Hindawi Publishing Corporation
Boundary Value Problems
Volume 2010, Article ID 584843, 20 pages
doi:10.1155/2010/584843
Research Article
Existence of Solutions for the px-Laplacian
Problem with Singular Term
Fu Yongqiang and Yu Mei
Department of Mathematics, Harbin Institute of Technology, Harbin 15000, China
Correspondence should be addressed to Yu Mei, yumei301796@gmail.com
Received 19 August 2009; Accepted 7 March 2010
Academic Editor: Raul F. Manásevich
Copyright q 2010 F. Yongqiang and Y. Mei. 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.
We study the following px-Laplacian problem with singular term: −div|∇u|px−2 ∇u |u|px−2 u λ|u|αx−2 u fx, u, x ∈ Ω, u 0, x ∈ ∂Ω, where Ω ⊂ RN is a bounded domain,
1,px
1 < p− ≤ px ≤ p < N. We obtain the existence of solutions in W0
Ω.
1. Introduction
After Kováčik and Rákosnı́k first discussed the Lpx Ω spaces and W k,px Ω spaces in 1, a
lot of research has been done concerning these kinds of variable exponent spaces, for example,
see 2–5 for the properties of such spaces and 6–9 for the applications of variable exponent
spaces on partial differential equations. Especially in W 1,px Ω spaces, there are a lot of
studies on px-Laplacian problems; see 8, 9. In the recent years, the theory of problems
with px-growth conditions has important applications in nonlinear elastic mechanics and
electrorheological fluids see 10–14.
In this paper, we study the existence of the weak solutions for the following pxLaplacian problem:
− div |∇u|px−2 ∇u |u|px−2 u λ|u|αx−2 u fx, u,
u 0,
x ∈ Ω,
1.1
x ∈ ∂Ω,
where Ω ⊂ RN is a bounded domain, 0 < λ ∈ R, px is Lipschitz continuous on Ω, and
1 < p− ≤ px ≤ p < N.
2
Boundary Value Problems
Let PΩ be the set of all Lebesgue measurable functions p : Ω → 1, ∞. For all
px ∈ PΩ, we denote p supx∈Ω px, p− infx∈Ω px, and denote by p1 x p2 x the
fact that inf{p2 x − p1 x} > 0.
We impose the following condition on f:
F fx, t bxgx, t, 0 < bx ∈ Lrx Ω, 1 rx ∈ CΩ and for gx, t ∈ CΩ ×
R, there exist M > 0 such that |gx, t| ≤ c1 c2 |t|qx−1 , whenever |t| ≥ M.
A typical example of 1.1 is the following problem involving subcritical SobolevHardy exponents of the form
− div |∇u|px−2 ∇u |u|px−2 u λ|u|αx−2 u u 0,
λ
|x|sx
|u|qx−2 u,
x ∈ Ω,
1.2
x ∈ ∂Ω,
where 0 < λ ∈ R, sx, qx ∈ CΩ, 0 ≤ s− ≤ s < N, 1 qx < N − sx/Np∗ x,
sx
, gx, t |t|qx−2 t, and rx ≡ 1/21 N/s , then
for all x ∈ Ω. In fact, take bx λ/|x|
it is easy to verify that F is satisfied.
Our object is to obtain the existence of solutions in the following four cases:
1 αx > px, qx > px;
2 αx < px, qx > px;
3 αx < px, qx < px;
4 αx > px, qx < px.
When bx 1, the solution of the p-Laplacian equations without singularity has been
studied by many researchers. The study of problem 1.1 with variable exponents is a new
topic.
The paper is organized as follows. In Section 2, we present some necessary preliminary
knowledge of variable exponent Lebesgue and Sobolev spaces. In Section 3, we prove our
main results.
2. Preliminaries
In this section we first recall some facts on variable exponent Lebesgue space Lpx Ω and
variable exponent Sobolev space W 1,px Ω, where Ω ⊂ RN is an open set; see 1–4, 8, 15 for
the details.
Let px ∈ PΩ and
u px
up inf λ > 0 :
dx ≤ 1 .
Ω λ
2.1
The
variable exponent Lebesgue space Lpx Ω is the class of functions u such that
|ux|px dx < ∞. Lpx Ω is a Banach space endowed with the norm 2.1.
Ω
Boundary Value Problems
3
For a given px ∈ PΩ, we define the conjugate function p x as
px
.
px − 1
p Theorem 2.1. Let px ∈ PΩ. Then the inequality
fx · gxdx ≤ rp f g p
p
2.2
2.3
Ω
holds for every f ∈ Lpx Ω and g ∈ Lp x Ω with the constant rp depending on px and Ω only.
Theorem 2.2. The dual space of Lpx Ω is Lp x Ω if and only if p ∈ L∞ Ω. The space Lpx Ω
is reflexive if and only if
1 < p− ≤ p < ∞.
2.4
Theorem 2.3. Suppose that px satisfies 2.4. Let meas Ω < ∞, p1 x, p2 x ∈ PΩ, then
necessary and sufficient condition for Lp2 x Ω ⊂ Lp1 x Ω is that for almost all x ∈ Ω one has
p1 x ≤ p2 x, and in this case, the imbedding is continuous.
Theorem 2.4. Suppose that px satisfies 2.4. Let ρu Ω |ux|px dx. If u, uk ∈ Lpx Ω,
then
1 up < 1 1; > 1 if and only if ρu < 1 1; > 1,
p−
p
p
p−
2 if up > 1, then up ≤ ρu ≤ up ,
3 if up < 1, then up ≤ ρu ≤ up ,
4 limk → ∞ uk p 0 if and only if limk → ∞ ρuk 0,
5 uk p → ∞ if and only if ρuk → ∞.
We assume that k is a given positive integer.
Given a multi-index α α1 , . . . , αn ∈ Nn , we set |α| α1 · · ·αn and Dα D1α1 · · · Dnαn ,
where Di ∂/∂xi is the generalized derivative operator.
The generalized Sobolev space W k,px Ω is the class of functions f on Ω such that
α
D f ∈ Lpx for every multi-index α with |α| ≤ k, endowed with the norm
Dα f .
f k,p
p
2.5
|α|≤k
k,px
By W0
Ω we denote the subspace of W k,px Ω which is the closure of C0∞ Ω with
respect to the norm 2.5.
In this paper we use the following equivalent norm on W 1,px Ω:
u1,p
∇u px u px
inf λ > 0 :
dx ≤ 1 .
λ λ
Ω
Then we have the inequality 1/2∇up up ≤ u1,p ≤ 2∇up up .
2.6
4
Boundary Value Problems
k,px
Theorem 2.5. The spaces W k,px Ω and W0
satisfies 2.4.
Ω are separable reflexive Banach spaces, if px
Theorem 2.6. Suppose that px satisfies 2.4. Let Iu W 1,px Ω, then
Ω
|∇ux|px |ux|px dx. If u, uk ∈
1 u1,p < 1 1; > 1 if and only if Iu < 1 1; > 1,
p−
p
p
p−
2 if u1,p > 1, then u1,p ≤ Iu ≤ u1,p ,
3 if u1,p < 1, then u1,p ≤ Iu ≤ u1,p ,
4 limk → ∞ uk 1,p 0 if and only if limk → ∞ Iuk 0,
5 uk 1,p → ∞ if and only if Iuk → ∞.
k,px
We denote the dual space of W0
Ω by W −k,p x Ω, then we have the following.
Theorem 2.7. Let p ∈ PΩ ∩ L∞ Ω. Then for every G ∈ W −k,p x Ω there exists a unique system
of functions {gα ∈ Lp x Ω : |α| ≤ k} such that
Dα fxgα xdx,
G f |α|≤k
Ω
k,px
f ∈ W0
Ω.
2.7
The norm of W −k,p x Ω is defined as
G−k,p
G f k,px
sup
: f ∈ W0
Ω \ {0} .
f k,p
2.8
Theorem 2.8. Let Ω be a domain in Rn with cone property. If p : Ω → R is Lipschitz continuous
and 1 < p− ≤ p < N/k, qx : Ω → R is measurable and satisfies px ≤ qx ≤ p∗ x :
Npx/N − kpx a.e. x ∈ Ω, then there is a continuous embedding W k,px Ω → Lqx Ω.
1,px
Theorem 2.9. Let Ω be a bounded domain. If px ∈ L∞ Ω and u ∈ W0
up ≤ C∇up ,
Ω, then
2.9
where C is a constant depending on Ω.
Next let us consider the weighted variable exponent Lebesgue space. Let ax ∈ PΩ,
and ax > 0 for x ∈ Ω. Define
px
Lax Ω
u ∈ P Ω :
px
Ω
ax|ux|
dx < ∞
2.10
Boundary Value Problems
5
with the norm
|u|Lpx Ω up,a
ax
ux px
dx ≤ 1 ,
inf λ > 0 :
ax
λ Ω
2.11
px
then Lax Ω is a Banach space.
Theorem 2.10. Suppose that px satisfies 2.4. Let ρu px
Lax Ω,
Ω
ax|ux|px dx. If u,uk ∈
then
1 for u /
0, up,a λ if and only if ρu/λ 1,
2 up,a < 1 1; > 1 if and only if ρu < 1 1; > 1,
p−
p
p
p−
3 if up,a > 1, then up,a ≤ ρu ≤ up,a ,
4 if up,a < 1, then up,a ≤ ρu ≤ up,a ,
5 limk → ∞ uk p,a 0 if and only if limk → ∞ ρuk 0,
6 uk p,a → ∞ if and only if ρuk → ∞.
Theorem 2.11. Assume that the boundary of Ω possesses the cone property and px ∈ CΩ.
Suppose that 0 < ax ∈ Lrx Ω, and 1 rx ∈ CΩ, for x ∈ Ω. If qx ∈ CΩ
and 1 ≤ qx < rx − 1/rxp∗ x, for all x ∈ Ω, then there is a compact embedding
qx
W 1,px Ω → Lax Ω.
Theorem 2.12. Let Ω ⊂ Rn be a measurable subset. Let g : Ω × R → R be a Caracheodory function
and satisfies
gx, u ≤ ax b|u|p1 x/p2 x
for any x ∈ Ω, t ∈ R,
2.12
where pi x ≥ 1, i 1, 2, ax ∈ Lp2 x Ω, ax ≥ 0, b ≥ 0 is a constant, then the Nemytsky
operator from Lp1 x Ω to Lp2 x Ω defined by Ng ux gx, ux is a continuous and bounded
operator.
3. Existence and Multiplicity of Solutions
Let
1 |u|αx
− Fx, udx,
|∇u|px |u|px − λ
αx
Ω px
t
t
Fx, t fx, sds,
Gx, t gx, sds.
Iu 0
0
3.1
6
Boundary Value Problems
The critical points of Iu, that is,
0 I uϕ Ω
|∇u|px−2 ∇u∇ϕ |u|px−2 uϕ − λ|u|αx−2 uϕ − fx, uϕ dx
3.2
1,px
Ω, are weak solutions of problem 1.1. So we need only to consider the
for all ϕ ∈ W0
existence of nontrivial critical points of Iu.
Denote by c, ci , C, Ci , K, and Ki the generic positive constants. Denote by |Ω| the
Lebesgue measure of Ω.
To study the existence of solutions for problem 1.1 in the first case, we additionally
impose the following conditions.
A-1 αx, qx ∈ CΩ and px αx < p∗ x, px qx < rx − 1/rxp∗ x.
B-1 There exists a function μx ∈ C1 Ω, such that px μx and 0 < μxGx, t ≤
tgx, t for x ∈ Ω, |t| ≥ M.
C-1 there exist δ > 0 such that |gx, t| ≤ c3 |t|qx−1
for x ∈ Ω, |t| ≤ δ, where qx
∈ CΩ
∗
and px qx
< rx − 1/rxp x.
D-1 gx, −t −gx, t, for all x ∈ Ω, t ∈ R.
Theorem 3.1. Under assumptions (F) and (A-1)–(C-1), problem 1.1 admits a nontrivial solution.
1,px
Proof. First we show that any P Sc sequence is bounded. Let {un } ⊂ W0
Ω and c ∈
R, such that Iun → c and I un → 0 in W −1,p x Ω. By A-1 and B-1, inf{1/px −
1/αx} a1 > 0, and inf{1/px − 1/μx} a2 > 0. Let a0 min{a1 , a2 } and lx 1/px − a0 /2, then lx > 1/αx, lx > 1/μx, |∇lx| < C, 1/px − lx a0 /2 and
lx − 1/αx ≥ a0 /2. Let Ω1 {x ∈ Ω : |ux| ≥ M} and Ω2 Ω \ Ω1 . Then
Iun − I un , lxun 1
1
− lx |∇un |px |un |px dx λ lx −
|un |αx dx
px
αx
Ω
Ω
lxfx, un un − Fx, un dx −
Fx, un dx
Ω1
Ω2
Ω2
lxfx, un un dx −
3.3
Ω
|∇un |px−2 ∇un ∇l · un dx.
By B-1, we get
Ω1
lxfx, un un − Fx, un dx ≥
Ω1
lx −
1
fx, un un dx > 0.
μx
3.4
Boundary Value Problems
7
By F, we get gx, t ∈ CΩ × −M, M, so there exist K > 0 such that |gx, t| ≤ K on
Ω × −M, M. Note fx, t bxgx, t, so we have
Fx, un dx ≤
MKbxdx < ∞,
Ω2
Ω2
fx, un un dx ≤
MKbxdx < ∞.
Ω2
Ω2
3.5
By Young’s inequality, for ε1 ∈ 0, 1, we get
Ω
|∇un |px−2 ∇un ∇l · un dx ≤ Cε1
1−p
Ω
|∇un |px dx Cε1
Ω
|un |px dx.
3.6
Take ε1 sufficiently small so that a0 /2 > Cε1 .
Note that px αx, by Young’s inequality again and for ε2 ∈ 0, 1, we get
Ω
|∇un |px−2 ∇un ∇l · un dx
≤ Cε1
Ω
|∇un |
px
dx 1−p
Cε1 ε2
Ω
1−p
Take ε2 sufficiently small so that λa0 /2 ≥ Cε1
From the above remark, we have
Iun − I un , lxun
|un |
αx
dx 1−p −p /α−p−
Cε1 ε2
|Ω|.
3.7
ε2 .
a0
− Cε1 |∇un |px |un |px dx − C1 .
≥
Ω 2
3.8
As lxun p ≤ l un p , lx∇un p ≤ l ∇un p and ∇lx · un p ≤ Cun p , we have
lxun 1,p ≤ 4Cl un 1,p . Since Iun → c and I un → 0, by Theorem 2.6 we have
c o1 4Cl un 1,p ≥
a
0
2
p−
− Cε1 un 1,p − C1 ,
3.9
when un 1,p ≥ 1 and n is sufficiently large. Then it is easy to see that {un } is bounded in
1,px
W0
Ω. Next we show that {un } possesses a convergent subsequence still denoted by
{un }.
8
Boundary Value Problems
Note that
|∇un |px−2 ∇un − |∇u|px−2 ∇u ∇un − ∇udx
Ω
≤ I un , un − u I u, un − u Ω
λ |un |αx−2 un − |u|αx−2 u un − udx
Ω
fx, un − fx, u un − udx
I1 I2 I3 I4 .
3.10
1,px
Because {un } is bounded in W0
Ω, there exists a subsequence {un } still denoted
1,px
by {un }, such that un u weakly in W0
embeddings W
1,px
Ω → L
αx
Ω and W
1,px
Ω. By Theorem 2.11, there are compact
qx
Ω → Lbx Ω, then un → u in Lαx Ω
qx
and Lbx Ω. So we get
I3 λ |un |αx−2 un − |u|αx−2 u un − udx
Ω
3.11
≤ 2λ
|un |αx−1 un − uα 2λ
|u|αx−1 un − uα .
α
α
Hence I3 → 0 as n → ∞.
By F, we have
Ω
fx, uun − udx ≤
≤
Ω
C2 bx|un − u|dx Ω2
Ω1
C2 bx|un − u|dx Ω
bx|un − u||u|
qx−1
bx|un − u||u|qx−1 dx
3.12
dx,
and similarly for every n,
Ω
fx, un un − udx ≤
Ω
C2 bx|un − u|dx Ω
bx|un − u||un |qx−1 dx.
3.13
Since
Ω
bx|un − u||u|
qx−1
dx Ω
bx1/qx bx1/q x |un − u||u|qx−1 dx
≤ 2
bx1/q x |u|qx−1 bx1/qx |un − u|
q
3.14
q
Boundary Value Problems
9
qx
and un → u in L1bx Ω and Lbx Ω, we obtain
u||u|qx−1 dx → 0. Similarly,
Ω
Ω
bx|un − u|dx → 0 and
Ω
bx|un −
bx|un − u||un |qx−1 dx ≤ 2
bx1/q x |un |qx−1 bx1/qx |un − u|
.
q
q
3.15
Because bx1/q x |un |qx−1 q is bounded,we get Ω bx|un − uun |qx−1 dx → 0, as n → ∞.
From the above remark, we conclude I4 Ω |fx, un − fx, uun − u|dx → 0, as n → ∞.
Thus I1 I2 I3 I4 → 0, as n → ∞. Then we get Ω |∇un |px−2 ∇un −|∇u|px−2 ∇u∇un −
∇udx → 0. As in the proof of Theorem 3.1 in 6, 7, we divide Ω into the following two parts:
Ω3 x ∈ Ω : 1 < px < 2 ,
Ω4 x ∈ Ω : px ≥ 2 .
3.16
On Ω3 , we have
|∇un − ∇u|px dx ≤
Ω3
Ω3
K1
px/2
|∇un |px−2 ∇un − |∇u|px−2 ∇u ∇un − ∇u
2−px/2
× |∇un |px |∇u|px
dx
px/2 px−2
px−2
≤ 2K1 |∇un |
∇un − |∇u|
∇u ∇un − ∇u
2−px/2 × |∇un |px |∇u|px
Then
2/px,Ω3
.
2/2−px,Ω3
|∇un − ∇u|px dx → 0, as n → ∞.
On Ω4 , we have
Ω3
Ω4
3.17
|∇un − ∇u|px dx ≤ K2
Ω4
|∇un |px−2 ∇un − |∇u|px−2 ∇u ∇un − ∇udx,
3.18
|∇un − ∇u|px dx → 0, as n → ∞.
1,px
Ω.
Thus we get Ω |∇un − ∇u|px dx → 0. Then un → u in W0
μx
− c5 , for all x ∈ Ω, t ∈ R. So we get
From F and B-1 we have Gx, t ≥ c4 |t|
Fx, t ≥ c4 bx|t|μx − c5 bx, for all x ∈ Ω, t ∈ R. for all x0 ∈ Ω, take ε 1/6αx0 − px0 ,
then px0 < αx0 −3ε. Since px and αx ∈ CΩ, there exists δ > 0 such that |px−px0 | <
so
Ω4
10
Boundary Value Problems
ε, and |αx − αx0 | < ε, for x ∈ Bx0 , δ. Let ϕ ∈ C0∞ Bx0 , δ, such that ϕx 1 for
x ∈ Bx0 , δ/2, ϕx 0 for x ∈ Ω \ Bx0 , δ, and 0 ≤ ϕx ≤ 1 in Bx0 , δ. Then we have
I sϕ ≤
px
Ω
s
px
∇ϕ
px
≤
px
s
px
∇ϕ
≤ spB
Bx0 ,δ
px
c5
px
px
∇ϕ
px
Bx0 ,δ
px ϕ
− λs
px ϕ
px
px ϕ
px
αx
αx
ϕ
− λs
αx
αx
μx
− c4 bxsμx ϕ
c5 bxdx
αx
ϕ
αx
−
dx c5
dx − sαB
λ
Bx0 ,δ
bxdx
Bx0 ,δ
αx
ϕ
αx
3.19
dx
bxdx,
Bx0 ,δ
where pB < α−B . So if s is sufficiently large, we obtain Isϕ < 0.
cδ|t|qx , then |Fx, t| ≤ c3 bx|t|qx
From F and C-1, we have |Gx, t| ≤ c3 |t|qx
qx
cδbx|t| . So we get
Iu ≥
1
p
Ω
λ
− cδbx|u|qx dx. 3.20
|u|αx − c3 bx|u|qx
|∇u|px |u|px dx − −
α Ω
Let θx min{αx, qx,
qx}, then θx ∈ CΩ. By Theorem 2.11, uα ≤ c6 u1,p ,
uq,b
≤ c7 u1,p , and uq,b ≤ c8 u1,p . When u1,p is sufficiently small, uα < 1, uq,b
< 1
and uq,b < 1. For any x ∈ Ω, as px, θx ∈ CΩ, for any ε > 0, we can find
QR x {y y1 , . . . , yn : |yk − xk | < R, k 1, . . . , n} such that |py − px| < ε and
|θy−θx| < ε whenever y ∈ QR x∩Ω. Take ε 1/4θx−px, then supy∈QR x∩Ω py <
infy∈QR x∩Ω θy. {QR x}x∈Ω is an open covering of Ω. As Ω is compact, we can pick a finite
⊂ Ω we define
subcovering {QRk xk }m
k1 for Ω from the covering {QR x}x∈Ω . If QRk xk /
u 0 on QRk xk \ Ω. We can use all the hyperplanes, for each of which there exists at
least one hypersurface of some QRk xk lying on it, to divide m
k1 QRk xk into finite open
Q
Q
hypercubes {Qi }i1 which mutually have no common points. It is obvious that Ω ⊆ i1 Qi
and for each Qi there exists at least one QRk xk such that Qi ⊆ QRk xk . Let Ωi Qi ∩ Ω, then
pi supy∈Ωi py < infy∈Ωi θy θi− and we have
Q
1
λ
px
px
αx
qx
qx
Iu ≥
dx − −
|u|
− c3 bx|u|
− cδbx|u| dx
|u|
|∇u|
p Ωi
α Ωi
i1
Q 1
λ α−i
pi
α−i
qi−
qi−
qi−
qi−
≥
u1,p,Ωi − − c6 u1,p,Ωi − c3 c7 u1,p,Ωi − cδc8 u1,p,Ωi .
p
α
i1
3.21
Boundary Value Problems
11
If u1,p k ≥ u1,p,Ωi is sufficiently small such that
1
λ α−
pi
α−i
q−
qi−
q−
qi−
− − c6 i u1,p,Ω
− c3 c7i u1,p,Ω
− cδc8i u1,p,Ω
> 0,
u1,p,Ω
i
i
i
i
p
α
3.22
we have Iu > 0.
The mountain pass theorem guarantees that I has a nontrivial critical point u.
1,px
1,px
W0
Since W0
Ω is a separable and reflexive Banach space, there exist {en }∞
n1 ⊂
−1,p x
Ω and {en∗ }∞
Ω such that
n1 ⊂ W
1,px
W0
ei , ej∗ ⎧
⎨1,
i j,
⎩0,
i/
j,
3.23
Ω span{en : n 1, 2, . . .},
W −1,p x Ω span{en∗ : n 1, 2, . . .}.
For k 1, 2, . . . , denote Xk span{ek }, Yk ⊕kj1 Xj , Zk ⊕∞
X.
jk j
Theorem 3.2. Under assumptions (F), (A-1)–(D-1), problem 1.1 admits a sequence of solutions
1,px
Ω such that Iun → ∞.
{un } ∈ W0
Proof. Let ϕu Ω λ|u|αx /αx Fx, udx. We first show that ϕu is weakly strongly
1,px
1,px
Ω. By the compact embedding W0
Ω →
continuous. Let un u weakly in W0
αx
on
Ω.
By
the
inequality
|u
≤
Lαx Ω, we have Ω |un − u|αx dx → 0 and un → u a.e.
n|
α −1
αx
αx
αx
αx
|un − u|
|u| and the Vitali Theorem, we get Ω |un | dx → Ω |u| dx.
2
Note that
Ω
|Fx, un − Fx, u|dx Ω
bx|Gx, un − Gx, u|dx
3.24
≤ 2bxr Gx, un − Gx, ur .
When |u| ≥ M,
q x
|u|qx
|u|
|u|qx c1
|u|qx
c2
≤ c
|Gx, u| ≤ c1
2
μx
μx
q x μxqx qx
μx
≤ C3 1 |u|qx C3 1 |u|r xqx/r x .
3.25
When |u| ≤ M, Gx, u is bounded. So we get
|Gx, u| ≤ C4 1 |u|r xqx/r x .
3.26
12
Boundary Value Problems
1,px
r xqx
Ω → Lr xqx Ω, we get
Ω. So
By the compact embedding W0
un → u in L
by Theorem 2.12 we obtain |Gx, un − Gx, u|r → 0, that is, Ω |Fx, un − Fx, u|dx → 0.
Hence we obtain that ϕu is weakly strongly continuous. By Proposition 3.5 in 8, βk βk r supu∈Zk ,u1,p ≤r |ϕu| → 0 as k → ∞ for r > 0. For all n, there exists a positive integer
kn such that βk n ≤ 1 for all k ≥ kn . Assume kn < kn1 for each n. Define {rk : k 1, 2, . . .} in
the following way:
rk ⎧
⎨n,
kn ≤ k < kn 1,
⎩1,
1 ≤ k < k1 .
3.27
Note that rk → ∞ as k → ∞. Hence for u ∈ Zk with u1,p rk , we get
Iu Ω
1
1 p−
|∇u|px |u|px dx − ϕu ≥ u1,p − 1.
px
p
3.28
Iu −→ ∞ as k −→ ∞.
3.29
So
inf
u∈Zk ,u1,p rk
Note that Fx, u bxGx, u ≥ c4 bx|u|μx −c5 bx and px, αx ∈ CΩ. Since the
dimension of Yk is finite, any two norms on Yk are equivalent, then c9 |u|1,p ≤ |u|α ≤ c10 |u|1,p .
If |u|1,p ≤ 1, it is immediate that |u|α ≤ c10 . If |u|1,p > 1/c9 , then |u|α > 1. As in the proof of
Q
Theorem 3.1 we can find hypercubes {Qi }i1 which mutually have no common points such
Q
that Ω ⊆ i1 Qi and pi supy∈Ωi py < infy∈Ωi αy α−i , where Ωi Qi ∩ Ω. Then we need
only to consider the case: |u|1,p,Ωi > max{1, 1/c9 } for every i. We have
Iu ≤
Ω
≤
Ω
|∇u|px |u|px
px
px
|∇u|px |u|px
px
px
−λ
|u|αx
− c4 bx|u|μx c5 bxdx
αx
−λ
|u|αx
dx c5
αx
Ω
bxdx
Q λ α−i
1
pi
α−i
≤
bxdx
u1,p,Ωi − c9 u1,p,Ωi c5
p−
α
Ω
i1
⎛
⎞
Q
λ α−i ⎝ α−
pi
α−i
⎠
c
c
−
bxdx.
u
u
5
1,p,Ωi
1,p,Ωi
α−
α 9
Ω
i1
p− λc i
3.30
9
α−
−
Let c0 α /p− λc9 i , fi t tαi − c0 tpi . Let fi si mint≥0 fi t < 0 and fi ti 0.
−
"Q
Denote t i1 u1,p,Ωi . Let t be sufficiently large such that t > max{Q, Qti , Q2c0 1/αi −pi ,
Boundary Value Problems
13
i 1, 2, . . . , Q}. There at least exists one i0 such that u1,p,Ωi0 ≥ 1/Q
We have
"Q
i1
u1,p,Ωi t/Q.
Q
Q
pi
α−i −pi
fi u1,p,Ωi − c0
u1,p,Ω
u1,p,Ω
i
i
i1
i1
i
/ i0
≥
pi
u1,p,Ω
i
α−i −pi
u1,p,Ω
i
fi si t
Q
fi si c0
t,
Q
i/
i0
≥
Q
i1
− c0 u1,p,Ωi
pi 0
t
Q
pi
0
α−i −pi
0
0
α−i −pi
0
u1,p,Ωi − c0
0
0
0
3.31
− c0
"Q
and i1 fi u1,p,Ωi → ∞ as t → ∞. Hence we obtain that Iu → −∞ as u1,p → ∞. Thus
for each k, there exists ρk > rk such that Iu < 0 for u ∈ YK ∩ Sρk . From Theorem 3.1 Iu
satisfies P Sc condition. In view of D-1, by Fountain Theorem see 16, we conclude the
result.
In the second case, we additionally impose the following condition:
A-2 αx, qx ∈ CΩ and 1 ≤ αx px qx < rx − 1/rxp∗ x.
Theorem 3.3. Under assumptions (F), (A-2), (B-1), and (C-1) there exist λ∗ > 0 such that when
λ ∈ 0, λ∗ , problem 1.1 admits a nontrivial solution.
Proof. It is obvious that α− < p− . Let ε0 > 0 be such that α− ε0 < p− . Since αx ∈ CΩ, there
exists δ > 0 and x0 ∈ Ω such that |αx − α− | < ε0 , for all x ∈ Bx0 , δ. Thus αx ≤ α− ε0
for all x ∈ Bx0 , δ. Let ϕ be as defined in Theorem 3.1. By C-1, |Gx, t| ≤ c3 |t|qx
, and
qx
|Fx, t| ≤ c3 bx|t| , when t < δ. Then for any t ∈ 0, 1 and t < δ, we have
I tϕ Ω
p−
t
px
px
∇ϕ
px
px
∇ϕ
≤t
px
Bx0 ,δ
−
tq
px ϕ
px
− λt
αx
αx
ϕ
αx
− F x, tϕ dx
px αx
ϕ
ϕ
α
dx − t
dx
λ
px
αx
Bx0 ,δ
qx
c3 bxϕ dx
Bx0 ,δ
p−
px
∇ϕ
≤t
px
Bx0 ,δ
−
tq
px αx
ϕ
ϕ
α− ε0
dx − t
dx
λ
px
αx
Bx0 ,δ
qx
c3 bxϕ dx.
Bx0 ,δ
If t is sufficiently small, Itϕ < 0.
3.32
14
Boundary Value Problems
cδ|t|qx and |Fx, t| ≤ c3 bx|t|qx
From F and C-1, we have |Gx, t| ≤ c3 |t|qx
qx
cδbx|t| . By Theorems 2.8 and 2.11, there exist positive constants k1 , k2 , k3 such that
≤ k3 |u|1,p . When u1,p is sufficiently small, we have
|u|α ≤ k1 |u|1,p , |u|q,b ≤ k2 |u|1,p , |u|q,b
<
1,
and
u
<
1.
As
in the proof of Theorem 3.1 we can find hypercubes
uα < 1, uq,b
q,b
Q
Q
{Qi }i1 which mutually have no common points such that Ω ⊆ i1 Qi , pi supy∈Ωi py <
infy∈Ωi qy qi− and pi < qi− , where Ωi Qi ∩ Ω. Then
Iu ≥
Q
i1
≥
Ωi
1
px
px
− c3 bx|u|qx
− cδbx|u|qx dx
|u|
|∇u|
p
λ
− −
α
Ω
|u|αx dx
Q −
1
λ −
pi
qi−
qi−
qi−
qi−
−
c
k
−
cδk
− − k1α uα1,p .
u
u
u
3 3
2
1,p,Ωi
1,p,Ωi
1,p,Ωi
p
α
i1
3.33
p
Since u1,p ≥ u1,p,Ωi , for all i, when u1,p ρ, u1,p,Ωi ρi < ρ. Fix ρ such that 1/p ρi i −
q− q−
q− q−
c3 k3 i ρi i − cδk2 i ρi i > 0. Then we have
N 1 pi
λ − −
qi− qi−
qi− qi−
Iu ≥
ρ − c3 k3 ρi − cδk2 ρi
− − k1α ρα l1 ρ − λl2 ρ .
i
p
α
i1
3.34
Let λ∗ l1 ρ/2l2 ρ. When λ ∈ 0, λ∗ , Iu ≥ l1 /2 > 0. As in the proof of Theorem 2.1
1,px
Ω : u1,p < ρ}, we have inf∂Bρ 0 Iu > 0 and −∞ <
in 17, denote Bρ 0 {u ∈ W0
c infBρ 0 Iu < 0. Let 0 < ε < inf∂Bρ 0 Iu − infBρ 0 Iu. Applying Ekeland’s variational
principle to the functional Iu : Bρ 0 → R, we find uε ∈ Bρ 0 such that Iuε < infBρ 0 Iu
ε, Iuε < Iuεu−uε 1,p , uε / u ∈ Bρ 0, and I uε −1,p ≤ ε. Thus we get a sequence {un } ⊂
1,px
Bρ 0 such that Iun → c and I un → 0. It is clear that {un } is bounded in W0
Ω. As
in the proof of Theorem 3.1, we get a subsequence of {un }, still denoted by {un }, such that
1,px
Ω. So Iu c < 0 and I u 0.
un → u in W0
Theorem 3.4. Under assumptions (F), (A-2), and (B-1)–(D-1), problem 1.1 has a sequence of
1,px
Ω such that Iun → ∞.
solutions {un } ∈ W0
1,px
Proof. First we show that any P Sc sequence is bounded. Let {un } ∈ W0
Ω and c ∈ R,
−1,p x
Ω. By B-1, there exist σ > 0 such that
such that Iun → c and I un → 0 in W
inf{1/px − 1 σ/μx} μ1 > 0. From F, A-2, and B-1–D-1, we have
1
fx, un un − Fx, un dx > 0,
Ω1 μx
1
fx,
u
−
Fx,
u
u
n n
n dx < C1 ,
μx
Ω2
1
μx
fx, un un dx C2 ≥
Fx, un dx ≥
c4 bx|un | dx −
c5 bxdx.
Ω μx
Ω
Ω
Ω
3.35
Boundary Value Problems
15
Thus we have
1 |un |αx
− Fx, un dx
|∇un |px |un |px − λ
αx
Ω px
1σ 1
|un |αx
−
− Fx, un dx
|∇un |px |un |px − λ
μx
αx
Ω px
1σ
1σ
1σ
αx
λ|un | dx fx, un un |∇un |px |un |px
μx
Ω μx
Ω μx
Iun 1σ
1σ
λ|un |αx −
fx, un un dx
μx
μx
1σ 1σ
1
px
px
dx −
−
un dx
≥
|un |
|∇un |px−2 ∇un ∇
|∇un |
μx
μx
Ω px
Ω
|un |αx
μx
−
dx λ
σc4 bx|un | dx −
σc5 bxdx
αx
Ω
Ω
Ω
#
$
1σ
I un ,
un − C3 .
μx
3.36
−
By Young’s inequality, for ε1 , ε2 , ε3 ∈ 0, 1, we get
Ω
|∇un |
px−2
≤ C1 ε1
1σ
· un dx
∇un ∇
μx
px
Ω
|∇un |
Ω
dx 1−p
C1 ε1 ε2
|un |αx
dx ≤ ε3
αx
Ω
1−p −p /μ−p−
ε2
|Ω|,
|un |μx dx C1 ε1
−α /p−α−
Ω
|un |px dx ε3
3.37
|Ω|.
1−p
Take ε1 , ε2 , and ε3 sufficiently small so that μ1 −C1 ε1 > 0, μ1 −λε3 ≥ 0 and σc4 bx−C1 ε1
0, then
$ #
1σ
un ≥
C4 |∇un |px dx − C5 .
Iun − I un ,
μx
Ω
ε2 ≥
3.38
1,px
Therefore by Theorems 2.6 and 2.9, we get that {un } is bounded in W0
Ω. Then as in the
proof of Theorem 3.1{un } possesses a convergent subsequence {un } still denoted by {un }.
By Theorem 3.2, we can also have
inf
u∈Zk ,u1,p rk
Iu −→ ∞ as k −→ ∞.
3.39
16
Boundary Value Problems
Q
As in the proof of Theorem 3.1 we can find hypercubes {Qi }i1 which mutually have
Q
no common points such that Ω ⊆ i1 Qi and pi supy∈Ωi py < infy∈Ωi μy μ−i , where
Ωi Qi ∩ Ω. Since the dimension of Yk is finite, any two norms on Yk are equivalent. Then we
need only to consider the cases |u|1,p,Ωi > 1 and uμ,b,Ωi > 1 for every i. We have
Iu ≤
Ω
≤
Ω
|∇u|px |u|px
px
px
|∇u|px |u|px
px
px
−λ
|u|αx
− c4 bx|u|μx c5 bxdx
αx
− c4 bx|u|μx dx c5
Ω
3.40
bxdx
Q 1
pi
μ−i
bxdx.
≤
u1,p,Ωi − C4 u1,p,Ωi c5
p
Ω
i1
Hence Iu → −∞ as u1,p → ∞. As in the proof of Theorem 3.2, we complete the proof.
In the third case, we additionally impose the following condition:
A-3 αx, qx ∈ CΩ, and 1 ≤ αx, qx px,
B-3 there exist θ > 0 such that Gx, t ≥ 0 for t ∈ 0, θ.
Theorem 3.5. Under assumptions (F), (A-3), and (B-3), problem 1.1 admits a nontrivial solution.
Proof. By Young’s inequality, for ε ∈ 0, 1, we get |un |αx /αx ≤ ε|un |px ε−α
we have Fx, u ≤ Cbx|u|qx C1 bx. Thus
Iu Ω
|∇u|px |u|px
px
px
−λ
/p−α−
. By F,
|u|αx
− Fx, udx
αx
|∇u|px |u|px
≥
dx − λε
|u|px dx
px
px
Ω
Ω
qx
−α /p−α−
bx|u| dx − λε
bxdx
−C
|Ω| − C1
≥
Ω
Ω
1
− λε
p
|∇u|px |u|px dx − C
3.41
Ω
Ω
bx|u|qx dx − C2 .
Take ε sufficiently small so that 1/p − λε > 0. From Theorem 2.11, uq,b ≤ cu1,p . If u1,p ≤
1, uq,b is bounded. Then we need only to consider the case u1,p > 1. As in the proof of
Boundary Value Problems
17
Q
Theorem 3.1 we can find hypercubes {Qi }i1 which mutually have no common points such
Q
that Ω ⊆ i1 Qi and qi supy∈Ωi qy < infy∈Ωi py pi− , where Ωi Qi ∩ Ω. Then we have
Iu ≥
M
i1
pi−
c1 u1,p,Ω
i
−
qi
c2 u1,p,Ω
i
J pj−
qj−
c1 u1,p,Ωj − c3 u1,p,Ωj C3
j1
3.42
M
pi−
qi
C4 ,
≥
c1 u1,p,Ω
−
c
u
4
1,p,Ω
i
i
i1
where Ωi bx|u|qx dx > 1, i 1, 2, . . . M, and Ωj bx|u|qx dx ≤ 1, i 1, 2, . . . J, M J Q.
As in the proof of Theorem 3.2, we obtain that Iu is coercive, that is, Iu → ∞ as u1,p →
∞. Thus I has a critical point u such that Iu infu∈W 1,px Ω Iu and further u is a weak
0
solution of 1.1.
Next we show that u is nontrivial. Let Bx0 , δ be the same as that in Theorem 3.3. By
B-3, Fx, tϕ ≥ 0. Then
I tϕ Ω
p−
≤t
≤t
p−
t
px
px
∇ϕ
px
px
∇ϕ
Ω
px
px
∇ϕ
Ω
px
px ϕ
px
− λt
αx
αx
ϕ
αx
− F x, tϕ dx
px αx
ϕ
ϕ
dx
dx − tα
λ
px
αx
Ω0
3.43
px αx
ϕ
ϕ
α− ε0
dx − t
dx.
λ
px
αx
Ω0
If t is sufficiently small, Itϕ < 0.
In the fourth case, we additionally impose the following condition:
A-4 αx, qx ∈ CΩ and 1 ≤ qx px αx < p∗ x.
Theorem 3.6. Under assumptions (F), (A-4), and (D-1), problem 1.1 admits a sequence of solutions
1,px
Ω such that Iun → ∞.
{un } ∈ W0
1,px
Proof. First we show that any P Sc sequence is bounded. Let {un } ∈ W0
Ω and c ∈ R,
such that Iun → c and I un → 0 in W −1,p x Ω. Denote inf{1/px − 1/αx} a > 0
and lx 1/px − a/2. We have 1/px − lx a/2 and lx − 1/αx ≥ a/2.
18
Boundary Value Problems
We can get
Iun − I un , lxun 1 |un |αx
− Fx, un dx
|∇un |px |un |px − λ
αx
Ω px
1
|un |αx
− lx |∇un |px |un |px − λ
− Fx, un dx
αx
Ω px
lxλ|un |αx dx lxfx, un un − |∇un |px−2 ∇un ∇lx · un dx
Ω
≥
3.44
Ω
|∇un |px−2 ∇un ∇lx · un dx
|∇un |px |un |px dx −
a
2
Ω
Ω
Ω
lx −
1
λ|un |αx lxfx, un un − Fx, un dx.
αx
By Young’s inequality, for ε1 , ε2 ∈ 0, 1, we get
Ω
|∇un |px−2 ∇un ∇lx · un dx
≤ Cε1
Ω
|∇un |
px
dx 1−p
Cε1 ε2
Ω
|un |
αx
dx 1−p −p /α−p−
Cε1 ε2
|Ω|.
1−p
Take ε1 and ε2 sufficiently small so that Cε1 < a/2 and Cε1
3.45
ε2 ≤ a/2. Then
∞ > Iun − I un , lxun a
− Cε1 |∇un |px |un |px − lxfx, un un |Fx, un | dx − C1
≥
2
Ω
a
px
px
dx − C2
− Cε1 |∇un |
≥
|un |
bx|un |qx dx − C3 .
Ω 2
Ω
3.46
As in the proof of Theorem 3.5, Iun − I un , lxun → ∞, when un 1,p → ∞. Thus,
1,px
we conclude that {un } is bounded in W0
Ω. Then as in the proof of Theorem 3.1{un }
possesses a convergent subsequence {un } still denoted by {un }. By Theorem 3.2, we can
also get
inf
u∈Zk ,u1,p rk
Iu −→ ∞ as k −→ ∞.
3.47
Q
As in the proof of Theorem 3.1 we can find hypercubes {Qi }i1 which mutually have
Q
no common points such that Ω ⊆ i1 Qi and pi supy∈Ωi py < infy∈Ωi αy α−i , where
Boundary Value Problems
19
Ωi Qi ∩ Ω. Since the dimension of Yk is finite, any two norms on Yk are equivalent. Then we
need only to consider the cases |u|1,p,Ωi > 1|u|α,Ωi > 1 and |u|q,b,Ωi > 1 for every i. We have
Iu ≤
Ω
|∇u|px |u|px
px
px
|u|αx
−λ
dx C4
αx
Ω
bx|un |qx dx C5
Q pi
qi
α−i
c0 u1,p,Ω
−
cu
C6 .
≤
u
1,p,Ω
1,p,Ω
i
i
i
3.48
i1
Hence we obtain Iu → −∞ as u1,p → ∞. As in the proof of Theorem 3.2, we complete
the proof.
Acknowledgments
This work is supported by the Science Research Foundation in Harbin Institute of Technology
HITC200702, The Natural Science Foundation of Heilongjiang Province A2007-04, and the
Program of Excellent Team in Harbin Institute of Technology.
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