Hindawi Publishing Corporation
Mathematical Problems in Engineering
Volume 2010, Article ID 640841, 10 pages
doi:10.1155/2010/640841
Research Article
Positive Solution for the Elliptic Problems with
Sublinear and Superlinear Nonlinearities
Chunmei Yuan, Shujuan Guo, and Kaiyu Tong
College of Physics and Mathematics, Changzhou University, Changzhou, Jiangsu 213164, China
Correspondence should be addressed to Chunmei Yuan, chunmeiyuan@126.com
Received 8 October 2010; Revised 13 December 2010; Accepted 13 December 2010
Academic Editor: Jyh Horng Chou
Copyright q 2010 Chunmei Yuan et al. 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.
This paper deals with the existence of positive solutions for the elliptic problems with sublinear
and superlinear nonlinearities −Δu λaxup bxuq in Ω, u > 0 in Ω, u 0 on ∂Ω, where λ > 0
is a real parameter, 0 < p < 1 < q. Ω is a bounded domain in ÊN N ≥ 3, and ax and bx are
some given functions. By means of variational method and super-subsolution method, we obtain
some results about existence of positive solutions.
1. Introduction
In this paper, we consider the elliptic problems with sublinear and superlinear nonlinearities
−Δu λaxup bxuq
u>0
u0
in Ω,
in Ω,
1λ
on ∂Ω,
where λ > 0 is a real parameter, 0 < p < 1 < q. Ω is a bounded domain in ÊN N ≥
3, and ax and bx are some given functions which satisfies the following assumptions:
H1 ax, bx ∈ L∞ Ω, ax ≥ c0 , bx ≤ −c1 , where c0 , c1 are positive constants,
or
H2 ax, bx ∈ L∞ Ω, ax, bx ≥ c0 , where c0 is a positive constant.
For convenience, we denote 1λ with hypothesis H1 or H2 by 1−λ and 1λ ,
respectively.
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Such problems occur in various branches of mathematical physics and population
dynamics, and sublinear analogues or superlinear analogues of 1λ have been considered
by many authors in recent years see 1–9 and their references. But most of such studies
have been concerned with equations of the type involving sublinear nonlinearity see 3–
6, 8, 9, with only few references dealing with the elliptic problems with sublinear and
superlinear nonlinearities. In 1, Ambrosetti et al. deal with the analogue of 1λ with
ax bx ≡ 1. It is known from 2 that there exist λ∗ ∈ 0, ∞, such that problem 1λ has
a solution if λ ≤ λ∗ and has no solution if λ > λ∗ , provided bx ≡ 1 on Ω.
Our goal in this paper is to show how variational method and super-subsolution
method can be used to establish some existence results of problem 1λ . We work on the
Sobolev space H01 Ω equipped with the norm x Ω |∇u|2 dx1/2 . For u ∈ H01 Ω we
define Iλ : H01 Ω → Ê by
Iλ u 1
2
Ω
|∇u|2 dx −
λ
p1
Ω
ax|u|p1 dx −
1
q1
Ω
bx|u|q1 dx.
1.1
Let λ1 be the first eigenvalue of
−Δu λu,
u 0,
x ∈ Ω,
x ∈ ∂Ω.
1.2
ϕ1 denotes the corresponding eigenfunction satisfying 0 ≤ ϕ1 x ≤ 1. Lp Ω, 1 ≤ p ≤ ∞,
denotes Lebesgue spaces, and the norm in Lp is denoted by · p .
2. The Existence of Positive Solution of 1−λ
It is well known that
∇ϕ1 x / 0,
∀x ∈ ∂Ω.
2.1
Define a min∂Ω |∇ϕ1 |2 ; from 2.1 we know a > 0, so we can split the domain Ω into
two parts: Ωε and Ω \ Ωε , where Ωε {x ∈ Ω : |∇ϕ1 |2 ≥ a/2} {x ∈ Ω : ϕ1 x ≤
ε, ε is small enough}. Let b infΩ\Ωε ϕ1 x; we obtain that b ≥ ε by the positivity of ϕ1 in Ω,
and Ω \ Ωε is nonempty when ε is small enough.
Theorem 2.1. Let ax, bx satisfy assumption H1 , and 0 < p < 1 < q < 2∗ − 1, where 2∗ 2N/N−2 is the limiting exponent in the Sobolev embedding. Then there exists a constant λ > 0 such
that 1−λ possesses at least a weak positive solution u∗ x ∈ H01 Ω for λ ≥ λ.
Proof. Let ex denote the positive solution of the following equation:
−Δe 1, x ∈ Ω,
e 0, x ∈ ∂Ω.
2.2
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Here and hereafter we use the following notations: A a∞ , B b∞ , E e∞ . Since
0 < p < 1, for all λ ∈ Ê , there exists T Tλ > 0 satisfying
2.3
T ≥ λAT p Ep .
Observing that bx ≤ −c1 < 0, as a consequence, the function Te verifies
T −ΔTe ≥ λATep ≥ λaxTep bxTeq ,
2.4
and hence it is a supersolution of 1λ . Let vx ϕl1 , x ∈ Ω, l > 1. For x ∈ Ω, we have x ∈ Ωε
or x ∈ Ω \ Ωε . We will discuss it from two conditions.
I For all x ∈ Ωε , observing that l > 1 and when ε is small enough, we have
all − 1
s−2
− Bslq−1 > λ1 l,
2
∀s ∈ 0, ε.
2.5
Since x ∈ Ωε , then it follows that ϕ1 x ≤ ε, |∇ϕ1 |2 ≥ a/2. From 2.5 we infer
2
lq−1
,
λ1 l ≤ ll − 1ϕ−2
1 ∇ϕ1 − Bϕ1 x
∀x ∈ Ωε .
2.6
Multiplying 2.6 with ϕl1 , we get
∇ϕ1 2 λ1 lϕl ≤ −Bϕlq .
l1 − lϕl−2
1
1
1
2.7
p
q
−Δ ϕl1 ≤ λax ϕl1 − bx ϕl1 .
2.8
It follows that
and we have
II For all x ∈ Ω \ Ωε , there exists λ > 0, such that for all λ ≥ λ,
λc0 spl − Bsql ≥ λ1 lsl ,
∀s ∈ R, b ≤ s ≤ 1.
2.9
Since x ∈ Ω \ Ωε , then we have ϕ1 x ≥ b and ϕ1 x ≤ 1. From 2.9, it follows that
p
q
lp
−Δ ϕl1 ≤ λ1 lϕl1 ≤ λc0 ϕ1 − Bsql ≤ λax ϕl1 bx ϕl1 .
2.10
From 2.8 and 2.10, we derive that there exists λ > 0 such that for all x ∈ Ω, for all λ ≥ λ,
q
p
−Δ ϕl1 ≤ λax ϕl1 bx ϕl1 ,
2.11
that is, vx ϕl1 x is a subsolution of 1−λ . Taking T as sufficiently large, we also have
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Mathematical Problems in Engineering
Te > ϕl1 by minimal principle. Define wx Tex, and let K {u ∈ H01 Ω: vx ≤ ux ≤
wx, for all x ∈ Ω}, then K is closed and convex and weakly closed. Let fs λaxsp bxsq , for all s ∈ Ê, s > 0. We consider the function
1
Iλ u 2
2
Ω
|∇u| dx −
u
Ω
fsds dx.
2.12
0
Observe that bx < 0, 0 < p < 1 < q < 2∗ − 1; we infer that Iλ is coercive, bounded, since it is
blow and weakly lower semicontinuous. Using this fact, we conclude that there exists u∗ ∈ K,
such that Iλ u∗ infK Iλ see 10. In the following, we will prove that u∗ is a solution of
problem 1−λ .
For φ ∈ K, define h : 0, 1 → Ê, such that
ht I tφ 1 − tu∗ .
2.13
Clearly, ht achieves its minimum at t 0, and
h tt0 Ω
∗ ∇u ∇ φ − u∗ dx −
Ω
fu∗ φ − u∗ dx ≥ 0.
2.14
For all ϕ ∈ H01 Ω, η > 0, define
⎧
⎪
v,
⎪
⎪
⎨
Ψx u∗ ηϕ,
⎪
⎪
⎪
⎩
w,
when u∗ ηϕ < v,
when v ≤ u∗ ηϕ ≤ w,
when u∗ ηϕ > w.
Obviously, Ψ ∈ K, and inserting 2.15 into 2.14, we find
0≤
v≤u∗ ηϕ≤w
u∗ ηϕ>w
u∗ ηϕ<v
∗
∇u · ∇ ηφ − fu∗ ηϕ dx
∗
∇u ∇w − u∗ − fu∗ w − u∗ dx
∗
∇u ∇v − u∗ − fu∗ v − u∗ dx
2.15
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∗
η
∇u · ∇ϕ − fu∗ ϕ dx
5
v≤u∗ ηϕ≤w
u∗ ηϕ>w
u∗ ηϕ<v
u∗ ηϕ>w
u∗ ηϕ>w
−
∇w · ∇w − u∗ − fww − u∗ dx
∇v · ∇v − u∗ − fvv − u∗ dx
∗ 2
u∗ ηϕ<v
|∇w − ∇u | dx −
u∗ ηϕ<v
|∇v − ∇u∗ |2 dx
fw − fu∗ w − u∗ dx
fv − fu∗ v − u∗ dx.
2.16
Since wx and vx are supersolution and subsolution, respectively, then
u∗ ηϕ>w
∇w · ∇w − u∗ − fww − u∗ dx ≤ η
u∗ ηϕ<v
∇v · ∇v − u∗ − fvv − u∗ dx ≤ η
u∗ ηϕ>w
u∗ ηϕ<v
∇w · ∇ϕ − fwϕ dx,
∇v · ∇ϕ − fvϕ dx.
2.17
Observe that measu∗ ηϕ > w → 0, measu∗ ηϕ < v → 0, as η → 0,
u∗ ηϕ>w
∇w · ∇ϕ − fwϕ dx −→ 0,
u∗ ηϕ<v
∇v · ∇ϕ − fvϕ dx −→ 0.
2.18
Since u∗ ∈ K, bx < 0, it follows that
fw − fu∗ w − u∗ dx
u∗ ηϕ>w
u∗ ηϕ>w
≤
u∗ ηϕ>w
p
∗p
∗
λax w − u w − u dx p
λax wp − u∗ w − u∗ dx.
u∗ ηϕ>w
q
bx wq − u∗ w − u∗ dx
2.19
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Mathematical Problems in Engineering
Similar to 2.19, we have
u∗ ηϕ<v
fv − fu∗ v − u∗ dx
≤
u∗ ηϕ<v
u∗ ηϕ<v
p
λax vp − u∗ v − u∗ dx u∗ ηϕ<v
q
bx vq − u∗ v − u∗ dx
2.20
p
λax vp − u∗ v − u∗ dx.
Similar to 2.18, as η → 0, it follows that
u∗ ηϕ>w
p
λax wp − u∗ w − u∗ dx −→ 0,
u∗ ηϕ<v
λax v − u
p
∗p
2.21
∗
v − u dx −→ 0.
As η → 0, we also have
v≤u∗ ηϕ≤w
∗
∇u · ∇ϕ − fu∗ ϕ dx −→
Ω
∇u∗ · ∇ϕ − fu∗ ϕ dx.
2.22
Inserting 2.17, 2.19, and 2.20 into 2.16, we find
0≤η
v≤u∗ ηϕ≤w
u∗ ηϕ<v
u∗ ηϕ>w
u∗ ηϕ<v
∗
∇u · ∇ϕ − fu∗ ϕ dx ∇v · ∇ϕ − fvϕ dx
u∗ ηϕ>w
∇w · ∇ϕ − fwϕ dx
2.23
p
λax wp − u∗ w − u∗ dx
p
λax vp − u∗ v − u∗ dx.
Dividing by η and letting η → 0, using 2.18, 2.21, and 2.22, we derive
Ω
∗
∇u · ∇ϕ − fu∗ ϕ dx ≥ 0.
2.24
Noting that ϕ is arbitrary, this holds equally for −ϕ, and it follows that u∗ is indeed a weak
solution of 1−λ , and the strong maximum principle yields u∗ > ϕl1 , in Ω. Therefore it is a
weak positive solution of 1−λ .
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3. The Existence of Positive Solution of 1λ
Theorem 3.1. Let ax, bx satisfy assumption H2 , and 0 < p < 1 < q < ∞. Then there exists
Λ ∈ Ê, Λ > 0, such that
i for all λ ∈ 0, Λ problem 1λ has a minimal solution uλ such that Iλ uλ < 0. Moreover
uλ is increasing with respect to λ;
ii for λ Λ problem 1λ has at least one weak solution u ∈ H ∩ Lp1 ;
iii for all λ > Λ problem 1λ has no solution.
To prove Theorem 3.1, let us define
Λ sup λ > 0 : 1λ has a solution .
3.1
First of all we prove a useful lemma.
Lemma 3.2. One has 0 < Λ < ∞.
Proof. Let ex denote the solution of the following equation:
−Δe 1,
e 0,
x ∈ Ω,
x ∈ ∂Ω.
3.2
Since 0 < p < 1 < q, we can find λ0 > 0 such that for all 0 < λ ≤ λ0 there exists T Tλ >
0 satisfying
T ≥ λAT p Ep BT q Eq .
3.3
As a consequence, the function Te verifies
T −ΔTe ≥ λATep BTeq ≥ λaxTep bxTeq ,
3.4
and hence it is a supersolution of 1λ . Moreover, let u0 denote the solution of the following
problem:
p
−Δu λaxu0 ,
u0 0,
x ∈ Ω,
x ∈ ∂Ω.
3.5
From 3 we know that u0 exists. Then εu0 is a subsolution of 1λ , provided
p
−Δεu0 λεaxu0 ≤ λaxεu0 p bxεu0 q ,
3.6
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Mathematical Problems in Engineering
which is satisfied for all ε > 0 small enough and all λ. Taking ε as possibly smaller, we also
have
3.7
εu0 < Te.
It follows that 1λ has a solution u, εu0 ≤ u ≤ Te whenever λ ≤ λ0 , and thus Λ ≥ λ0 .
Next, let λ∗ be such that
c0 λ∗ tp tq > λ1 t,
∀t > 0.
3.8
If λ is such that 1λ has a solution u, multiplying 1λ by ϕ1 and integrating over Ω we find
λ1
Ω
uϕ1 dx λ
axu ϕ1 dx p
Ω
bxu ϕ1 dx ≥ c0
q
Ω
Ω
p
λu ϕ1 uq ϕ1 dx .
3.9
This and 3.5 immediately imply that λ < λ∗ and show that Λ ≤ λ∗ , hence 0 < Λ < ∞.
We are now ready to give the proof of Theorem 3.1.
Proof. i From the proof of lemma, it follows that, for all λ ∈ 0, Λ, problem 1λ has a
solution uλ . Let u0 satisfy 3.5; the iteration
p
q
−Δun1 λaxun bxun
3.10
satisfies un ↑ uλ by making use of Lemma 3.3 of 1 and maximum principle. It is easy to
check that uλ is a minimal solution of 1λ . Indeed, if u is any solution of 1λ , then u ≥ u0 and
u is a supersolution of 1λ . Thus un ≤ u, for all n, by induction, and uλ ≤ u. Next, we will
prove that Iλ uλ < 0. Indeed,
Iλ u 1
2
Ω
|∇u|2 dx −
λ
p1
Ω
ax|u|p1 dx −
1
q1
Ω
bx|u|q1 dx.
3.11
Since uλ is a solution of 1λ we have
2
Ω
|∇uλ | dx Ω
p1
λaxuλ dx
q1
Ω
bxuλ dx.
3.12
From Lemma 3.5 of 1, we know
2 ∇ϕ − λpaxup−1 qbxuq−1 ϕ2 dx ≥ 0,
λ
Ω
λ
∀ϕ ∈ H01 .
3.13
In particular with ϕ uλ , we infer
2
Ω
|∇uλ | dx − λp
Ω
p1
axuλ dx
−q
q1
Ω
bxuλ dx ≥ 0.
3.14
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Combining 3.12 and 3.14, we obtain
1
1
−
Iλ uλ λ
2 p1
Ω
p1
axuλ dx
1
1
−
2 q1
q1
Ω
bxuλ dx
3.15
1−p
1
1
p1
−
axuλ dx < 0.
≤
2
p1 q1
Ω
To complete the proof of i, it remains to show that
uλ < uλ 1
3.16
whenever λ < λ1 .
Indeed, if λ < λ1 then uλ1 is a supersolution of 1λ . Since, for ε > 0 small, εu0 is a subsolution
of 1λ and εu0 < uλ1 , then 1λ possesses a solution v, with
εu0 ≤v ≤ uλ1 .
3.17
Since uλ is the minimal solution of 1λ , we infer that uλ ≤ v ≤ uλ1 . Moreover
p
q
p
q
−Δuλ1 − uλ λ1 axuλ1 bxuλ1 − λaxuλ bxuλ
≥
p
λaxuλ1
q
bxuλ1
−
p
axuλ
−
q
bxuλ
3.18
≥ 0.
Since uλ1 /
uλ because λ < λ1 , then the Hopf Maximum principle yields uλ < uλ1 .
ii Let λn be a sequence such that λn ↑ Λ; then from Iλn uλn < 0 we deduce that there
exists C > 0 such that
∇un 2 ≤ C,
p1
un p1 ≤ C.
3.19
Then there exists u∗ ∈ H01 such that un → u∗ > 0 a.e. in Ω, strongly in Lp1 and weakly in H01 .
Such a u∗ is thus a weak solution of 1λ for λ Λ.
iii This follows from the definition of Λ.
Acknowledgment
This work supported by the Physics and Mathematics Foundation of Changzhou University
ZMF10020065.
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