Hindawi Publishing Corporation
Abstract and Applied Analysis
Volume 2010, Article ID 435083, 10 pages
doi:10.1155/2010/435083
Research Article
Existence and Nonexistence Results for Classes of
Singular Elliptic Problem
Peng Zhang and Jia-Feng Liao
Department of Mathematics, Zunyi Normal College, Zunyi 563002, China
Correspondence should be addressed to Peng Zhang, gzzypd@sina.com
Received 14 January 2010; Accepted 19 July 2010
Academic Editor: Norimichi Hirano
Copyright q 2010 P. Zhang and J.-F. Liao. This is an open access article distributed under the
Creative Commons Attribution License, which permits unrestricted use, distribution, and
reproduction in any medium, provided the original work is properly cited.
The singular semilinear elliptic problem −Δu kxu−γ λup in Ω, u > 0 in Ω, u 0 on ∂Ω, is
α
Ω ∩ CΩ, and
considered, where Ω is a bounded domain with smooth boundary in RN , k ∈ Cloc
γ, p, λ are three positive constants. Some existence or nonexistence results are obtained for solutions
of this problem by the sub-supersolution method.
1. Introduction and Main Results
In this paper, we study the existence or the nonexistence of solutions to the following singular
semilinear elliptic problem
−Δu kxu−γ λup ,
u > 0,
u 0,
in Ω,
in Ω,
1.1
on ∂Ω,
where Ω ⊂ RN N ≥ 1 is a bounded domain with C2α boundary for some α ∈ 0, 1,
α
k ∈ Cloc
Ω ∩ CΩ, and γ, p, and λ are three nonnegative constants. This problem arises
in the study of non-Newtonian fluids, chemical heterogeneous catalysts, in the theory of heat
conduction in electrically conducting materials see 1–7 and their references.
Many authors have considered this problem. For examples, when kx < 0 in Ω,
problem 1.1 was studied in 3, 8–11; when kx > 0 in Ω, problem 1.1 was considered in
12–14. Particularly, when kx ≡ 1, it has been established in Zhang 14 that there exists
λ > 0 such that problem 1.1 has at least one solution in C2α Ω ∩ CΩ for all λ > λ and
2
Abstract and Applied Analysis
has no solution in C2 Ω ∩ CΩ if λ < λ. After that Shi and Yao in 13 have also obtained
the same results with k ∈ C2,α Ω and kx > 0 in Ω. Recently, Ghergu and Rădulescu in 12
considered more general sublinear singular elliptic problem with k ∈ Cα Ω.
α
In this paper, we consider the case that k ∈ Cloc
Ω ∩ CΩ, and k may have zeros in
Ω. The following main results are obtained by the sub-supersolution method with restriction
on the boundary in Cui 15.
α
Ω ∩ CΩ, k ≥ 0, and k /
0. Assume that 0 < γ < 1 and
Theorem 1.1. Suppose that k ∈ Cloc
0 < p < 1, then there exists λ ∈ 0, ∞ such that problem 1.1 has at least one solution uλ ∈
−γ
C2α Ω ∩ CΩ and uλ ∈ L1 Ω for all λ > λ, and problem 1.1 has no solution in C2 Ω ∩ CΩ
if λ < λ. Moreover, problem 1.1 has a maximal solution vλ which is increasing with respect to λ for
all λ > λ.
Remark 1.2. Theorem 1.1 generalizes Theorem 1.2 in 13 in coefficient kx of the singular
term. Consequently, it also generalizes Theorem 1 in 14. Moreover, there are functions k
satisfying our Theorem 1.1 and not satisfying Theorem 1.2 in 13. For example, let
⎧
⎨−
1
,
ln|x − x0 |/2d
kx ⎩
0,
x ∈ Ω \ {x0 },
x x0 ,
1.2
where x0 ∈ ∂Ω, and
Δ
d diamΩ max x − y | x, y ∈ Ω .
1.3
Certainly, this example does not satisfy Theorem 1.2 in 12 yet.
α
Ω ∩ CΩ and kx > 0 in Ω. If γ ≥ 1, problem 1.1 has no
Theorem 1.3. Suppose that k ∈ Cloc
2
solution in C Ω ∩ CΩ for all λ > 0 and p > 0.
Remark 1.4. Obviously, Theorem 1.3 is a generalization of Theorem 2 in 14. There are also
functions kx satisfying our Theorem 1.3 and not satisfying Theorem 2 in 14 and Theorem
1.1 in 12. For example, let
kx ⎧
⎨−
⎩
ε,
1
ε,
ln|x − x0 |/2d
x ∈ Ω \ {x0 },
x x0 ,
where x0 ∈ ∂Ω, ε is any positive constant and d diamΩ is the diameter of Ω.
1.4
Abstract and Applied Analysis
3
2. Proof of Theorems
Consider the more general semilinear elliptic problem
−Δu fx, u,
in Ω,
u > 0,
u 0,
in Ω,
2.1
on ∂Ω,
where the function fx, s is locally Hölder continuous in Ω × 0, ∞ and continuously
differentiable with respect to the variable s. A function u is called to be a subsolution of
problem 2.1 if u ∈ C2 Ω ∩ CΩ, and
−Δu ≤ f x, u ,
2.2
in Ω,
u > 0,
u 0,
in Ω,
on ∂Ω.
A function u is called to be a supersolution of problem 2.1 if u ∈ C2 Ω ∩ CΩ, and
−Δu ≥ fx, u,
2.3
in Ω,
u > 0,
u 0,
in Ω,
on ∂Ω.
According to Lemma 3 in the study of Cui 15, we can easily have the following basic
existence of classical solution to problem 2.1.
α
Lemma 2.1. Let f ∈ Cloc
Ω × 0, ∞ be continuously differentiable with respect to the variable s.
Suppose that problem 2.1 has a supersolution u and a subsolution u such that
ux ≤ ux,
in Ω,
2.4
then problem 2.1 has at least one solution u ∈ C2α Ω ∩ CΩ satisfying
ux ≤ ux ≤ ux,
in Ω.
2.5
Let λ1 be the first eigenvalue of the eigenvalue problem
−Δu λu,
u 0,
in Ω,
on ∂Ω,
2.6
and ϕ1 > 0 in Ω the corresponding eigenfunction. Then ϕ1 ∈ C2α Ω. Moreover one has the
following lemma.
4
Abstract and Applied Analysis
Lemma 2.2 see 10. One has
Ω
ϕr1 dx < ∞
2.7
if and only if r > −1.
Now we give the proof of our theorems.
Proof of Theorem 1.1. Let p ∈ 0, 1, and let u∗ denote the unique solution of
in Ω,
−Δu up ,
in Ω,
u > 0,
u 0,
2.8
on ∂Ω,
where u∗ belongs to C2 Ω see 16. Then u λ1/1−p u∗ is a solution of
−Δu λup ,
in Ω,
u > 0,
u 0,
in Ω,
2.9
on ∂Ω,
where 0 < p < 1 and λ > 0. Then fix λ > 0 and set
u λ1/1−p u∗ ,
2.10
thus we can easily obtain that u is a supersolution of problem 1.1.
Now, we want to find a subsolution of problem 1.1. Let
2/1γ
u Mϕ1
2.11
,
where M is a positive constant; now we will prove that u is a subsolution of problem 1.1.
By Hopf’s maximum principle in 17, there exist δ > 0 and ε0 > 0 such that
∇ϕ1 ≥ δ,
ϕ1 ≥ δ,
on Ω \ Ω ,
2.12
on Ω ,
Δ
where Ω {x ∈ Ω | distx, ∂Ω > ε0 }. On Ω , we choose M ≥ M1 k∞ 1 γ/λ1 δ2 1/1γ , then we have
kx
2γ/1γ
Mγ ϕ1
≤
λ1 M 2/1γ
ϕ
,
1γ 1
2.13
Abstract and Applied Analysis
5
Δ
where k∞ max{|kx| | x ∈ Ω} for k ∈ CΩ. On Ω \ Ω , we choose M ≥ M2 k∞ 1 γ2 /21 − γδ2 1/1γ , then one obtains
kx
2γ/1γ
Mγ ϕ1
2
2 1 − γ M∇ϕ1 ≤ 2 2γ/1γ .
1 γ ϕ1
2.14
Δ
Thus, we choose M ≥ max{M1 , M2 }, then fixing M, let λ > λ 3λ1 M1−p /1 21−p/1γ
, it follows from 2.13 and 2.14 that
γ ϕ1 ∞
−γ
−Δu kxuλ
2/1γ
−MΔϕ1
kx
2γ/1γ
Mγ ϕ1
2 −2γ/1γ
2 1−γ kx
2
1−γ/1γ
ϕ1
−M Δϕ1 2 ∇ϕ1 ϕ1
2γ/1γ
1γ
1γ
Mγ ϕ1
2
2 1 − γ M∇ϕ1 2λ1 M 2/1γ
kx
ϕ
− 2 2γ/1γ
2γ/1γ
1γ 1
Mγ ϕ
1γ ϕ
1
1
3λ1 M 2/1γ
ϕ
1γ 1
2/1γ p
≤ λ Mϕ1
≤
p
λuλ .
2.15
2/1γ
Thus we proved that u Mϕ1
is a subsolution of problem 1.1 for all λ > λ .
According to Lemma 4 in 14, there exists a positive constant C such that
ϕ1 x ≤ Cu∗ x,
Δ
1−γ/1γ 1−p
Set λ ≥ λ MCϕ1 ∞
2.16
in Ω.
, then we have
2/1γ
u λ1/1−p u∗ ≥ u Mϕ1
,
in Ω.
2.17
Thus we choose λ∗ max{λ , λ }; via Lemma 2.1, problem 1.1 has at least one solution
uλ ∈ C2α Ω ∩ CΩ and satisfying
ux ≤ uλ x ≤ ux,
for all λ ≥ λ∗ .
in Ω,
2.18
6
Abstract and Applied Analysis
2/1γ
Since uλ ≥ Mϕ1
one has
Ω
in Ω for all λ ≥ λ∗ and −2γ/1 γ > −1, according to Lemma 2.2
−γ
uλ xdx ≤
1
Mγ
−2γ/1γ
Ω
ϕ1
xdx < ∞.
2.19
−γ
So we obtain uλ ∈ L1 Ω.
Let Ωj {x ∈ Ω | distx, ∂Ω > r/2j}, j 1, 2, 3, . . . , and let uj be the unique solution
of
−γ
p
−Δu kxuj−1 λuj−1 ,
u uj−1 ,
in Ωj ,
2.20
on Ω \ Ωj ,
for j 1, 2, 3, . . ., and with u0 u λ1/1−p u∗ , where
r max minx − y.
2.21
x∈Ω y∈∂Ω
We claim that uj is nonincreasing with respect to j in Ω for all j ∈ N. Indeed, since u is a
supersolution of problem 1.1 for all λ > 0, then we have
−Δu0 − u1 −Δu0 Δu1
−γ
p
−Δu0 kxu0 − λu0
−γ
p
−Δu kxuλ − λuλ
2.22
> 0,
for all x ∈ Ω1 . Since u1 u0 in Ω \ Ω1 , so by the maximum principle, one has u0 ≥ u1 in Ω. So
when j 0 our claim is true. We assume that our claim is true when j n; that is, un ≤ un−1
in Ω. Then we obtain
−Δun − un1 −Δun Δun1
−γ
p
p
−γ
λ un−1 − un kx un − un−1
2.23
> 0,
for all x ∈ Ωn1 . Since un un1 in Ω \ Ωn1 , so by the maximum principle, one has un ≥ un1
in Ω. Thus by the induction, one obtains
uj1 ≤ uj ,
in Ω,
2.24
Abstract and Applied Analysis
7
for all j ∈ N. Then by the monotonicity of uj , we have
−γ
p
−Δuj λuj−1 − kxuj−1
2.25
−γ
p
≥ λuj − kxuj ,
for all x ∈ Ωj and j ∈ N . According to the definitions of uj and u0 , we obtain that uj is a
supersolution of problem 1.1 for all j ∈ N . Let uλ be a classical solution of problem 1.1,
thus one has
uλ x ≤ uj1 x ≤ uj x ≤ u0 x,
in Ω.
2.26
Assume that vλ x limj → ∞ uj x for all x ∈ Ω, then by standard elliptic arguments see
17 it follows that vλ is a solution of problem 1.1, and vλ ≥ uλ in Ω for any uλ . Therefore,
vλ is the maximal solution of problem 1.1. According to the above arguments, problem 1.1
has a maximal solution for λ ≥ λ∗ .
To complete the proof of Theorem 1.1, setting
σ λ > 0 | problem 1.1 has at least one solution uλ ,
λ inf σ,
2.27
then λ∗ , ∞ ⊂ σ, λ ≤ λ∗ . It suffices to prove that if λ0 ∈ σ, then λ0 , ∞ ⊂ σ; that is, assume
that λ > λ0 , then problem 1.1 has at least one solution. Let uλ0 be a solution of problem 1.1
corresponding to λ0 , then uλ0 is a subsolution of problem 1.1 with every fixed λ > λ0 . Since
u λ1/1−p u∗ is a supersolution of problem 1.1 for any λ > 0, then one has
1/1−p ∗
λ1/1−p u∗ ≥ λ0
u ≥ uλ 0 ,
in Ω,
2.28
for all λ > λ0 . According to Lemma 2.1, problem 1.1 has at least one solution uλ ∈ C2α Ω ∩
CΩ for all λ > λ0 . Moreover,
uλ0 x ≤ uλ x ≤ ux,
in Ω.
2.29
Consequently, the maximal solution vλ of problem 1.1 is increasing with respect to λ for all
λ > λ. So the proof of Theorem 1.1 is completed.
Proof of Theorem 1.3. Suppose to the contrary that there exists λ > 0 such that problem 1.1
has one solution uλ ∈ C2 Ω ∩ CΩ. Let e be the unique solution of
−Δu 1,
u 0,
in Ω,
on ∂Ω,
2.30
8
Abstract and Applied Analysis
e ∈ C2α Ω. By the maximum principle, e > 0 in Ω. We claim that for any solution uλ of
problem 1.1, there exists a constant M Mλ > 0 such that
in Ω.
Mex > uλ x,
2.31
p
Indeed, let M λuλ ∞ 1, then one obtains
−ΔMe − uλ −MΔe Δuλ
p
−γ
p
2.32
λuλ ∞ 1 − λuλ x kxuλ
> 0,
for all x ∈ Ω. Since Me − uλ |∂Ω 0, by the maximum principle we have
in Ω.
Mex > uλ x,
2.33
According to Lemma 4 in 14, there exists a positive constant C such that
ex ≤ Cϕ1 x,
in Ω.
2.34
−γ
2.35
uλ dx ∞.
2.36
Since γ ≥ 1, from Lemma 2.2, it follows that
Ω
−γ
uλ xdx
1
≥
CMγ
Ω
ϕ1 xdx ∞.
Thus we obtain
−γ
Ω
Set
r
Ωi x ∈ Ω | distx, ∂Ω > , i ∈ N ,
2i
and Ω ∞
i1
2.37
Ωi , then Ωi ⊂ Ω and uλ ∈ C2 Ωi , satisfying
−γ
p
2.38
−Δuλ kxuλ λuλ ,
for all x ∈ Ωi and i ∈ N . Consequently, integrating 2.38 we have
−
Ωi
Δuλ dx −γ
Ωi
kxuλ dx λ
p
Ωi
uλ dx ≤ λ
p
Ω
uλ dx,
2.39
Abstract and Applied Analysis
9
noting that
Ωi
Δuλ dx ∂Ωi
∂uλ
ds,
∂n
2.40
where n denotes the outward normal to ∂Ωi . From 2.39 and 2.40, letting i → ∞, one has
Ω
−γ
kxuλ dx
−
∂Ω
∂uλ
p
ds ≤ λuλ ∞ |Ω|,
∂n
2.41
where |Ω| denotes the Lebesgue measure of Ω. According to 2.36 and kx > 0 in Ω, one
obtains
∂Ω
∂uλ
ds ∞.
∂n
2.42
But this is impossible, by Hopf’s maximum principle, we have
∂uλ
< 0,
∂n
2.43
for all x ∈ ∂Ω, where n denotes the outward normal to ∂Ω at x. Therefore Theorem 1.3 is
true.
Acknowledgment
This paper is supported by NNSF of China under Grant 10771173 and the Natural Science
Foundation of Education of Guizhou Province under Grant 2008067.
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