Nonexistence Results For The Elliptic Equations With
Fractional Laplacian and Variable Exponents Nonlinearities
MOHAMMED YAHIAOUI, ALI HAKEM
Department of technology, Laboratory ACEDP,
Djillali Liabes university,
P. O. Box 89, 22000 Sidi Bel Abbes,
ALGERIA
Abstract: The main motivation behind this paper is to investigate the nonexistence of weak solution for
the following Cauchy problem (−∆)γ
2wl+ ∆wl=|w|m(x), x ∈IRn,
where γ∈(0,2), l ≥1,m:IRn→(1,+∞)is a measurable function, and (−∆)γ
2is the fractional
Laplacian operator of order γ
2. Then, this result is extended to the case of 2×2-system of the same type.
The proof of our results is based on a contradiction argument by using the so-called test function method.
The results obtained in this paper extend several contributions and we focus on new nonexistence result
which is due to the presence of variable-exponents
Key-Words: - Variable-exponent nonlinearities, nonexistence, test functions, fractional Laplacian, elliptic
equations, blow-up .
Received: October 13, 2023. Revised: May 11, 2024. Accepted: July 2, 2024. Published: July 29, 2024.
1 Introduction
In this paper, we are rst concerned with the
nonexistence of weak solutions for the following
Cauchy problem
(−∆)γ
2wl+ ∆wl=|w|m(x), x ∈IRn,(1)
where n≥1, l ≥1, γ ∈(0,2), m :IRn→
(1,+∞)are measurable functions and (−∆)γ
2is
the fractional Laplacian operator of order γ
2. Then
we extend our analysis to the 2×2system of the
same type
(−∆)γ
2wl+ ∆wl=|ϑ|k(x), x ∈IRn,
(−∆)β
2ϑl+ ∆ϑl=|w|m(x), x ∈IRn.
(2)
Let us start by noting some well-known problems
related to (1) and (2). Firstly, [1], investigated the
nonexistence of local solutions to the correspond-
ing problems:
(−∆)γ
2w+λ∆w=|w|m(x), x ∈IRn,(3)
(−∆)γ
2w+λ∆w=|ϑ|k(x), x ∈IRn,
(−∆)β
2ϑ+µ∆ϑ=|w|m(x), x ∈IRn.
(4)
Namely, it was shown that
1. if 1< m−≤m+< mcwhere
mc=
∞if n≤γ,
n
n−γif n > γ,
then the problem (3) admits no nontrivial lo-
cal weak solution.
2. If n < m+k+
m+k+−1max β+γ
k+, γ +β
m+
then (4) admits no nontrivial weak solution.
Nonlocal operators have been receiving increased
attention in recent years due to their usefulness
in modeling complex systems with long-range in-
teractions or memory eects, which cannot be de-
scribed properly via standard dierential opera-
tors.
To overcome the diculty caused by the nonlo-
cal property of the fractional Laplacian operator,
[2], introduced an extension method which con-
sists of localizing the fractional Laplacian by con-
structing a Dirichlet to Neumann operator of a
degenerate elliptic equation. More precisely, [3],
established a nonexistence result for the following
equation
(−∆)γ
2w=wm, x ∈IRn,
w≥0, x ∈IRn.
(5)
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It was shown that, if 1≤γ < 2and 1< m < n+γ
n−γ
, then 5 has no nontrivial bounded solution. In
[4], the authors studied the fractional Lane-Emden
system
(−∆)γ
2w=ϑk, x ∈IRn,
(−∆)γ
2ϑ=wm, x ∈IRn,
w≥0, x ∈IRn
ϑ≥0, x ∈IRn,
(6)
where 0< γ < 2, n > γ and m, k > 0. Using
the method of moving planes, it was shown that,
if mk > 1,
α(k+ 1)
mk −1,γ(m+ 1)
mk −1∈n−γ
2, n −γ,
and
γ(k+ 1)
mk −1,γ(m+ 1)
mk −1=n−γ
2,n−γ
2,
then for some σ > 0, there exists no positive solu-
tion to 6 in Xσ,σ(IRn), where
Xσ,σ(IRn) =
Cγ+σ(IRn)if 0< γ < 1,
C1,γ+σ−1(IRn)if 1≤γ < 2.
For other contributions related to inhomogeneous
problems, see, [5], [6], [7], for example and the ref-
erences therein. Motivated by the above works,
the main objective of this paper is to obtain the
nonexistence of weak solutions for problems (1)
and (2). Before stating our main results, we in-
troduce some important fundamental preliminary
knowledge that will be needed for obtaining our
results in the next sections.
Denition 1.1 ([7], [8]) Let δ∈(0,1). Let Ybe
a suitable set of functions dened on IRn. Then,
the fractional Laplacian (−∆)δin IRnis a non-
local operator given by
(−∆)δ:χ∈ Y → (−∆)δχ(x),
where
Cn,δ P.V ZIRn
χ(x)−χ(y)
|x−y|n+2δdy,
as long as the right-hand side exists, and P.V
stands for the Cauchy’s principal value and
Cn,δ =4δΓn
2+δ
πn
2Γ(−δ)
is the normalization constant and Γdenotes the
Gamma function.
It will be assumed throughout the paper that the
exponents k(x)and m(x)are continuous in IRn
such that
1< k−=ess inf
x∈Ωk(x)≤k(x)≤k+
=ess sup
x∈Ω
k(x)<∞,
(7)
1< m−=ess inf
x∈Ωm(x)≤m(x)≤m+
=ess sup
x∈Ω
m(x)<∞.
(8)
Therefore, we introduce the variable exponent
Lebesgue space Lm(x)(IRn)dened by
Lm(x)(Ω) = w(x) : wis measurable in IRn,
Bm(.)(w) = ZΩ
|w(x)|m(x)dx < ∞,
with the following Luxembourg-type norm
∥w∥m(.)= inf ξ > 0, Bm(.)(w/ξ)≤1.
Lemma 1.1 ([9]) For ∈Lm(x)(Ω), the following
relations hold:
1. ∥w∥m(.)<1(= 1; >1) ⇐⇒
Bm(.)(w)<1(= 1; >1).
2. If ∥w∥m(.)<1then
∥w∥m+
m(.)≤Bm(.)(w)≤ ∥w∥m−
m(.).
3. If ∥w∥m(.)>1then
∥w∥m+
m(.)≥Bm(.)(w)≥ ∥w∥m−
m(.).
4. ∥w∥m(.)→0⇐⇒ Bm(.)(w)→0.
5. ∥w∥m(.)→ ∞ ⇐⇒ Bm(.)(w)→ ∞.
6. minBm(.)(g)1
m−, Bm(.)(g)1
m+≤ ∥g∥m(.).
7. ∥g∥m(.)≤maxBm(.)(g)1
m−, Bm(.)(g)1
m+.
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2 Main Results
2.1 Case of the single equation
Denition 2.1 We say that wis a weak solution
of (1) if
w∈IL2l(IRn)∩IL2m(.)(IRn),
and ZIRn
w(x)l(−∆)γ
2φ(x)dx
+ZIRn
w(x)l∆φ(x)dx
≥ZIRn
|w(x)|m(x)φ(x)dx,
(9)
holds for all φ∈H2(IRn), φ ≥0.
Theorem 2.1 If
1< m−≤m+< mc,(10)
where
mc=
∞if n≤γ,
nl
n−γif n < γ,
(11)
then problem (1) has no nontrivial weak solutions.
The following Lemma is crucial in the proof of
Theorem 2.1.
Lemma 2.1 ([10]) Let δ∈(0,1) and µ:IRn→
(0,∞)be the function dened by
µ(x) = ⟨x⟩−ρ,⟨x⟩=1 + |x|21
2, x ∈IRn,(12)
where n<ρ≤n+ 2δ. Then µ∈IL1(IRn)∩
H2(IRn), and the following estimate holds
|(−∆)δµ(x)| ≤ Kµ(x), x ∈IRn,(13)
where Kis a constant independent of x.
Proof. [Theorem 2.1] Let ube a weak solution
to (1), then for all φ∈H2(IRn), one has
ZIRn
|w(x)|m(x)φ(x)dx
≤ZIRn
|w(x)|l∆φ(x)dx
+ZIRn
|w(x)|l|(−∆)γ
2φ(x)|dx.
(14)
Writing
|w(x)|l|(−∆)γ
2φ(x)|=φ(x)1
j(x)|w(x)|l
×φ(x)
−1
j(x)|(−∆)γ
2φ(x)|,
and invoking Young’s inequality, it holds that
|w(x)|l|(−∆)γ
2φ(x)|
≤1
j(x)φ(x)1
j(x)|w(x)|lj(x)
+1
j′(x)φ(x)
−1
j(x)|(−∆)γ
2φ(x)|j′(x),
(15)
where 1
j(x)+1
j′(x)= 1,and j(x) = m(x)
l. From
(15) we get
|w(x)|l|(−∆)γ
2φ(x)| ≤ m
m−|w(x)|m(x)φ(x)
+m+−l
p+φ(x)−l
m(x)−l|(−∆)γ
2φ(x)|
m(x)
m(x)−l.
Similarly, one obtains
|w(x)|l|∆φ(x)| ≤ l
m−|w(x)|m(x)φ(x)
+m+−l
p+φ(x)−l
m(x)−l|∆φ(x)|
m(x)
m(x)−l.
(16)
From (10), one can check easily that
ZIRn
|w(x)|m(x)φ(x)dx
≤2l
m−ZIRn
|w(x)|m(x)φ(x)dx
+m+−l
m+×
ZIRn
φ(x)−l
m(x)−l|(−∆)γ
2φ(x)|
m(x)
m(x)−ldx
+m+−l
m+×
ZIRn
φ(x)−l
m(x)−l|∆φ(x)|
m(x)
m(x)−l.
(17)
Throughout, Kdenotes a positive constant, whose
value may change from line to line. It follows from
(17) that
ZIRn
|w(x)|m(x)φ(x)dx ≤K(I(φ) + J(φ)) ,(18)
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where
I(φ) = ZIRn
φ(x)−l
m(x)−l|(−∆)γ
2φ(x)|
m(x)
m(x)−ldx,
J(φ) = ZIRn
φ(x)−l
m(x)−l|∆φ(x)|
m(x)
m(x)−l.
Now, for R > 1, we introduce the function
φ(x) = µx
R, x ∈IR,
where θis the function dened by (12) with ρ=
n+γ. At this stage, we use the change of variable
˜x=R−1x. A straight-forward calculation gives
I(φ)≤RnR
−γm+
m+−lZIRn
φ(˜x)d˜x
≤KRn−γm+
m+−l.
(19)
Similarly, one has
J(φ)≤RnR
−2m+
m+−lZIRn
φ(˜x)d˜x
≤KRn−2m+
m+−l.
(20)
Using the formula (18), one deduces that
ZIRn
|w(x)|m(x)φR(x)dx ≤KRn−γm+
m+−l+Rn−2m+
m+−l.
It follows from (10) by passing to the limit in the
above inequality as Rgoes to ∞, that
lim
R→∞ ZIRn
|w(x)|m(x)φR(x)dx = 0.
Using the Lebesgue dominated convergence theo-
rem and the fact that φR(x)→1as R→ ∞, we
arrive at
Bm(.)(w) = ZΩ
|w(x)|m(x)dx = 0,
which implies that ∥w∥m(.)= 0. We conclude
that wis the trivial solution. This completes the
proof.
2.2 Case of the system
Denition 2.2 We say that (w, ϑ)is a weak so-
lution of (2) if
(w, ϑ)∈IL2l(IRn)∩IL2m(.)(IRn)×IL2l(IRn)∩IL2k(.)(IRn),
and the following formulations hold
ZIRn
ϑ(x)l(−∆)β
2φ(x)dx
+ZIRn
ϑ(x)l∆φ(x)dx
≥ZIRn
|w(x)|m(x)φ(x)dx,
(21)
and ZIRn
w(x)l(−∆)γ
2φ(x)dx
+ZIRn
w(x)l∆φ(x)dx
≥ZIRn
|ϑ(x)|k(x)φ(x)dx,
(22)
for all φ∈H2(IRn), φ ≥0.
Theorem 2.2 Assume that m+k+−l2>0. If
n < m+k+
m+k+−l2maxβ+lγ
k+, γ +lβ
m+,(23)
then the only weak solution of (2) is the trivial
one,ie (w, ϑ) = (0,0).
Proof. Let (w, ϑ)∈IL2l(IRn)∩IL2m(.)(IRn)×
IL2l(IRn)∩IL2k(.)(IRn)be a weak solution to (2).
By (21) and (22), one has
Z=ZIRn
|ϑ(x)|k(x)φ(x)dx
≤ZIRn
|w(x)|l|∆φ(x)|dx
+ZIRn
|w(x)|l|(−∆)γ
2φ(x)|dx,
(24)
and
W=ZIRn
|w(x)|m(x)φ(x)dx
≤ZIRn
|ϑ(x)|l|(−∆)β
2φ(x)|dx
+ZIRn
|ϑ(x)|l|∆φ(x)|dx.
(25)
Invoking the Holder’s inequality, it follows imme-
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diately
ZIRn
|w(x)|l|(−∆)γ
2φ(x)|dx
≤ZIRn
|w(x)|lj+φ(x)dx1
j+
+ZIRn
|w(x)|j−φ(x)dx1
j−
×
ZIRn
φ(x)−1
j−−1
|(−∆)γ
2φ(x)|j−
j−−1dxj−−1
j−
=Wl
m+ZIRn
φ(x)−l
m+−l
|(−∆)γ
2φ(x)|m+
m+−ldxm+−l
m+
+Wl
p−ZIRn
φ(x)−l
m−−l
|(−∆)γ
2φ(x)|m−
m−−ldxm−−l
m−
.
(26)
Now we dene the functional for σ∈(0,2], r >
1, ϕ > 0, ω ∈H2(IRn)as follows:
G(s, ϱ, ω) = ZIRn
ω(x)−l
s−l|(−∆)ϱ
2ω(x)|s
s−l
Then from (26) we get
ZIRn
|w(x)|l|(−∆)γ
2φ(x)|dx
= [G(m+, γ, φ)]m+−l
p+Wl
m+
+ [G(m−, γ, φ)]m−−l
m−Wl
m−.
(27)
In the analogous way, one obtains
ZIRn
|ϑ(x)|l|(−∆)β
2φ(x)|dx
= [G(q+, β, φ)]k+−l
k+Zl
k+
+ [G(k−, β, ϕ)]k−−l
k−Zl
k−,
(28)
ZIRn
|w(x)|l|∆φ(x)|dx
= [G(2, m+, φ)]m+−l
m+Wl
m+
+ [G(2, m−, ϕ)]m−−1
m−Wl
l−,
(29)
ZIRn
|ϑ(x)|l|∆φ(x)|dx
= [G(2, k+, φ)] k+−l
k+Zl
k+
+ [G(k−,2, φ)]k−−l
k−Zl
k−.
(30)
Next, it follows from (24)-(30) that
W≤E(ϕ)Zm
k++F(ϕ)Zl
k−,
Z≤e
E(ϕ)Wm
m++e
F(ϕ)Wl
m−,(31)
where
E(φ) = [G(k+, β, φ)]k+−l
k++ [G(2, k+, φ)] k+−l
k+,
F(φ) = [G(k−, β, φ)]k−−l
k−+ [G(2, k−, φ)] k−−l
k−,
and
e
E(φ) = [G(m+, α, φ)]m+−l
m++ [G(2, m+, φ)]m+−l
m+,
e
F(φ) = [G(m−, α, φ)]m−−l
m−+ [G(2, m−, φ)]m−−l
m−.
From (31), one deduces that
W≤E(φ)e
E(φ)Wl
m+
+e
F(φ)Wl
m−l
k+
+F(φ)he
E(φ)Wm
m++e
F(φ)Wm
m−il
k−,
Z≤e
E(φ)hE(φ)Zl
k++F(φ)Zl
k−il
m+
+e
F(φ)hE(φ)Zl
k++F(φ)Zl
k−il
m−.
(32)
Applying the following inequality
(a+b)m≤2m(am+bm); a, b ≥0, m > 0,
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we get easily from (32) that
W≤KE(φ)[ e
E(φ)] l
k+Wl2
m+k+
+E(φ)[ e
F(φ)] l
k+Wl2
p−q+
+F(ϕ)[ e
E(φ)] l
k−Wl2
m+k−
+F(φ)[ e
F(φ)] l
k−Wm2
m−k−,
Z≤Ke
E(φ)[E(φ)] l
m+Zl2
k+m+
+e
E(φ)[F(φ)] l
m+Zl2
k−m+
+e
F(ϕ)[E(φ)] l
m−Zm2
k+m−
+e
F(φ)[F(φ)] m
m−Zl2
k−m−.
(33)
Using the ε−Young inequality, one has
E(ϕ)[ e
E(φ)] l
k+Wl2
m+k+≤εW
+KE(φ)[ e
E(φ)] m
k+m+k+
m+k+−l2
,
E(ϕ)[ e
F(φ)] l
k+Wl2
m−k+≤εW
+KE(φ)[ e
F(φ)] m
k+m−k+
m−k+−l2
,
F(ϕ)[ e
E(φ)] l
k−Wl2
m+k−≤εW
+KF(φ)[ e
E(φ)] m
k−m+k−
m+k−−l2
,
F(ϕ)[ e
F(φ)] l
k−Wl2
m−k−≤εW
+KF(φ)[ e
F(φ)] m
k−m−k−
m−k−−l2
,
(34)
and
e
E(ϕ)[E(φ)] l
m+Zl2
k+m+≤εZ
+Ke
E(φ)[E(φ)] m
m+k+m+
k+m+−l2
,
e
E(ϕ)[F(φ)] l
m+Zl2
k−m+≤εZ
+Ke
E(φ)[F(φ)] m
m+k−m+
k−m+−l2
,
e
F(ϕ)[E(φ)] l
m−Zl2
k+m−≤εZ
+Ke
F(φ)[E(φ)] m
m−k+m−
k+m−−l2
,
e
F(ϕ)[F(φ)] l
m−Zl2
k−m−≤εZ
+Ke
F(φ)[F(φ)] m
m−k−m−
k−m−−l2
.
(35)
Moreover, it follows from (34)-(35) that
W≤KE(φ)[ e
E(ϕ)] l
k+m+k+
m+k+−l2
+E(ϕ)[ e
F(ϕ)] l
k+m−k+
m−k+−l2
+F(φ)[ e
E(φ)] l
k−m+k−
m+k−−l2
+F(φ)[ e
F(φ)] l
k−m−k−
m−k−−l2,
Z≤Ke
E(φ)[E(ϕ)] l
m+k+m+
k+m+−l2
×e
E(φ)[F(φ)] m
m+k−m+
k−m+−l2
+e
F(φ)[E(φ)] m
m−k+m−
k+m−−l2
+e
F(φ)[F(φ)] m
m−k−m−
k−m−−l2.
(36)
We choose φas in Lemma 1.1 with ρ=n+
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min{γ, β}. We have the estimates
E(φ)≤KRnk+−l
k+−β,
F(φ)≤KRnk−−l
k−−β,
(37)
and
e
E(φ)≤KRnm+−l
m+−γ;
e
F(φ)≤KRnm−−l
m−−γ.
(38)
Finally, using (37)-(38) and the estimates (36), one
obtains
W≤K(Rζ1+Rζ2+Rζ3+Rζ4),(39)
Z≤K(Rλ1+Rλ2+Rλ3+Rλ4),(40)
where
ζ1=m+k+
m+k+−l2hnm+k+−l2
m+k+−β−lγ
k+i,
ζ2=m−k+
m−k+−l2hnm−k+−l2
m−k+−β−lγ
k+i,
ζ3=m+k−
m+k−−l2hnm+k−−l2
m+k−−β−lγ
k−i,
ζ4=m−k−
m−k−−l2hnm−k−−l2
m−k−−β−lγ
k−i,
and
λ1=k+m+
k+m+−l2hnm+k+−l2
m+k+−γ−lβ
m+i,
λ2=k−m+
k−m+−l2hnk−m+−l2
k−m+−γ−lβ
m+i,
λ3=k+m−
k+m−−l2hnk+m−−l2
k+m−−γ−lβ
m−i,
λ4=k−m−
k−m−−l2hnk−m−−l2
k−m−−γ−lβ
m−i.
Moreover, one has
ζ1= maxnζ1, ζ2, ζ3, ζ4o,
λ1= maxnλ1, λ2, λ3, λ4o.
(41)
1. If ζ1<0, the right hand side of (39) goes to
0, as Rgoes to innity, one can easily see that
Bm(.)(w) = ZΩ
|w(x)|m(x)dx = 0,
which implies that ∥w∥m(.)= 0 ie w= 0.
Then by 24 one obtains Y= 0 which yields
ϑ= 0. Hence (w, ϑ) = (0,0).
2. If λ1<0, the right hand side of (40) goes to
0, as Rgoes to innity, one obtains
Bm(.)(ϑ) = ZΩ
|ϑ(x)|k(x)dx = 0,
which implies that ∥ϑ∥p(.)= 0 ie ϑ= 0. Then
by 25 one obtains W= 0 which yields w= 0.
Hence (w, ϑ) = (0,0). Then, one can check
easily that under the condition 23, the only
weak solution to 2 is (w, ϑ) = (0,0). This
completes the proof of Theorem 2.2.
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