q(x)|φk(x)|2 dx∫

Ω
γk(x)|φk(x)|2 dx

. (3.13)

By the compact embedding of Hs
0(Ω) into L2(Ω) and ‖φk‖Hs0 (Ω) ≤ ‖ϕ1,q‖Hs0 (Ω),

we can find a subsequence φkj → ψ in L2(Ω), where ψ is some element of L2(Ω).

Then, φ2
kj
→ ψ2 in L1(Ω). Since q ∈ L∞(Ω),

∫
Ω
q(x)|φkj (x)|2 dx→

∫
Ω
q(x)|ψ(x)|2 dx.

Now we show that ∫
Ω

γkj (x)|φkj (x)|2 dx→
∫

Ω

q(x)|ψ(x)|2 dx

as well. Indeed,∫
Ω

γkj (x)|φkj (x)|2 dx =

∫
Ω

(γkj (x)− q(x))|φkj (x)|2 dx+

∫
Ω

q(x)|φkj (x)|2 dx .

By (3.10), q(x)− γkj (x) ≤ 1/kj , thus∣∣ ∫
Ω

(γkj (x)− q(x))|φkj (x)|2 dx
∣∣ ≤ 1/kj

∫
Ω

|φkj |2 dx ≤ C

kj
‖ϕ1,q‖2Hs0 (Ω) → 0

as kj → +∞, where C is the constant of the embedding of Hs
0(Ω) into L2(Ω).

Observe that ψ ≥ δs > 0 a.e. in Ω since φkj ≥ δs a.e. in Ω, and hence∫
Ω
q(x)|φk(x)|2 dx∫

Ω
γk(x)|φk(x)|2 dx

→
∫

Ω
q(x)|ψ(x)|2 dx∫

Ω
q(x)|ψ(x)|2 dx

= 1

as kj → +∞. Thus, (3.13) establishes that λ1,γkj
→ λ1,q. Since λ1,γk is monotone

sequence, it must hold for entire sequence λ1,γk ↘ λ1,q. �

4. Proof of Theorem 2.3

We follow the idea of proof from Clement-Sweers [14], where this result was
proven for the Laplacian case (s = 1) using the Schauder fixed point theorem.

Proof of Theorem 2.3. Consider the modified function g∗ : Ω× R→ R defined as

g∗(x, t) :=


g(x, u(x)) if t < u(x) ,

g(x, t) if u(x) ≤ t ≤ u(x) ,

g(x, u(x)) if t > u(x)

and note that g∗ is clearly a Carathéodory function. We observe that any weak
solution u of (2.7) satisfying u ≤ u ≤ u a.e. in Ω is also a weak solution of the
modified problem

(−∆)su = g∗(x, u) in Ω;

u = 0 in RN \ Ω .
(4.1)

Moreover, using the definition of g∗, it follows from the claim below that any weak
solution of (4.1) is a weak solution of (2.7).

Claim: If u is a weak solution of (4.1), then u ≤ u ≤ u a.e. in Ω.
First, we establish u ≤ u a.e. in Ω by showing that meas(A) = 0, where A :=
{x ∈ RN : u(x) < u(x)}. Clearly A is measurable (in the sense of Lebesgue) since
u ∈ Hs

0(Ω) and u ∈ Hs(RN ). Assume to the contrary that meas(A) > 0. We note
that meas

(
A∩(RN \Ω)

)
= 0 since u ≥ 0 = u a.e. in RN \Ω. Hence, meas(A∩Ω) > 0.

Setting z+ := max{0, z} ≥ 0, we see that [u−u]+ ∈ Hs
0(Ω) since [u−u]+ ∈ Hs(RN )



EJDE-2020/81 FRACTIONAL LAPLACIAN EQUATIONS 9

and it vanishes almost everywhere outside A ⊂ Ω. Taking φ := [u − u]+ as a test
function in (2.4) and (2.5), and using the definitions of g∗ and A, we obtain

E(u, [u− u]+) =

∫
Ω

g∗(x, u(x))[u− u]+(x) dx

=

∫
A

g∗(x, u(x))[u− u]+(x) dx

=

∫
A

g(x, u(x))[u− u]+(x) dx

=

∫
Ω

g(x, u(x))[u− u]+(x) dx

≤ E(u, [u− u]+) .

(4.2)

On one hand, subtracting the right-hand side from the left-hand side in (4.2) and
rearranging the terms yields the inequality

E((u− u), [u− u]+) ≤ 0 . (4.3)

On the other hand, by taking v = u− u, it follows from [24, Lem. 3.3] that

[v(x)− v(y)][v+(x)− v+(y)] ≥ [v+(x)− v+(y)]2 for a.e. x, y ∈ RN . (4.4)

Using (2.2) and (2.3), the pointwise estimate (4.4) yields

E(v, v+) ≥ CN,s
2
‖v+‖2Hs0 (Ω) > 0 ,

since meas(A) > 0, v > 0 in A, and ‖ · ‖Hs0 (Ω) is a norm on Hs
0(Ω). This is a

contradiction to (4.3). Hence, meas(A) = 0, that is, u(x) ≤ u(x) for a.e. x ∈ Ω.
Similarly, by letting φ := [u − u]+ as a test function, and repeating the argument
above we can show that meas(B) = 0, where B := {x ∈ RN : u(x) > u(x)}. Hence,
u(x) ≥ u(x) a.e. x ∈ Ω. This proves the claim.

Therefore, it suffices to show the existence of solution of (4.1) using the Schauder
fixed point theorem. To construct a compact operator, consider the following linear
problem

(−∆)sw = θ(x) in Ω;

w = 0 in RN \ Ω .
(4.5)

For each θ ∈ H−s(Ω) (the dual of Hs
0(Ω)), there exists a unique weak solution

w ∈ Hs
0(Ω) of (4.5), see [20, Thm. 12] for N ≥ 2 and [9, Prop. 2.1] for N = 1.

Moreover, if θ ∈ L∞(Ω), then there exists C > 0 such that

‖w‖C0,s(Ω) ≤ C‖θ‖L∞(Ω) , (4.6)

see [28, Prop. 7.2]. Then, the solution operator K : L∞(Ω) → L∞(Ω) given by
θ 7→ w is well defined, continuous, and compact since the following holds for some
s′ ∈ (0, s)

L∞(Ω)
K−→ C0,s(Ω) ↪→↪→ C0,s′(Ω) ↪→ L∞(Ω) .

The Nemytskii operator H : L∞(Ω) → L∞(Ω) defined by u 7→ g∗(x, u(x)) is con-
tinuous (see [4, Thm. 3.17, p. 110]) since g∗ satisfies (H1) and (H2). Then,
KH : L∞(Ω) → L∞(Ω) is continuous and compact, and fixed points of KH are
solutions of (4.1).



10 M. CHHETRI, P. GIRG, E. HOLLIFIELD EJDE-2020/81

Next, we find a nonempty, closed, and convex subset of L∞(Ω) to apply the
Schauder fixed point theorem. Without loss of generality, assume that u 6≡ u in Ω.
Then, using u, u ∈ L∞(Ω), we define

r∗ := max{‖u‖L∞(Ω), ‖u‖L∞(Ω)} > 0 .

Then, it follows from (H1), applied to g∗, that there exists ar∗ ∈ L∞(Ω) such that
|g∗(x, t)| ≤ ar∗(x) for all |t| ≤ r∗. Therefore, for any u ∈ L∞(Ω), we have

‖KH(u)‖L∞(Ω) ≤ ‖K‖‖H(u)‖L∞(Ω) ≤ ‖K‖‖ar∗‖L∞(Ω) ,

and hence the operator KH maps BR(0) to itself where R := ‖K‖‖ar∗‖L∞(Ω) and
‖ · ‖ is the operator norm. Hence, by the Schauder fixed point theorem, KH has
a fixed point u ∈ BR(0) ⊂ L∞(Ω). This implies that the modified problem (4.1)
and hence the original problem (2.7) has a weak solution u ∈ L∞(Ω) such that
u ≤ u ≤ u. By the definition of K it follows that u ∈ Hs

0(Ω) as well. Hence, the
proof of Theorem 2.3 is complete. �

5. Proofs of Theorems 2.5 and 2.6

To prove Theorems 2.5 and 2.6, we employ Theorem 2.3. For each case, we
construct an ordered pair of weak sub- and supersolutions in Hs

0(Ω) ⊂ Hs(RN ) of
(1.1) where f(x, t) = f(t).

Proof of Theorem 2.5. Since f(0) > 0, it follows that u ≡ 0 ∈ Hs
0(Ω) is a weak

subsolution of (1.1). Now let λ > 0 be fixed and e ∈ Hs
0(Ω) be the positive weak

solution of (3.1). We show that there exists Mλ > 0 such that u := Me is a weak
supersolution of (1.1) for all M ≥Mλ. We observe that while f is not assumed to
be nondecreasing, f(t) := maxσ∈[0,t] f(σ) is nondecreasing. Moreover, f(t) ≤ f(t)

for all t ≥ 0, and f satisfies the sublinear condition at infinity

lim
t→+∞

f(t)

t
= 0 .

Therefore, there exists Mλ > 0 such that for all M ≥Mλ,

f(M‖e‖L∞(Ω))

M‖e‖L∞(Ω)
≤ 1

λ‖e‖L∞(Ω)
or equivalently λf(M‖e‖L∞(Ω)) ≤M .

Then u = Me ∈ Hs
0(Ω) satisfies

ME(e, φ) = M

∫
Ω

φ(x) dx

≥ λ
∫

Ω

f(M‖e‖L∞(Ω))φ(x) dx

≥ λ
∫

Ω

f(Me(x))φ(x) dx

≥ λ
∫

Ω

f(Me(x))φ(x) dx = λ

∫
Ω

f(u)φ(x) dx

for all φ ∈ Hs
0(Ω) with φ ≥ 0 a.e. in Ω. Therefore, u is a weak supersolution of

(1.1) for each λ > 0. Clearly u = Me ≥ 0 = u a.e. in Ω. Hence, by Theorem 2.3
and [28, Lem. 7.3], there exists a weak solution u of (1.1) such that 0 < u ≤ Me
a.e. in Ω for any λ > 0. This completes the proof of Theorem 2.5. �



EJDE-2020/81 FRACTIONAL LAPLACIAN EQUATIONS 11

Proof of Theorem 2.6. Let λ > λ1

f ′(0) be fixed, where λ1 is the principal eigenvalue

of (3.3) with q ≡ 1 and ϕ1 > 0 is the corresponding principal eigenfunction. Note
that since f(0) = 0, u ≡ 0 is a solution and hence a subsolution of (1.1). Hence, to
complete the proof, we must construct a positive weak subsolution. We show that
an appropriate constant multiple of ϕ1 is a weak subsolution of (1.1). We find this
constant by analyzing the function Θ(t) := λ1t− λf(t) for t ≥ 0. Clearly Θ(0) = 0
and Θ′(t) = λ1 − λf ′(t). Therefore, Θ′(0) < 0 since λ > λ1

f ′(0) and hence there

exists θ(λ) > 0 such that Θ(t) < 0 for any t ∈ (0, θ(λ)).
Now we show that u := mϕ1 ∈ Hs

0(Ω) is a positive weak subsolution of (1.1)

for any m ∈ (0,mλ), where mλ := θ(λ)
‖ϕ1‖L∞(Ω)

. Indeed, by (3.6) and the discussion

above, u satisfies

m E(ϕ1, φ) = λ1

∫
Ω

mϕ1(x)φ(x) dx

≤ λ
∫

Ω

f(mϕ1)φ(x) dx = λ

∫
Ω

f(u)φ(x) dx

for all φ ∈ Hs
0(Ω) with φ ≥ 0 a.e. in Ω. Hence, for any λ > λ1

f ′(0) and any

m ∈ (0,mλ), u = mϕ1 is a weak subsolution of (1.1).
As in the proof of Theorem 2.5, for any λ > λ1

f ′(0) there exists Mλ > 0 such

that for M ≥ Mλ, the function u = Me ∈ Hs
0(Ω) is a weak supersolution of (1.1).

Using the left estimate of e in (3.2) and the right estimate of ϕ1 in (3.7), and
by choosing M sufficiently large and/or choosing m sufficiently small, we obtain
u ≤ u a.e. in Ω. Hence, by Theorem 2.3, (1.1) has a positive weak solution u
satisfying mϕ1 ≤ u ≤Me a.e. in Ω for any λ > λ1

f ′(0) . This completes the proof of

Theorem 2.6. �

6. Proofs of Theorems 2.8 and 2.9

Here also we construct an ordered pair of weak sub- and supersolution of (1.1)
in Hs

0(Ω) ⊂ Hs(RN ).

Proof of Theorem 2.8. Let λ > λ1,q be fixed. First, we construct a positive weak
subsolution of (2.9). Since λ > λ1,q and λ1,γk ↘ λ1,q as k → +∞ (by Proposi-
tion 3.2), there exists l ∈ N such that λ1,q ≤ λ1,γl < λ, where λ1,γl is the principal
eigenvalue of (3.9) with γl defined by (3.9). Let ϕ1,γl ∈ Hs

0(Ω) be a positive eigen-
function corresponding to λ1,γl , and e ∈ Hs

0(Ω) be a positive weak solution of
(3.1)

We show that there exist mλ > 0 and ε > 0 such that for all m ∈ (0,mλ),
u := m(ϕ1,γl − εe) ∈ Hs

0(Ω) is a positive weak subsolution of (2.9). Set α :=√
λ1,γl

λ ∈ (0, 1). Then, using (3.2), and (3.7) with q = γl, there exists ε > 0 such

that

ϕ1,γl − εe > αϕ1,γl > 0 a.e. in Ω . (6.1)

Define

mλ := min
{ ε

λα‖ϕ1,γl(ϕ1,γl − εe)‖L∞(Ω)
,

1− α
l‖ϕ1,γl − εe‖L∞(Ω)

}



12 M. CHHETRI, P. GIRG, E. HOLLIFIELD EJDE-2020/81

and let m ∈ (0,mλ). Using the weak formulation of e and ϕ1,γl , we see that
u = m(ϕ1,γl − εe) ∈ Hs

0(Ω) satisfies

E(u, φ) = m

∫
Ω

[λ1,γl γl(x)ϕ1,γl(x)− ε]φ(x) dx

for all φ ∈ Hs
0(Ω). Therefore, u is a weak subsolution of (2.9) if

m

∫
Ω

[λ1,γl γl(x)ϕ1,γl(x)− ε]φ(x) dx

≤ λm
∫

Ω

[ϕ1,γl(x)− εe(x)]
[
q(x)−m(ϕ1,γl(x)− εe(x))

]
φ(x) dx

(6.2)

for all φ ∈ Hs
0(Ω) with φ ≥ 0 a.e. in Ω. Using the definition of γl and (6.1), we

obtain

λmαϕ1,γl

[
γl −m(ϕ1,γl − εe)

]
≤ λm(ϕ1,γl − εe)

[
q −m(ϕ1,γl − εe)

]
a.e. in Ω .

Therefore, (6.2) holds if

λ1,γl γlϕ1,γl − ε ≤ λαϕ1,γl

[
γl −m(ϕ1,γl − εe)

]
a.e. in Ω . (6.3)

Let Ωl := {x ∈ Ω : q(x) < 1/l}. If x ∈ Ωl, then γl(x) = 0. In this case, the
inequality in (6.3) holds a.e. in Ωl since

m < mλ ≤
ε

λα‖ϕ1,γl(ϕ1,γl − εe)‖L∞(Ω)
.

If x ∈ Ω \ Ωl, then γl(x) = q(x) ≥ 1/l. In this case, the inequality in (6.3) holds
a.e. in Ω \ Ωl since the inequality

λ1,γl γlϕ1,γl ≤ λαϕ1,γl

[
γl −m(ϕ1,γl − εe)

]
holds by choosing

m < mλ ≤
1− α

l‖ϕ1,γl − εe‖L∞(Ω)
.

Hence, u = m(ϕ1,γl−εe) is a positive weak subsolution of (2.9) for any m ∈ (0,mλ).
Now we construct a positive weak supersolution. Since q ∈ L∞(Ω) with 0 ≤ q ≤ 1

a.e. in Ω and maxy∈R y(1− y) = 1/4, the inequality

1 ≥M e(q −Me)

holds for all M ≥Mλ = λ a.e. in Ω. Then u = M e ∈ Hs
0(Ω) satisfies

E(u, φ) ≥ λ
∫

Ω

φ(x) dx

≥ λ
∫

Ω

Me(x)
(
q(x)−Me(x)

)
φ(x) dx

= λ

∫
Ω

u
(
q(x)− u

)
φ(x) dx

for all φ ∈ Hs
0(Ω) with φ ≥ 0 a.e. in Ω. Therefore, u = Me is a weak supersolution

of (2.9) for any M ≥ λ.
Finally, using the left estimate of (3.2), and the right estimate of (3.7) combined

with (6.1), we can choose M ≥ λ sufficiently large and/or 0 < m < mλ sufficiently
small, so that u ≤ u a.e. in Ω. Hence, by Theorem 2.3, (2.9) has a positive weak
solution u satisfying m(ϕ1,γl − εe) ≤ u ≤ Me a.e. in Ω for any λ > λ1,q. This
completes the proof of Theorem 2.8 �



EJDE-2020/81 FRACTIONAL LAPLACIAN EQUATIONS 13

Proof of Theorem 2.9. First, we construct a positive weak subsolution for (2.10).

Let λ > λ1 be fixed and define β :=
√

λ1

λ ∈ (0, 1). Then, as in the proof of

Theorem 2.8 there exists ε > 0 such that ϕ1 − εe > βϕ1 > 0 a.e. in Ω. Define
u := m∗(ϕ1 − εe) ∈ Hs

0(Ω) with fixed m∗ := 1−β
2‖ϕ1−εe‖L∞(Ω)

. We show that u is a

positive weak subsolution of (2.10) for any 0 < a < a∗ := εm∗

λ . Then u satisfies

E(e, φ) = m∗
∫

Ω

(λ1 ϕ1(x)− ε)φ(x) dx ,

for all φ ∈ Hs
0(Ω). Hence, u is a weak subsolution of (2.10) if

m∗(λ1 ϕ1 − ε) ≤ λm∗(ϕ1 − εe)
[
1−m∗(ϕ1 − εe)

]
− λa a.e. in Ω . (6.4)

Since ϕ1 − εe > βϕ1 > 0, (6.4) is satisfied if

λ1ϕ1 ≤ λβϕ1

[
1−m∗(ϕ1 − εe)

]
+ ε− λa

m∗
a.e. in Ω . (6.5)

But, εm∗ − λa ≥ 0 since εm∗ − λa∗ = 0 by our choice of a∗, and a < a∗. Then,
using ϕ1 > ϕ1 − εe > 0 a.e. in Ω and m∗ = 1−β

2‖ϕ1−ε e‖L∞(Ω)
, (6.5) follows from the

inequality

λ1ϕ1 ≤ λβϕ1

[
1−m∗‖ϕ1 − εe‖L∞(Ω)

]
a.e. in Ω .

Hence, u = m∗(ϕ1 − εe) is a subsolution of (2.10) for a < a∗.
As in the proof of Theorem 2.8, u = M e ∈ Hs

0(Ω) is a supersolution of (2.10)
for any M ≥Mλ = λ since 1 ≥Me

(
1−M e

)
− a a.e. in Ω.

Again, using the estimates (3.2) and (3.7), we can further refine the choice of
M ≥ λ to be sufficiently large such that u ≤ u in Ω. Therefore, by Theorem 2.3, for
any λ > λ1, (2.10) has a positive weak solution u satisfying m∗(ϕ1− εe) ≤ u ≤Me
a.e. in Ω for 0 < a < a∗. This completes the proof of Theorem 2.9. �

7. Finite element method for the fractional Laplacian in one
dimension

The finite element approximation of the linear one dimensional problem

(−∆)su = z(x) in (0, 1)

u = 0 in R \ (0, 1) ,
(7.1)

for s ∈ (0, 1) was investigated, including convergence results with z in appropriate
function spaces, see [2, 8]. This motivated the investigation of numerical positive
weak solutions for the nonlinear fractional Laplacian problems

(−∆)su = λf(x, u) in (0, 1)

u = 0 in R \ (0, 1) ,
(7.2)

where λ > 0 and f : (0, 1)×[0,+∞)→ R is a Carathéodory function. For numerical

experiments, we further assume that ft(x, t) := ∂f
∂t (x, t) is continuous a.e. in (0, 1),

and f satisfies certain Hölder type conditions with respect to x ∈ (0, 1), specified
below. We consider examples of nonlinearity f satisfying the respective hypotheses
of Theorems 2.5 - 2.9 with these additional assumptions.

We use the one dimensional finite element method (FEM) developed for the lin-
ear fractional Laplacian problems of the form (7.1) in [8] to construct a numerical
solution u (often positive) with λ > 0 of the nonlinear problem (7.2). Moreover, us-
ing the branch following technique of [25], we construct bifurcation diagrams ‖u‖∞



14 M. CHHETRI, P. GIRG, E. HOLLIFIELD EJDE-2020/81

vs. λ, where ‖u‖∞ = ‖u‖L∞(0,1). Additionally, we formulate relevant conjectures
based on the qualitative features of the bifurcation diagrams from our numerical
experiments.

As in [8], we use the weak formulation of (7.2) to seek solutions u ∈ Hs
0(0, 1)

such that for all φ ∈ Hs
0(0, 1)

C1,s

2

∫
R

∫
R

(u(x)− u(y))(φ(x)− φ(y))

|x− y|1+2s
dx dy = λ

∫ 1

0

f(x, u(x))φ(x)dx .

Remark 7.1. It is known that (see [9, 15, 30]) for u from a suitable class of
functions,

lim
s→1−

(−∆)su = −∆u and lim
s→0+

(−∆)su = I , (7.3)

where I is the identity operator. Furthermore, it was shown in [9] that the weak
solution of the Poisson equation for (−∆)s with homogeneous Dirichlet condition
on RN \ Ω approaches to the weak solution of Poisson equation for −∆ with ho-
mogeneous Dirichlet condition on ∂Ω as s → 1−. We utilize the limiting behavior
information (7.3) as a hint for the correctness of our numerical scheme. In particu-
lar, throughout this section, we use the finite difference or the quadrature method
to generate the bifurcation diagram for the Laplacian case (s = 1) and then com-
pare to the fractional Laplacian case (s = 0.99) using the finite element method,
before proceeding with other values of s ∈ (0, 1).

We describe our method below.

7.1. Discretization. We introduce a uniform partition 0 = x0 < x1 < x2 . . . <
xn+1 = 1, of [0, 1], with step size h = xi − xi−1 for i = 1, . . . , n + 1. Let Vh be an
n-dimensional subspace of Hs

0(0, 1) spanned by {φ1, . . . , φn}, where

φi(x) :=

{
1− |x− xi|/h if x ∈ [xi−1, xi+1] ,

0 if x ∈ R \ [xi−1, xi+1]
(7.4)

for i = 1, . . . , n. The finite element approximation uh ∈ Vh of weak solution
u ∈ Hs

0(0, 1) of (7.2) is expressed as

uh(x) :=

n∑
i=1

uiφi(x) ,

where ui ∈ R are unknowns and uh satisfies the system of n equations

C1,s

2

∫
R

∫
R

[uh(x)− uh(y)][φj(x)− φj(y)]

|x− y|1+2s
dxdy = λ

∫ 1

0

f (x, uh(x))φj(x) dx

(7.5)
for j = 1, . . . , n. To implement the finite element scheme, we express (7.5) in matrix
notation. For a column vector u := [u1, . . . , un]T , the left hand side of (7.5) can
be expressed as Au, where A is the n×n stiffness matrix corresponding to the left
hand side of (7.5) derived in [8].

To numerically compute the integral on the right hand side of (7.5), we assume
that there exists L > 0 such that for any y1, y2 ∈ (0, 1) and any t1, t2 ≥ 0

|f(y2, t2)− f(y1, t1)| ≤ L(|y2 − y1|s + |t2 − t1|) . (7.6)

Then, the expectation that ‖uh‖C0,s([0,1]) ≤ K ′ holds (independent of h), yields

|f(x, uh(x))− f(xj , uh(xj))| ≤ L(|x− xj |s + |uh(x)− uh(xj)|) ≤ L(1 +K ′)hs .



EJDE-2020/81 FRACTIONAL LAPLACIAN EQUATIONS 15

Therefore, for all j = 1, . . . , n, using the definition (7.4) of φj one has

∫ 1

0

f (x, uh(x))φj(x) dx

=

∫ xj+1

xj−1

f (x, uh(x))φj(x) dx

=

∫ xj+1

xj−1

[
f (xj , uh(xj))φj(x) + f (x, uh(x))φj(x)− f (xj , uh(xj))φj(x)

]
dx

= f(xj , uj)

∫ xj+1

xj−1

φj(x) dx+

∫ xj+1

xj−1

[
f (x, uh(x))φj(x)− f (xj , uh(xj))φj(x)

]
dx

= hf(xj , uj) +O(h1+s) ,

where (more precisely)

0 ≤ |O(h1+s)| ≤ L(1 +K ′)h1+s .

Then, defining the column vector F : Rn → Rn by

F(u) := h[f(x1, u1) , f(x2, u2), . . . , f(xn, un)]T ,

we rewrite (7.5) as a matrix equation

Au = λF(u) . (7.7)

We solve this system (7.7) for a given nonlinearity f and λ > 0 with Newton’s
method provided we have a suitable initial guess for iteration. A multiple of the
solution of the linear problem (−∆)se = 1 in (0, 1) with u = 0 in R \ (0, 1) served
as a good candidate for an initial guess in many cases.

Now we describe the pseudo-code for constructing numerical solutions and nu-
merical bifurcation diagrams, where | · |∞ will denote the maximum norm in Rn.

Input:

s ∈ (0, 1) (real parameter in (−∆)s)

0 ≤ λmin < λmax (range of values of λ in the bifurcation diagram)

m ∈ N (number of partitions of the interval [λmin, λmax])

n ∈ N (number of interior nodes in partition of interval [0, 1])

6 < r < 15 (10−r is the tolerance in the Newton iteration)

Output:

S (list of points of the form (λ, |u|∞))



16 M. CHHETRI, P. GIRG, E. HOLLIFIELD EJDE-2020/81

Begin

% Initialization

01 Create interior nodes of the uniform partition P of [0, 1]

by setting xj ← j/(n+ 1), j = 1, . . . , n

02 C1,s ← 22s sΓ(1/2+s)√
πΓ(1−s)

03 Assemble the n× n stiffness matrix A for the partition P
and parameter s using the algorithm described in [8, page 12]

04 Create a uniform partition Λ of [λmin, λmax]

by setting µi ← λmin + λmax−λmin

m i, i = 0, . . . ,m

05 uinit ← Solution of A e = 1

% Here 1 stands for n× 1 column vector of 1s.

06 S ← Empty list

% End of Initialization

% Main Loop

07 For i := 0 : m do

% Apply Newton iterations to: Au = µiF(u)

08 u← uinit

% Compute F(u) componentwise (represented by column vector b)

09 [b]j ← hf(xj , uj) for j = 1, . . . , n

10 res← Au− µib
% Newton loop

11 While |res|∞ > 10−r do

% Compute JF, the Jacobian matrix of F(u) componentwise

12 [JF]j,j ← hft(xj , uj) for j = 1, . . . , n, and

[JF]i,j ← 0 for i 6= j, i, j = 1, . . . n

% Compute J, the Jacobian matrix of the system (7.7)

13 J← A− µiJF

% Newton’s update of u

14 u← u− J−1res

% Update of F(u) componentwise

15 [b]j ← hf(xj , uj) for j = 1, . . . , n

% Update of res

16 res← Au− µib
17 EndWhile

18 S ← Append(S, (µi, |u|∞))

19 EndFor

% End of Main Loop

20 Return S
End



EJDE-2020/81 FRACTIONAL LAPLACIAN EQUATIONS 17

7.2. Construction of bifurcation diagrams. Let S be a closed connected set of
(λ, u) ∈ R×L∞(0, 1) such that u is a weak solution of (7.2) corresponding to λ > 0.
In each example below, we will discuss the shape of S via the bifurcation diagram
obtained numerically in the ‖u‖∞ vs. λ plane. These bifurcation diagrams are in
agreement with the results obtained in Section 2 and furthermore, help formulate
conjectures.

For each of the bifurcation diagrams, we also give a numerical positive solution
for a specific value of λ for which existence is guaranteed by the results in Section 2.

We observe from the proofs of Theorems 2.5-2.9 that these positive weak solu-
tions u satisfy c1δ

s(x) ≤ u(x) ≤ c2δ
s(x) for x ∈ [0, 1]. Hence, the influence of s on

the behavior of positive solutions near the boundary (x = 0 and x = 1) becomes
more pronounced for s ∈ (0, 1) small.

Example 7.2. Let f(t) = e
t

1+t for t ≥ 0 (satisfying the hypotheses of Theorem 2.5).
For existence result and the bifurcation diagram for the Laplacian case (s = 1),

see [22, Sec. 2.2]. For the fractional Laplacian case, (7.2) has a positive weak
solution, for each λ > 0, by Theorem 2.5. Clearly u ≡ 0 is a solution of (7.2) for
λ = 0. Figure 1 shows the bifurcation diagrams for (A) s = 1 (B) s = 0.99 (C)
s = 0.9 (D) s = 0.7 (E) s = 0.5 and (F) s = 0.3. We observe that the bifurcation
diagrams are qualitatively similar for all the s values considered. The inset in
each bifurcation diagram shows the typical profile of a numerical positive solution
corresponding to λ = 55 and the influence of s on the behavior of positive solutions,
in particular near the boundary points x = 0 and x = 1.

We see from the bifurcation diagrams in Figure 1 that the solution set S emanates
from the origin and increases with respect to λ (hence there is a unique positive
solution for each λ > 0). Moreover, ‖u‖∞ → 0 as λ → 0+ and ‖u‖∞ → +∞
as λ → +∞. In [20], the authors prove uniqueness of positive solutions if f(t)

t

is decreasing in t. Note that this condition is satisfied by f(t) = e
t

1+t and our
bifurcation diagram confirms the uniqueness result in [20, Thm. 20].

Example 7.3. f(t) = e
5t

5+t for t ≥ 0 (satisfying the hypotheses of Theorem 2.5).
Investigation of the bifurcation diagrams for the perturbed Gelfand problem

f(t) = e
κt
κ+t for κ > 0 and t ≥ 0 has been of interest since the paper of Keller

and Cohen [18]. It was shown in [11] that a sufficient condition for the bifurcation
diagram to be S-shaped is satisfied if κ ≥ 4.07 for the Laplacian case (s = 1).
Indeed, we see in Figure 2 that the numerical bifurcation diagram is S-shaped for
both s = 1 (obtained using quadrature method) and s = 0.99. As in Example 7.2,
‖u‖∞ → 0 as λ → 0+ and ‖u‖∞ → +∞ as λ → +∞. However, the solution set S
is not monotone with respect to λ. Additionally, there is a range of λ for which we
see three numerical positive solutions.

In Figure 3, the bifurcation diagrams are given in (A) s = 0.9, (C) s = 0.7, (E)
s = 0.5, s = 0.3 and corresponding profiles of three positive solutions are given in
(B) λ = 2.75, (D) λ = 1.5, (F) λ = 1.5 and (H) λ = 0.6, respectively. We also
observe that the interval of λ for which three solutions exist shifts to the left as s
decreases. With these observations, we state the following conjecture.

Conjecture 7.4. Let f(t) > 0 and t/f(t) be strictly increasing for small t ≥ 0 as
well as for large t > 0, and decreasing somewhere, then the bifurcation diagram is
S-shaped.



18 M. CHHETRI, P. GIRG, E. HOLLIFIELD EJDE-2020/81

(A) s = 1 (B) s = 0.99

(C) s = 0.9 (D) s = 0.7

(E) s = 0.5 (F) s = 0.3

Figure 1. Bifurcations diagrams for f(t) = e
t

1+t and numerical
positive solutions with λ = 55

For nonlinearities like f(t) = e
5t

5+t /tη, η ∈ (0, 1), it was shown in [17] that there
is a range of λ for which there exist three positive solutions and unique positive
solution for λ large. Their result suggests the existence of the S-shaped bifurcation
diagram. For existence and multiplicity results using critical point theory, see
[5]. However, the connectedness of such solution set remains an important open
question.



EJDE-2020/81 FRACTIONAL LAPLACIAN EQUATIONS 19

(A) s = 1 (B) s = 0.99

Figure 2. Bifurcation diagrams for f(t) = e
5t

5+t for (A) s = 1 and
(B) s = 0.99

Example 7.5. f(t) = 3(1 + t)1/3 − 3 for t ≥ 0 (satisfying the hypotheses of
Theorem 2.6).

The bifurcation diagram for the Laplacian case (s = 1) was discussed in detail
in [22, Sec. 1.2]. In particular, if f(0) = 0 and f ′(0) > 0, then the positive solution
bifurcates from the line of trivial solutions at λ = λ1

f ′(0) . Here f ′(0) = 1, so the

bifurcation from the trivial branch of solutions occurs at λ = λ1 = π2 for s = 1,
see Figure 4 (A). The inset of Figure 4 (A) shows a numerical positive solution for
λ = 55.

In Figure 4, (B)-(F) show the bifurcation diagrams and the insets give the nu-
merical positive solution corresponding to λ = 55 for (B) s = 0.99, (C) s = 0.9, (D)
s = 0.7 and (E) s = 0.5 and (F) s = 0.3, respectively. We observe again that the
bifurcation diagrams for any s ∈ (0, 1) are qualitatively similar to those for s = 1.
For s = 0.99, the bifurcation of positive solutions from the line of trivial solutions
occurs near π2 ≈ 9.8696, see Figure 4 (B). The influence of s ∈ (0, 1) is noticeable
in the location of the point of bifurcation from the line of trivial solutions. This
can be justified by the estimate of the principal eigenvalue of (−∆)s on (0, 1), see
[19]. Also, the profile of the numerical positive solutions corresponding to λ = 55
for values of s ∈ (0, 1] exhibit the boundary behavior similar to δs.

Conjecture 7.6. If f satisfies the hypotheses of Theorem 2.6, then there exists
a continuum of positive weak solutions that bifurcates from the branch of trivial
solutions at λ = λ1

f ′(0) and from infinity at infinity in the positive λ direction.

Example 7.7. Logistic reaction term f(t) = t(1 − t) for t ≥ 0 (corresponding to
Theorem 2.8 with q(x) ≡ 1).

The logistic reaction term considered here is essentially a sublinear nonlinearity
at infinity with f(0) = 0 and f ′(0) = 1. Hence, the bifurcation diagrams in Figure 5
resemble those obtained for Example 7.5. The L∞ norm of the solutions ‖u‖∞ are
bounded above by 1 for any s ∈ (0, 1]. Therefore, to understand the influence of
s ∈ (0, 1) on solutions, we compute the L1 norm ‖u‖L1(0,1) = ‖u‖1 of the numerical
positive solution u for λ = 55. We observe that ‖u‖1 increases as s decreases. It
appears, numerically, that ‖u‖1 ↗ 1 as s→ 0+.



20 M. CHHETRI, P. GIRG, E. HOLLIFIELD EJDE-2020/81

(A) s = 0.9 (B) 3 solutions

(C) s = 0.7 (D) 3 solutions

(E) s = 0.5 (F) 3 solutions

(G) s = 0.3 (H) 3 solutions

Figure 3. Bifurcation diagrams for f(t) = e
5t

5+t and three numer-
ical positive solutions for the λ specified



EJDE-2020/81 FRACTIONAL LAPLACIAN EQUATIONS 21

(A) s = 1 (B) s = 0.99

(C) s = 0.9 (D) s = 0.7

(E) s = 0.5 (F) s = 0.3

Figure 4. Bifurcation diagrams for f(t) = 3(1 + t)
1
3 − 3 and

numerical positive solutions with λ = 55



22 M. CHHETRI, P. GIRG, E. HOLLIFIELD EJDE-2020/81

(A) s = 1 (B) s = 0.99

(C) s = 0.9 (D) s = 0.7

(E) s = 0.5 (F) s = 0.3

s 1 0.99 0.9 0.7 0.5 0.3
‖u‖1 0.652287 0.660978 0.725085 0.842018 0.921431 0.960814

Figure 5. Bifurcation diagrams for f(t) = t(1 − t), numerical
positive solutions, and L1 norms of solutions with λ = 55



EJDE-2020/81 FRACTIONAL LAPLACIAN EQUATIONS 23

Example 7.8. Logistic reaction with constant yield harvesting f(t) = t(1−t)− .05
for t ≥ 0 (corresponding to Theorem 2.9).

For the Laplacian case (s = 1), the bifurcation diagram for the sublinear, semi-
positone problem was obtained in [13, Thm. 1.1 (B)] using the quadrature method.
For the fractional Laplacian, the bifurcation diagrams are given in the first column
of Figure 6 for (A) s = 0.99, (C) s = 0.7, (E) s = 0.5, and (G) s = 0.3 which retain
the qualitative behavior observed for s = 1. The solid part of the solution set S
contains positive solutions and the dashed part contains sign changing solutions.
On the solution set S, the markers ∆,©, and ∗ indicate the locations of a positive
solution, the last positive solution in the positive λ direction on the lower branch of
S, and a sign changing solution in (0, 1), respectively. The Locations of ∆ and ∗ are
chosen so that the L∞ norms of the solutions corresponding to these locations are
approximately same but always greater than the one for the solution corresponding
to ©.

The second column in Figure 6 shows three numerical solutions corresponding
to the location of ∆, © and ∗ on S for s ∈ (0, 1) corresponding to the bifurcation
diagrams in the first column. The solution corresponding to ∆ is given by a solid
line, the solution corresponding to© is given by a long dashed line, and the solution
corresponding to ∗ is given by a short dashed lines.

Based on the numerical experiments, we expect the following multiplicity and
uniqueness results for a general sublinear semipositone problem:

Conjecture 7.9. If f : [0,+∞) → R is such that f(0) < 0, eventually positive,
and satisfies (2.8), then there exist 0 < λ∗ < λ∗ such that (1.1) has two positive
solutions for λ ∈ (λ∗, λ

∗) and a unique positive solution for λ > λ∗.

We remark that it is straightforward to show that there is no nonnegative weak
solution for λ sufficiently small. Indeed, it follows from the assumption of Con-
jecture 7.9 that there exists a > 0 such that f(t) ≤ a t for all t ≥ 0. Let u be a
nonnegative solution of (1.1). Taking ϕ1 > 0 as test function in the definition of
weak solution, we obtain

λ1

∫
Ω

uϕ1 dx = E(u, ϕ1) = λ

∫
Ω

f(u)ϕ1 dx ≤ λ a
∫

Ω

uϕ1 dx ,

a contradiction if λ < λ1

a .

Example 7.10. Weighted logistic problem f(x, t) = t(q(x) − t) for t ≥ 0 and
x ∈ (0, 1) (q satisfying the hypothesis of Theorem 2.8).

For numerical experiments, we consider three specific cases of q. Namely, let
q1, q2, q3 : [0, 1]→ {0, 1} be given by the following:

(1) q1(x) = 1 for x ∈ [6/10, 7/10) and q1(x) = 0 otherwise,
(2) q2(x) = 1 for x ∈ [0, 1/5) ∪ [2/5, 4/5) and q2(x) = 0 otherwise,
(3) q3(x) = 1 for x ∈ [1/5, 2/5) ∪ [3/5, 4/5) and q3(x) = 0 otherwise.

If we consider nonlinearities f(x, t) that have discontinuities at finitely many points
on [0, 1], with q1 , q2 , q3 above, we partition the interval [0, 1] in such a way that the
possible points of discontinuity occur at xj with j ∈ {1, . . . , n}. In order to compute
the integral on the right hand side of (7.5), we modify the Hölder type assumption
(7.6) for any t1, t2 ≥ 0 to local type as follows. For any y1, y2 ∈ (xj−1, xj),

|f(y2, t2)− f(y1, t1)| ≤ L(|y2 − y1|s + |t2 − t1|) (7.8)



24 M. CHHETRI, P. GIRG, E. HOLLIFIELD EJDE-2020/81

(A) s = 0.99 (B) 3 solutions

(C) s = 0.7 (D) 3 solutions

(E) s = 0.5 (F) 3 solutions

(G) s = 0.3 (H) 3 solutions

Figure 6. Bifurcation diagrams for f(t) = t(1− t)− .05 and three
numerical solutions with λ corresponding to ∆, ©, and ∗

for all j = 1, . . . , n. We compute the integral on the right hand side of (7.5) as∫ 1

0

f (x, uh(x))φj(x) dx



EJDE-2020/81 FRACTIONAL LAPLACIAN EQUATIONS 25

=
(∫ xj

xj−1

+

∫ xj+1

xj

)
f (x, uh(x))φj(x) dx

=
h

2
[f
(xj−1 + xj

2
, uj
)

+ f
(xj + xj+1

2
, uj
)
]

+

∫ xj

xj−1

[f(x, uh(x))− f(
xj−1 + xj

2
, uh(xj))]φj(x) dx

+

∫ xj+1

xj

[f (x, uh(x))− f
(
xj + xj+1

2
, uh(xj)

)
]φj(x) dx

=
h

2
[f
(xj−1 + xj

2
, uj
)

+ f
(xj + xj+1

2
, uj
)
] +O(h1+s)

and proceed with the finite element scheme.
In Figures 7-9, (A) gives graph of qi(i = 1 , 2 , 3) and (B)-(F) show the bifurcation

diagrams for s = 0.99, 0.9, 0.7, 0.5 , 0.3 and the insets give numerical positive solu-
tions corresponding to the specified λ. The table at the end of each figure provides
comparison of the L1 norms of the positive solutions for s = 0.99, 0.9, 0.7, 0.5 and
0.3. We observe that on one hand, for the choices of the weight function q1 and
λ = 55, the L1 norms in Figure 7 attain maximal value for s between 0.7 and 0.3.
On the other hand, for the weight functions q2 and q3, the L1 norms in Figures 8-9
appear to be monotone for λ = 55 and λ = 25, respectively for s between 0.99 and
0.3.



26 M. CHHETRI, P. GIRG, E. HOLLIFIELD EJDE-2020/81

(A) graph of q1 (B) s = 0.99

(C) s = 0.9 (D) s = 0.7

(E) s = 0.5 (F) s = 0.3

s 0.99 0.9 0.7 0.5 0.3
‖u‖L1 0.106618 0.131814 0.1610100 0.156378 0.137046

Figure 7. Graph of q1, bifurcation diagrams for f(x, t) =
t(q1(x) − t), numerical positive solutions, and the L1 norms of
the solutions with λ = 55



EJDE-2020/81 FRACTIONAL LAPLACIAN EQUATIONS 27

(A) graph of q2 (B) s = 0.99

(C) s = 0.9 (D) s = 0.7

(E) s = 0.5 (F) s = 0.3

s 0.99 0.9 0.7 0.5 0.3
‖u‖L1 0.416029 0.463028 0.562381 0.617061 0.627710

Figure 8. Graph of q2, bifurcation diagrams for f(x, t) =
t(q2(x) − t), numerical positive solutions, and the L1 norms of
the solutions with λ = 55



28 M. CHHETRI, P. GIRG, E. HOLLIFIELD EJDE-2020/81

(A) graph of q3 (B) s = 0.99

(C) s = 0.9 (D) s = 0.7

(E) s = 0.5 (F) s = 0.3

s 0.99 0.9 0.7 0.5 0.3
‖u‖L1 0.122209 0.211938 0.357183 0.433505 0.449811

Figure 9. Graph of q3, bifurcation diagrams for f(x, t) =
t(q3(x) − t), numerical positive solutions, and the L1 norms of
the solutions with λ = 25



EJDE-2020/81 FRACTIONAL LAPLACIAN EQUATIONS 29

8. Appendix

We show that the norms generated by (2.1) and (2.2) are equivalent in dimension
N = 1 as well. Let Ω = (0, 1) ⊂ R.

Lemma 8.1. Norms generated by 〈·, ·〉Hs0 (0,1) and 〈·, ·〉Hs(R) are equivalent in Hs
0(0, 1).

Proof. Let s ∈ (0, 1) and v ∈ Hs
0(0, 1). Then,∫

R

∫
R

|v(x)− v(y)|2

|x− y|1+2s
dxdy = ‖v‖2Hs0 (0,1) ≤ ‖v‖

2
Hs(R)

=

∫ 1

0

|v(x)|2 dx+

∫
R

∫
R

|v(x)− v(y)|2

|x− y|1+2s
dxdy .

To establish the reverse inequality, we compute the integral below using v = 0 in
R \ (0, 1),∫

R

∫
R

|v(x)− v(y)|2

|x− y|1+2s
dxdy

=

∫
R

∫ 1

0

|v(x)− v(y)|2

|x− y|1+2s
dxdy +

∫
R

∫
R\(0,1)

|v(x)− v(y)|2

|x− y|1+2s
dxdy

=
[ ∫ 1

0

∫ 1

0

+

∫
R\(0,1)

∫ 1

0

+

∫
R\(0,1)

∫
R\(0,1)

+

∫ 1

0

∫
R\(0,1)

] |v(x)− v(y)|2

|x− y|1+2s
dxdy

=

∫ 1

0

∫ 1

0

|v(x)− v(y)|2

|x− y|1+2s
dxdy + 2

∫ 1

0

|v(y)|2 y
−2s + (1− y)−2s

2s
dy

≥
∫ 1

0

|v(y)|2ω(y) dy

where ω(y) := y−2s+(1−y)−2s

s . Letting B :=
(

miny∈(0,1) ω(y)
)−1

> 0, we obtain∫ 1

0

|v(y)|2 dy ≤ B
∫
R

∫
R

|v(x)− v(y)|2

|x− y|1+2s
dxdy .

Then,

‖v‖Hs(R) ≤ (1 +B)1/2‖v‖Hs0 (0,1)

as desired. Hence, the two norms are equivalent in Hs
0(0, 1). �

Acknowledgements. M. Chhetri would like to acknowledge the project LO1506
of the Czech Ministry of Education, Youth and Sports for supporting her visit at
the research centre NTIS - New Technologies for the Information Society of the
Faculty of Applied Sciences, University of West Bohemia. P. Girg was supported
by the Grant Agency of the Czech Republic Project No. 18-03253S. All authors
wish to express their gratitude to Prof. John M. Neuberger for the insight and
encouragement which lead to the completion of numerical experiments of the paper.

References

[1] N. Abatangelo; Large S-harmonic functions and boundary blow-up solutions for the fractional
Laplacian, Disc. Contin. Dyn. Syst., 35 (2015), no. 12, 5555–5607.

[2] G. Acosta, J. P. Borthagaray; A fractional Laplace equation: regularity of solutions and finite

element approximations, SIAM J. Numer. Anal., 55 (2017), no. 2, 472–495.



30 M. CHHETRI, P. GIRG, E. HOLLIFIELD EJDE-2020/81

[3] W. Alt, G. Hoffmann; Biological motion: Proceedings of a workshop held in Königswinter,

Germany, March 16-–19, 1989, Lecture Notes in Biomathematics 89, Springer-Verlag Berlin

Heidelberg, 1990.
[4] J. Appell, P. P. Zabrejko; Nonlinear superposition operators, Cambridge Tracts in Mathe-

matics, vol. 95, Cambridge University Press, Cambridge, 1990.

[5] J.-H. Bae, Y.-H. Kim; Multiple solutions for discontinuous elliptic problems involving the
fractional Laplacian, Electron. J. Diff. Eq., 2019 (2019), no. 18, 1–16.

[6] A. Bahrouni; Comparison and sub-supersolution principles for the fractional p(x)-Laplacian,

J. Math. Anal. Appl., 458 (2018), no. 2, 1363–1372.
[7] J. Bebernes, D. Eberly; Mathematical problems from combustion theory, Applied Mathemat-

ical Sciences, vol. 83, Springer-Verlag, New York, 1989.

[8] U. Biccari, V. Hernández-Santamaŕıa; A finite element approximation of the one-dimensio-
-nal fractional Poisson equation with applications to numerical control, arXiv preprint

arXiv:1707.06769 (2017).
[9] U. Biccari, V. Hernández-Santamaŕıa; The Poisson equation from non-local to local, Electron.

J. Diff. Eq., 2018 (2018), no. 145, 1–13.

[10] D. Brockmann, L. Hufnagel; Front propagation in reaction-superdiffusion dynamics: Taming
Lévy flights with fluctuations, Phy. Rev. Lett. 98 (2007), no. 17, 178301.

[11] K. J. Brown, M. M. A. Ibrahim, R. Shivaji; S-shaped bifurcation curves, Nonlinear Anal., 5

(1981), no. 5, 475–486.
[12] L. Caffarelli, S. Dipierro, E. Valdinoci; A logistic equation with nonlocal interactions, Kinet.

and Relat. Models 10 (2017), no. 1, 141–170.

[13] A. Castro, R. Shivaji; Nonnegative solutions for a class of nonpositone problems, Proc. Roy.
Soc. Edin. Sect. A, 108 (1988), no. 3-4, 291–302.

[14] P. Clément, G. Sweers; Getting a solution between sub- and supersolutions without monotone

iteration, Rend. Istit. Mat. Univ. Trieste, 19 (1987), no. 2, 189–194.
[15] E. Di Nezza, G. Palatucci, E. Valdinoci; Hitchhiker’s guide to the fractional Sobolev spaces,

Bull. Sci. Math., 136 (2012), no. 5, 521–573.
[16] Y. Fang, D. Tang; Method of sub-super solutions for fractional elliptic equations, Disc. Con-

tin. Dyn. Syst. 23 (2018), no. 8, 3153–3165.

[17] J. Giacomoni, T. Mukherjee, K. Sreenadh; Existence of three positive solutions for a nonlocal
singular Dirichlet boundary problem, Adv. Nonlinear Stud. 19 (2019), no. 2, 333–352.

[18] H. B. Keller, D. S. Cohen; Some positone problems suggested by nonlinear heat generation,

J. Math. Mech. 16 (1967), 1361–1376.
[19] M. Kwaśnicki; Eigenvalues of the fractional Laplace operator in the interval, J. Func. Anal.

262 (2012), no. 5, 2379–2402.

[20] T. Leonori, I. Peral, A. Primo, F. Soria; Basic estimates for solutions of a class of nonlocal
elliptic and parabolic equations, Disc. Contin. Dyn. Syst., 35 (2015), no. 12, 6031–6068.

[21] M. Levandowsky, B. S. White, F. L. Schuster; Random movements of soil amebas, Acta

Protozool., 36 (1997), 237–248.
[22] P.-L. Lions; On the existence of positive solutions of semilinear elliptic equations, SIAM

Rev., 24 (1982), no. 4, 441–467.
[23] R. Metzler, J. Klafter; The random walk’s guide to anomalous diffusion: a fractional dy-

namics approach, Phy. Rep., 339 (2000), no. 1, 1–77.

[24] G. Molica Bisci, V.D. Radulescu, R. Servadei; Variational methods for nonlocal fractional
problems, Encyclopedia of Mathematics and its Applications, 162, vol. 162, Cambridge Uni-

versity Press, Cambridge, 2016, With a foreword by Jean Mawhin.
[25] J. M. Neuberger, N. Sieben, J.W. Swift; Symmetry and automated branch following for a

semilinear elliptic PDE on a fractal region, SIAM J. Appl. Dyn. Syst. 5 (2006), no. 3, 476–

507.

[26] S. Oruganti, J. Shi, R. Shivaji; Diffusive logistic equation with constant yield harvesting. I.
Steady states, Trans. Amer. Math. Soc. 354 (2002), no. 9, 3601–3619.

[27] C. Pozrikidis; The fractional Laplacian, CRC Press, Boca Raton, FL, 2016.
[28] X. Ros-Oton; Nonlocal elliptic equations in bounded domains: a survey, Publ. Mat., 60

(2016), no. 1, 3–26.

[29] X. Ros-Oton, J. Serra; The Dirichlet problem for the fractional Laplacian: regularity up to

the boundary, J. Math. Pures Appl., (9) 101 (2014), no. 3, 275–302.



EJDE-2020/81 FRACTIONAL LAPLACIAN EQUATIONS 31

[30] P. R. Stinga, J. L. Torrea; Extension problem and Harnack’s inequality for some fractional

operators, Comm. Par. Diff. Eqns., 35 (2010), no. 11, 2092–2122.

[31] V. V. Uchaikin; Fractional phenomenology of cosmic ray anomalous diffusion, Physics-
Uspekhi, 56 (2013), no. 11, 1074–1119.

Maya Chhetri

Department of Mathematics and Statistics, The University of North Carolina at Greens-

boro, NC 27402, USA
Email address: maya@uncg.edu

Petr Girg
Department of Mathematics, Faculty of Applied Sciences, University of West Bohemia,

Univerzitńı 8, 30100 Plzeň, Czech Republic.

NTIS - European Centre of Excellence, Faculty of Applied Sciences, University of
West Bohemia, Univerzitńı 8, 30100 Plzeň, Czech Republic

Email address: pgirg@kma.zcu.cz

Elliott Hollifield

Department of Mathematics and Statistics, The University of North Carolina at Greens-

boro, NC 27402 USA
Email address: ezhollif@uncg.edu


	1. Introduction
	2. Preliminaries and statement of results
	3. Auxiliary problems
	4. Proof of Theorem 2.3
	5. Proofs of Theorems 2.5 and 2.6
	6. Proofs of Theorems 2.8 and 2.9
	7. Finite element method for the fractional Laplacian in one dimension
	7.1. Discretization
	7.2. Construction of bifurcation diagrams

	8. Appendix
	Acknowledgements

	References