Hindawi Publishing Corporation
International Journal of Differential Equations
Volume 2011, Article ID 356356, 17 pages
doi:10.1155/2011/356356
Research Article
L∞ -Solutions for Some Nonlinear Degenerate
Elliptic Equations
Albo Carlos Cavalheiro
Department of Mathematics, State University of Londrina, 86051-990 Londrina, PR, Brazil
Correspondence should be addressed to Albo Carlos Cavalheiro, accava@gmail.com
Received 17 May 2011; Revised 6 October 2011; Accepted 6 October 2011
Academic Editor: Toka Diagana
Copyright q 2011 Albo Carlos Cavalheiro. 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 are interested in the existence
of solutions for Dirichlet problem associated to the degenerate
quasilinear elliptic equations − nj1 Dj ω2 xAj x, u, ∇u ω1 xgx, ux Hx, u, ∇u ω2 x 1,p
fx, on Ω in the setting of the weighted Sobolev spaces W0 Ω, ω1 , ω2 .
1. Introduction
In this paper we prove the existence of weak solutions in the weighted Sobolev spaces
1,p
W0 Ω, ω1 , ω2 for the homogeneous Dirichlet problem:
Lux fx,
ux 0,
on Ω,
on ∂Ω,
P where L is the partial differential operator:
Lux − divω2 xAx, u, ∇u gx, uω1 x Hx, u, ∇uω2 x,
1.1
where Ω is a bounded open set in RN N ≥ 2, ω1 and ω2 are two weight functions, and the
functions A : Ω × R × RN → RN , g : Ω × R → R, and H : Ω × R × RN → R are Carathéodory
functions.
By a weight, we will mean a locally integrable function ω on RN such that ωx > 0
for a.e. x ∈ RN . Every weight ωi i 1, 2 gives rise to a measure on the measurable subsets
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International Journal of Differential Equations
on RN through integration. This measure will be denoted by μi . Thus, μi E E ωi xdx
i 1, 2 for measurable sets E ⊂ RN .
In general, the Sobolev spaces Wk,p Ω without weights occur as spaces of solutions
for elliptic and parabolic partial differential equations. For degenerate partial differential
equations, that is, equations with various types of singularities in the coefficients, it is natural
to look for solutions in weighted Sobolev spaces see 1–4.
A class of weights, which is particularly well understood, is the class of Ap -weights
or Muckenhoupt class that was introduced by Muckenhoupt see 5. These classes have
found many useful applications in harmonic analysis see 6, 7. Another reason for studying
Ap -weights is the fact that powers of distance to submanifolds of RN often belong to Ap
see 8. There are, in fact, many interesting examples of weights see 4 for p-admissible
weights.
Equations like 1.1 have been studied by many authors in the nondegenerate case
i.e., with ωx ≡ 1 see, e.g., 9 and the references therein. The degenerate case with
different conditions has been studied by many authors. In 2 Drábek et al. proved that under
certain condition, the Dirichlet problem associated with the equation − divax, u, ∇u h,
1,p
1,p
h ∈ W0 Ω, ω∗ has at least one solution u ∈ W0 Ω, ω, and in 1 the author proved the
existence of solution when the nonlinear term Hx, η, ξ is equal to zero.
Firstly, we prove an L∞ estimate for the bounded solutions of P : we assume that
f/ω1 ∈ Lq Ω, ω1 , with r2 /r2 − 1 < q < ∞ where r2 > 1 as in Theorem 2.5, and we prove
1,p
that any u ∈ W0 Ω, ω1 , ω2 ∩ L∞ Ω that solves P satisfies uL∞ Ω ≤ C, where C depends
only on the data, that is, Ω, N, p, q, α1 , α2 , C0 , C1 and f/ω1 Lq Ω,ω1 . After that, we prove the
existence of solution for problem P if f/ω1 ∈ Lq Ω, ω1 , with p r2 /r2 − 1 < q < ∞.
Note that, in the proof of our main result, many ideas have been adapted from 9–11.
The following theorem will be proved in Section 3.
Theorem 1.1. Let ω1 and ω2 be Ap -weights, 1 < p < ∞, with ω1 ≤ ω2 . Suppose the following.
H1 x → Ax, η, ξ is measurable in Ω for all η, ξ ∈ R × RN : η, ξ → Ax, η, ξ is
continuous in R × RN for almost all x ∈ Ω.
H2 Ax, η, ξ − Ax, η , ξ · ξ − ξ > 0, whenever ξ, ξ ∈ RN , ξ / ξ .
H3 Ax, η, ξ · ξ ≥ α1 |ξ|p , with 1 < p < ∞, where α1 > 0.
H4 |Ax, η, ξ| ≤ K2 x h1 x|η|p/p h2 x|ξ|p/p , where K2 , h1 , and h2 are positive
functions, with h1 and h2 ∈ L∞ Ω, and K2 ∈ Lp Ω, ω2 (1/p 1/p 1).
H5 x → gx, η is measurable in Ω for all η ∈ R : η → gx, η is continuous in R for
almost all x ∈ Ω.
H6 |gx, η| ≤ K1 xh3 x|η|p/p , where K1 and h3 are positive functions, with h3 ∈ L∞ Ω
and K1 ∈ Lp Ω, ω1 .
H7 gx, η η ≥ α0 |η|p , for all η ∈ R, where α0 > 0.
H8 x → Hx, η, ξ is measurable in Ω for all η, ξ ∈ R × RN : η, ξ → Hx, η, ξ is
continuous in R × RN for almost all x ∈ Ω.
H9 |Hx, η, ξ| ≤ C0 C1 |ξ|p , where C0 and C1 are positive constants.
H10 f/ω1 ∈ Lq Ω, ω1 , with r2 /r2 − 1 < q < ∞ (where r2 > 1 as in Theorem 2.5) and
ω2 /ω1 ∈ Lq Ω, ω1 .
1,p
Let u ∈ W0 Ω, ω1 , ω2 ∩ L∞ Ω be a solution of problem P . Then there exists a constant C > 0,
which depends only on Ω, n, p, α1 , α0 , C0 , C1 and f/ω1 Lq Ω,ω1 , such that uL∞ Ω ≤ C.
International Journal of Differential Equations
3
The main result of this article is given in the next theorem, which is proved in
Section 4.
Theorem 1.2. Assume that (H1)–(H9) hold true and suppose that
H11 f/ω1 ∈ Lq Ω, ω1 , with p r2 /r2 − 1 < q < ∞;
H12 Hx, η, ξ η ≥ 0, for all η ∈ R.
1,p
Then there exists at least one solution u ∈ W0 Ω, ω1 , ω2 ∩ L∞ Ω of the problem P .
Theorem 1.2 will be proved by approximating problem P with the following
problems:
− divω2 Ax, u, ∇u gx, uω1 Hm x, u, ∇uω2 fx,
ux 0,
on ∂Ω,
on Ω,
Pm where Hm x, η, ξ Hx, η, ξ/1 1/m|Hx, η, ε|, for m ∈ N. Note that |Hm | ≤ |H| and that
|Hm | ≤ m.
2. Definitions and Basic Results
Let ω be a locally integrable nonnegative function in RN and assume that 0 < ωx < ∞
almost everywhere. We say that ω belongs to the Muckenhoupt class Ap , 1 < p < ∞, or that
ω is an Ap -weight, if there is a constant C Cp,ω such that
1
|B|
ωxdx
B
1
|B|
ω
1/1−p
xdx
p−1
≤ Cp,ω
2.1
B
for all balls B ⊂ RN , where | · | denotes the N-dimensional Lebesgue measure in RN . If 1 <
q ≤ p, then Aq ⊂ Ap see 4, 7, 12 or 13 for more information about Ap -weights. The
weight ω satisfies the doubling condition if μ2B ≤ CμB, for all balls B ⊂ Rn , where
μB B ωxdx and 2B denotes the ball with the same center as B which is twice as large.
If ω ∈ Ap , then ω is doubling see Corollary 15.7 in 4.
As an example of Ap -weight, the function ωx |x|α , x ∈ RN , is in Ap if and only if
−N < α < Np − 1 see Corollary 4.4, Chapter IX in 7. If ϕ ∈ BMORN , then ωx eαϕx ∈ A2 for some α > 0 see 6.
Definition 2.1. Let ω be a weight, and let Ω ⊂ RN be open. For 0 < p < ∞, we define Lp Ω, ω
as the set of measurable functions f on Ω such that
f p
L Ω,ω
Ω
fxp ωxdx
1/p
< ∞.
2.2
Remark 2.2. If ω ∈ Ap , 1 < p < ∞, then since ω−1/p−1 is locally integrable, we have Lp Ω, ω ⊂
L1loc Ω for every open set Ω see Remark 1.2.4 in 13. It thus makes sense to talk about weak
derivatives of functions in Lp Ω, ω.
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International Journal of Differential Equations
Definition 2.3. Let Ω ⊂ RN be open, 1 < p < ∞, and let ω1 and ω2 be Ap -weights, 1 < p < ∞.
We define the weighted Sobolev space W 1,p Ω, ω1 , ω2 as the set of functions u ∈ Lp Ω, ω1 with weak derivatives Dj u ∈ Lp Ω, ω2 , for j 1, . . . , N. The norm of u in W 1,p Ω, ω1 , ω2 is
given by
⎛
uW 1,p Ω,ω1 ,ω2 ⎝
⎞1/p
N p
Dj ux ω2 xdx⎠ .
|ux| ω1 x dx p
Ω
j1
Ω
2.3
1,p
The space W0 Ω, ω1 , ω2 is the closure of C0∞ Ω with respect to the norm uW 1,p Ω,ω1 ,ω2 .
1,p
The dual space of W0 Ω, ω1 , ω2 is the space
1,p
W0 Ω, ω1 , ω2 ∗
W −1,p Ω, ω1 , ω2 gj
f
p
p
T f − div g : g g1 , . . . , gN ,
∈ L Ω, ω1 ,
∈ L Ω, ω2 .
ω1
ω2
2.4
Remark 2.4. a If ω ∈ Ap , 1 < p < ∞, then C∞ Ω is dense in W 1,p Ω, ω W 1,p Ω, ω, ω see
Corollary 2.16 in 13.
b If ω1 ≤ ω2 , then
1,p
1,p
1,p
W0 Ω, ω2 ⊂ W0 Ω, ω1 , ω2 ⊂ W0 Ω, ω1 .
2.5
In this paper we use the following four results.
Theorem 2.5 The Weighted Sobolev Inequality. Let Ω be an open bounded set in RN N ≥ 2
and ω2 ∈ Ap 1 < p < ∞. There exist constants CΩ and δ positive such that for all u ∈ C0∞ Ω and
all r2 satisfying 1 ≤ r2 ≤ N/N − 1 δ,
uLp∗ Ω,ω2 ≤ CΩ ∇uLp Ω,ω2 ,
2.6
where p∗ p r 2 .
Proof. See Theorem 1.3 in 3.
The following lemma is due to Stampacchia see 14, Lemma 4.1.
Lemma 2.6. Let α, β, C, and k0 be real positive numbers, where β > 1.
Let ϕ : R → R be a decreasing function such that
ϕl ≤
β
C α ϕk
l − k
for all l > k ≥ k0 . Then ϕk0 d 0, where dα C ϕk0 β−1 2α β/β−1 .
2.7
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5
Lemma 2.7. If ω ∈ Ap , then |E|/|B|p ≤ Cp,ω μE/μB, whenever B is a ball in RN and E is a
measurable subset of B.
Proof. See Theorem 15.5 Strong doubling of Ap -weights in 4.
By Lemma 2.7, if μE 0, then |E| 0.
Lemma 2.8. Let ω1 and ω2 be Ap -weights, 1 < p < ∞, ω1 ≤ ω2 , and a sequence {un }, un ∈
1,p
W0 Ω, ω1 , ω2 satisfies the following:
1,p
i un u in W0 Ω, ω1 , ω2 and μ2 -a.e. in Ω;
ii Ω Ax, un , ∇un − Ax, un , ∇u, ∇un − u ω2 dx → 0 with n → ∞.
1,p
Then un → u in W0 Ω, ω1 , ω2 .
Proof. The proof of this lemma follows the line of Lemma 5 in 10.
1,p
Definition 2.9. We say that u ∈ W0 Ω, ω1 , ω2 ∩ L∞ Ω is a weak solution of problem P if
Ω
ω2 Ax, u, ∇u · ∇ϕ dx Ω
gx, uϕω1 dx Ω
Hx, u, ∇u ϕω2 dx Ω
fϕ dx,
2.8
1,p
for all ϕ ∈ W0 Ω, ω1 , ω2 ∩ L∞ Ω.
3. Proof of Theorem 1.1
Set λ C1 /α1 1 and define for k > 0 the functions φ ∈ C1 R and Gk ∈ W 1,∞ R by
φs ⎧
⎨eλs − 1,
if s ≥ 0,
⎩− e−λs 1,
if s ≤ 0,
⎧
⎪
s − k,
⎪
⎪
⎨
Gk s 0,
⎪
⎪
⎪
⎩
s k,
3.1
if s ≥ k,
if − k ≤ s ≤ k,
if s ≤ −k.
1,p
If u ∈ W0 Ω, ω1 , ω2 ∩ L∞ Ω is a solution of problem P , define the set Ak {x ∈
Ω : |ux| > k}. We will use the test functions vx φGk ux. Since u ∈ W01,2 Ω, ω1 , ω2 ∩
1,p
L∞ Ω and Ω is a bounded set in RN , then vx φGk ux ∈ W0 Ω, ω1 , ω2 ∩ L∞ Ω
and
vx φ |u| − k χAk signu,
∇v φ |u| − k6 χAk ∇u,
3.2
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International Journal of Differential Equations
where χAk is the characteristic function of the set Ak by signux we mean the function
equal to 1 for ux > 0 and to −1 for ux < 0.
1,p
1,p
Since u ∈ W0 Ω, ω1 , ω2 ∩ L∞ Ω, we have that v ∈ W0 Ω, ω1 , ω2 ∩ L∞ Ω.
Using the function v in 2.8 we obtain
Ω
ω2 Ax, u, ∇u · ∇v dx Ω
gx, uv ω1 dx Ω
Hx, u, ∇u v ω2 dx Ω
f v dx. 3.3
We have the following estimates.
i By H3 we obtain
Ω
φ |u| − k Ax, u, ∇u · ∇u ω2 dx
ω2 Ax, u, ∇u · ∇vdx Ak
p ≥ α1
3.4
|∇u| φ |u| − k ω2 dx.
Ak
ii By H7 we obtain
Ω
gx, u φ |u| − k ω1 dx
gx, uv ω1 dx Ak
p−1
≥ α0
|u|
3.5
φ |u| − k ω1 dx.
Ak
iii Using H9 we obtain
Hx, u, ∇uv ω2 dx ≤
|Hx, u, ∇u||v|ω2 dx
Ω
Ω
C0 C1 |∇u|p φ |u| − k ω2 dx.
≤
3.6
Ak
And we also have
f v dx ≤
Ω
f φ |u| − k dx.
3.7
Ak
Hence in 3.3 we obtain
p |∇u| φ |u| − k ω2 dx α0
α1
Ak
|u|p−1 φ |u| − k ω1 dx
Ak
p
f φ |u| − k dx.
C0 C1 |∇u| φ |u| − k ω2 dx ≤
Ak
Ak
3.8
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7
Since λ C1 /α1 1, we have for s ≥ 0
α1 φ s − C1 φs α1 λeλs − C1 eλs − 1
α1 λ − C1 eλs C1 α1 eλs C1
α1 λs/p p α1 s p
λs
≥ α1 e p λe
p φ
.
λ
λ
p
3.9
Hence in 3.8 we obtain
α1 |∇u|p φ |u| − k − C1 |∇u|p φ |u| − k ω2 dx
Ak
α0
|u|
p−1
f C0 ω2 φ |u| − k dx.
φ |u| − k ω1 dx ≤
Ak
3.10
Ak
Using 3.9 and k < |ux| if x ∈ Ak, we obtain
α1
λp
Ak
p
|u| − k
φ
ω2 dx α0 kp−1
∇u
p
≤
φ |u| − k ω1 dx
Ak
3.11
f C0 ω2 φ |u| − k dx.
Ak
Let us define the function ψk by ψk x φ|ux| − k /p. We have that ψk ∈ W0 Ω, ω1 , ω2 and
1,p
1 |u| − k
φ
χAk signu ∇u.
p
p
∇ψk 3.12
We have the following.
a For all s ≥ 0, eλs − 1 ≥ eλs/p − 1p .
b There exists, a constant C2 > 0 C2 C2 λ, p such that for all s ≥ 1
p
eλs − 1 ≤ C2 eλs/p − 1 ,
p
λeλs ≤ C2 λ eλs/p − 1 .
3.13
This implies the following.
I1 φ|u| − k eλ|u|−k − 1 ≥ eλ|u|−k
/p
− 1p | ψk |p a.e. on Ω.
I2 If x ∈ Ak 1, then
p
φ |u| − k eλ|u|−k − 1 ≤ C2 eλ|u|−k /p − 1 C2 ψk ,
p
p
φ |u| − k |u| − k eλ|u|−k ≤ C2 λ eλ|u|−k /p − 1 C2 λψk .
3.14
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International Journal of Differential Equations
Combining I1 and I2 with 3.11 and 3.12 we obtain
α1 p p
λp
Ω
∇ψk p ω2 dx α0 kp−1
Ω
p
ψk ω1 dx
f C0 ω2 φ |u| − k dx
≤
Ak
3.15
f C0 ω2 C2 ψk p dx
≤
Ak1
f C0 ω2 φ |u| − k dx.
Ak−Ak1
Define the function h |f| C0 ω2 . Since f/ω1 ∈ Lq Ω, ω1 and ω2 /ω1 ∈ Lq Ω, ω1 , we have
that h/ω1 ∈ Lq Ω, ω1 . Hence
Ω
p
hψk dx h p 1/q 1/q
ψk ω1 ω1 dx
Ω ω1
h p
ψk .
≤
ω q
Lp q Ω,ω1 1 L Ω,ω1 3.16
If x ∈ Ak − Ak 1, we have k < |u| < k 1. Hence
φ |u| − k eλ|u|−k − 1 ≤ eλ − 1,
3.17
and we obtain
f C0 ω2 φ |u| − k dx ≤
Ak−Ak1
eλ − 1 h dx
Ak−Ak1
≤ e
λ
3.18
h dx.
Ak
By Theorem 2.5, 3.15, and 3.18 we have
α1 pp 1
λp Cp
Ω
Ω
p ∗
ψk ω2 dx
≤ C2
Ω
p/p∗
p
h ψk dx eλ
p−1
α0 k0
Ω
p
ψk ω1 dx
h dx.
Ak
3.19
International Journal of Differential Equations
9
Therefore, there exist positive constants C3 and C4 depending only on Ω, α1 , p, λ, C2 , and
CΩ such that
C3
Ω
p ∗
ψk ω2 dx
≤
Ω
p/p∗
p
h ψk dx p−1
C4 α0 k0
p
ψk ω1 dx
Ω
3.20
h dx.
Ak
Since r2 /r2 − 1 < q, then q < r2 and p < p q < p∗ . For 0 < θ < 1 such that 1/p q θ/p 1 − θ/p∗ , using an interpolation inequality, Young’s inequality with 0 < γ < ∞,
and Hölder’s inequality with exponents q and q we thus obtain
p
h p
ψk hψk dx ≤ ω q
Lp q Ω,ω1 1 L Ω,ω1 Ω
h θp
1−θp
ψk p
ψk p∗
≤
ω q
L Ω,ω1 L Ω,ω1 1 L Ω,ω1 ≤ 1 − θγ
3.21
1/θ
p
p
−1/θ h ψk p
ψk Lp∗ Ω,ω θ γ
.
ω q
L Ω,ω1 1
1 L Ω,ω1 1/1−θ Hence in 3.20 we obtain
C3
Ω
p∗
ψk ω2 dx
p/p∗
p−1
C4 α0 k0
Ω
p
≤ 1 − θγ 1/1−θ ψk Lp∗ Ω,ω p
ψk ω1 dx
1
−1/θ h
θ γ
ω
1
Ak
1/θ
q
L Ω,ω1 p
ψk p
L Ω,ω1 3.22
p
h dx ≤ 1 − θγ 1/1−θ ψk Lp∗ Ω,ω 2
1/θ
h p
ψk p
θ γ −1/θ h dx.
ω q
L Ω,ω1 1 L Ω,ω1 Ak
Now, we can choose γ in order to have 1 − θ γ 1/1−θ C3 /2 and k0 such that
p−1
C4 α0 k0
θ γ
h
ω
−1/θ 1
1/θ
q
L Ω,ω1 3.23
.
We obtain, from 3.22, that for every k ≥ k0 it results
C3
2
Ω
p ∗
ψk ω2 dx
p/p∗
≤
Ak
h
h dx ≤ ω
1
Lq Ω,ω1 1/q
,
μ1 Ak 3.24
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International Journal of Differential Equations
where μ1 Ak ω1 dx. Hence for all k ≥ k0 we have
Ak
Ω
p ∗
ψk ω2 dx ≤
p∗ /p
1/q
2 h μ1 Ak
C3 ω1 Lq Ω,ω1 2
C3
p∗ /p h
ω
1
p∗ /p
q
L Ω,ω1 μ1 Ak
p∗ /pq
3.25
p∗ /pq
C5 μ1 Ak
.
Let us now take l > k ≥ k0 . Then we have
∗
∗ p ∗
l − k p
l−k p
ψk ω1 dx
μ1 Al λ
≤ μ1 Akφ
≤
p
p
Ak
p ∗
ψk ω2 dx.
≤
3.26
Ω
Therefore for all l > k ≥ k0 we obtain by 3.25 and 3.26
∗
∗
l − kp μ1 Al ≤
p∗ /pq
p∗ /pq
pp
C6 μ1 Ak
,
∗ C5 μ1 Ak
p
λ
∗
∗
3.27
that is, μ1 Al ≤ C6 /l − kp μ1 Akp /pq .
Let ϕk μ1 Ak. Since β p∗ /p q > 1, by Lemma 2.6 there exists a constant C7 > 0
such that
μ1 Ak 0,
3.28
∀k ≥ C7 .
Using Lemma 2.7 we have |Ak| 0 for all k ≥ C7 . Therefore any solution u of problem P satisfies the estimate uL∞ Ω ≤ C7 .
4. Proof of Theorem 1.2
Step 1. Let us define for m ∈ N the approximation
Hm
H x, η, ξ
x, η, ξ .
1 1/mH x, η, ξ 4.1
We have that |Hm x, η, ξ| ≤ |Hx, η, ξ|, |Hm x, η, ξ| < m, and Hm x, η, ξ satisfies the
conditions H9 and H12. We consider the approximate problem
− divω2 Ax, u, ∇u gx, u ω1 Hm x, u, ∇u ω2 ,
ux 0,
on ∂Ω.
on Ω,
Pm International Journal of Differential Equations
11
1,p
We say that u ∈ W0 Ω, ω1 , ω2 is weak solution of problem Pm if
Ω
Ax, u, ∇u · ∇ϕω2 dx Ω
Ω
Hm x, u, ∇u ϕω2 dx gx, u ϕω1 dx
4.2
Ω
f ϕ dx,
1,p
for all ϕ ∈ W0 Ω, ω1 , ω2 . We will prove that there exists at least one solution um of the
1,p
problem Pm . For u, v, ϕ ∈ W0 Ω, ω1 , ω2 we define
B u, v, ϕ Bm u, ϕ T ϕ ω2 Ax, u, ∇v · ∇ϕ dx,
Ω
gx, u ϕω1 dx Ω
Ω
Hm x, u, ∇u ϕω2 dx,
4.3
fϕ dx.
Ω
1,p
Then u ∈ W0 Ω, ω1 , ω2 is a weak solution of problem Pm if
B u, u, ϕ Bm u, ϕ T ϕ ,
1,p
4.4
for all ϕ ∈ W0 Ω, ω1 , ω2 .
Let au, v, ϕ Bu, v, ϕ Bm u, ϕ.
i Using H4 we obtain
B u, v, ϕ ≤
p/p
K2 Lp Ω,ω2 h1 L∞ Ω u
× ϕW 1,p Ω,ω1 ,ω2 .
p/p
1,p
W0 Ω,ω1 ,ω2 h2 L∞ Ω v
1,p
W0 Ω,ω1 ,ω2 4.5
0
ii Using H6 and |Hm x, η, ξ| ≤ m, we obtain
Bm u, ϕ ≤
p /p
K1 Lp Ω,ω1 h3 L∞ Ω u
× ϕW 1,p Ω,ω1 ,ω2 .
1,p
W0 Ω,ω1 ,ω2 1 /p
mCΩ μ2 Ω
4.6
0
Hence,
a u, v, ϕ ≤
p/p
K2 Lp Ω,ω2 h1 L∞ Ω h3 L∞ Ω u 1,p
W0 Ω,ω1 ,ω2 p/p
h2 L∞ Ω v
1,p
W0 Ω,ω1 ,ω2 m CΩ
K1 Lp Ω,ω1 1/p ϕ 1,p
.
μ2 Ω
W Ω,ω1 ,ω2 0
4.7
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International Journal of Differential Equations
1,p
1,p
For each u, v ∈ W0 Ω, ω1 , ω2 × W0 Ω, ω1 , ω2 we have that au, v, . is linear and
continuous. Hence, there exists a linear and continuous operator
∗
1,p
1,p
Au, v : W0 Ω, ω1 , ω2 −→ W0 Ω, ω1 , ω2 4.8
such that Au, v, ϕ au, v, ϕ. We set
Au Au, u,
1,p
∀ u ∈ W0 Ω, ω1 , ω2 .
4.9
The operator A : W0 Ω, ω1 , ω2 → W0 Ω, ω1 , ω2 ∗ is semimonotone; that is, by similar
arguments as in the proof of Theorem 2 in 11 we have the following.
1,p
1,p
1,p
i Au, u − Au, v, u − v ≥ 0 for all u, v ∈ W0 Ω, ω1 , ω2 .
1,p
ii For each u ∈ W0 Ω, ω1 , ω2 , the operator v → Au, v is hemicontinuous and
bounded from W0 Ω, ω1 , ω2 to W0 Ω, ω1 , ω2 ∗ and,
1,p
1,p
1,p
for each v ∈ W0 Ω, ω1 , ω2 the operator u → Au, v is hemicontinuous and bounded from
1,p
W0 Ω, ω1 , ω2 to W0 Ω, ω1 , ω2 ∗ .
1,p
1,p
iii If un u in W0 Ω, ω1 , ω2 and Aun , un −Aun , u, un −u → 0, then Aun , u iv
1,p
1,p
Au, v in W0 Ω, ω1 , ω2 ∗ as n → ∞ for all v ∈ W0 Ω, ω1 , ω2 .
1,p
1,p
If v ∈ W0 Ω, ω1 , ω2 , un u in W0 Ω, ω1 , ω2 , and Aun , v
1,p
W0 Ω, ω1 , ω2 ∗ , then Aun , v, un → v, u as n → ∞.
v in
v The operator A : W0 Ω, ω1 , ω2 → W0 Ω, ω1 , ω2 ∗ is bounded.
1,p
1,p
Hence the operator A : W0 Ω, ω1 , ω2 → W0 Ω, ω1 , ω2 ∗ is pseudomonotone see 15.
1,p
1,p
vi By H3, H7, and H12 we have
!
"
p
p
Au, u ≥ α1
|∇u| ω2 dx α0
|u|p ω1 dx ≥ α u
Ω
1,p
W0 Ω,ω1 ,ω2 Ω
,
4.10
where α min{α0 , α1 }. Since p > 1, we have
!
Au, u
"
uW 1,p Ω,ω1 ,ω2 −→ ∞
on uW 1,p Ω,ω1 ,ω2 −→ ∞;
0
4.11
0
that is, the operator A is coercive. Then, by Theorem 27.B in 15, for each T
1,p
W0 Ω, ω1 , ω2 ∗ , the equation
Au T,
1,p
u ∈ W0 Ω, ω1 , ω2 ∈
4.12
1,p
has a solution. Therefore, the problem Pm , has a solution um ∈ W0 Ω, ω1 , ω2 .
International Journal of Differential Equations
13
Step 2. We will show that um ∈ L∞ Ω and um L∞ Ω ≤ C, where C is independent of m. If
1,p
u ∈ W0 Ω, ω1 , ω2 is a solution of problem Pm , we define
⎧
⎪
ux,
⎪
⎪
⎨
un x Tn ux n,
⎪
⎪
⎪
⎩
−n,
if |ux| ≤ n,
4.13
if ux > n,
if ux < −n.
We have Di un Di u if |ux| ≤ n. For k > 0, let us define the function ψn x 1,p
signun x max{|un x| − k, 0}. We have ψn ∈ W0 Ω, ω1 , ω2 ∩ L∞ Ω.
Now consider the function
⎧
⎪
t k,
⎪
⎪
⎨
Φt 0,
⎪
⎪
⎪
⎩
t − k,
if t ≤ −k,
4.14
if | t| ≤ k,
if t ≥ k.
1,p
Since Φ is a Lipschitz function and Φ0 0, then Φψn ∈ W0 Ω, ω1 , ω2 . Moreover,
Di Φψn Φ ψn Di ψn and
Φ un ∇un −→ Φ u∇u,
μ2 − a.e. in Ω.
4.15
|∇un |p ω2 dx.
4.16
We also have, for all measurable subset E ⊂ Ω,
Φ un ∇un p ω2 dx ≤
E
E
By applying Vitali’s Convergence Theorem, with ψ Φu, we obtain
∇ψn −→ ∇ψ
4.17
in Lp Ω, ω2 .
Since ω1 ≤ ω2 , we obtain
ψn − ψ p
≤ ψn − ψ Lp Ω,ω2 ≤ CΩ ∇ψn − ∇ψ Lp Ω,ω2 .
L Ω,ω1 4.18
1,p
Hence ψn → ψ in Lp Ω, ω1 . Since u ∈ W0 Ω, ω1 , ω2 is a solution of problem Pm and
1,p
ψn ∈ W0 Ω, ω1 , ω2 ∩ L∞ Ω, we have
Ω
Ax, u, ∇u · ∇ψn ω2 dx Ω
gx, uψn ω1 dx Ω
Hm x, u, ∇uψn ω2 dx Ω
fψn dx.
4.19
14
International Journal of Differential Equations
Using H4, H6, |Hm x, η, ξ| ≤ m, 4.17, and 4.18, we obtain in 4.19 as n → ∞
Ω
Ax, u, ∇u · ∇ψω2 dx Ω
gx, uψω1 dx Ω
Hm x, u, ∇uψω2 dx Ω
fψ dx. 4.20
Using ϕ ψχAk in 4.2 where Ak {x ∈ Ω : |ux| > k} we obtain
Ax, u, ∇u · ∇ψω2 dx Ak
gx, uψω1 dx Ak
Hm x, u, ∇uψω2 dx
Ak
4.21
f ψ dx.
Ak
Since
⎧
⎪
u k,
⎪
⎪
⎨
ψ Φu 0,
⎪
⎪
⎪
⎩
u − k,
if u ≤ −k,
if |u| ≤ k,
4.22
if u ≥ k,
we obtain the following.
i By H7 we have gx, η η ≥ 0 for all η ∈ R, and
gx, uψω1 dx {u≤−k}
Ak
gx, uu kω1 dx
4.23
{u≥k}
gx, uu − kω1 dx ≥ 0.
ii Using H12 we have Hm x, η, ξ η ≥ 0 for all η ∈ R, and
Hm x, u, ∇uψ ω2 dx Ak
{u≤ −k}
Hm x, u, ∇uu kω2 dx
4.24
{u≥ k}
Hm x, u, ∇uu − kω2 dx ≥ 0.
We have ∇ψ ∇u in Ak. Using H3, i, and ii we obtain in 4.21
|∇u|p ω2 dx ≤
α1
Ak
fψ dx.
Ak
4.25
International Journal of Differential Equations
15
By Theorem 2.1.14 in 13 there is a positive constant C such that
p
ψ ω1 dx ≤
p
ψ ω2 dx ≤ C
Ω
ω
p
∇ψ ω2 dx.
Ω
4.26
Then we obtain
f ψ dx ≤
Ak
Ak
f
ω
1
p 1/p ω1
p
ψ ω1 dx
Ak
Ak
f
ω
p 1/p ω1
Ak
f
ω
p 1/p ω1
≤C
1
C
1
1/p
p
∇ψ ω2 dx
1/p
4.27
Ak
1/p
p
|∇u| ω2 dx
.
Ak
Using 4.27 and Young’s inequality we obtain in 4.25 for all ε > 0
p
|∇u| ω2 dx ≤ C
α1
Ak
Ak
# ≤C ε
f
ω
1
p 1/p ω1
1/p
p
|∇u| ω2 dx
Ak
|∇u|p ω2 dx Cε
Ak
Ak
f
ω
1
$
p ω1 dx ,
4.28
where Cε εp−p /p /p . We can choose ε > 0 so that Cε α1 /2, and there exists a constant
C8 such that
p
|∇u| ω2 dx ≤ C8
Ak
Ak
f
ω
1
p ω1 dx.
4.29
Using Sobolev’s inequality Theorem 2.5 and Hölder’s inequality with exponents q and q
we obtain since q > p r/r − 1 > p p/p∗
p∗
|u| − k ω1 dx
p ∗
ψ ω1 dx
Ak
p/p∗
Ak
p
∇ψ ω2 dx C
≤ C
Ak
p ∗
ψ ω2 dx
≤
f
ω
≤ C C8
1
Ak
f
≤ C9
ω
1
p/p∗
Ak
|∇u|p ω2 dx
Ak
Ω
p ω2 dx
q
p /q
1−p /q
ω1 dx
.
μ1 Ω
4.30
16
International Journal of Differential Equations
Let us now take l > k > 0 and observe that Al ⊂ Ak. Then, from the previous inequality, it
follows that
∗
∗
μ1 All − kp l − kp ω1 dx
Al
p∗
≤
∗
|u| − kp ω1 dx
|u| − k ω1 dx ≤
Al
4.31
Ak
f
≤ C9
ω
Ω
1
q
p p∗ /qp
1−p /qp∗ /p
ω1 dx
.
μ1 Ak
Hence we obtain
C9
μ1 Al ≤
∗
f/ω1 q ω1 dx p p /q p Ω
l − k
p∗
μ1 Ak
1−p /q p∗ /p
.
4.32
Since 1−p /qp∗ /p > 1, by Lemma 2.6 there exists a constant C10 > 0 such that μ1 Ak 0 for all k ≥ C10 , and using Lemma 2.7 we obtain |Ak| 0. Therefore if um is a solution of
problem Pm , we have um L∞ Ω ≤ C10 and C10 is independent of m.
1,p
Step 3. Since um ∈ W0 Ω, ω1 , ω2 ∩ L∞ Ω and um L∞ Ω ≤ C10 , then the sequence {um } is
1,p
relative compact in the strong topology of W0 Ω, ω1 , ω2 by apply the analogous results of
10 and Lemma 2.8. Then, by extracting a subsequence {umk } which strongly converges in
1,p
1,p
1,p
W0 Ω, ω1 , ω2 i.e., there exists u ∈ W0 Ω, ω1 , ω2 such that umk → u in W0 Ω, ω1 , ω2 ,
1,p
we have for all ϕ ∈ W0 Ω, ω1 , ω2 ∩ L∞ Ω
Ω
Ax, umk , ∇umk · ∇ϕω2 Ω
Ω
Hmk x, umk , ∇umk ϕω2 dx −→
Ω
gx, umk ϕω1 dx
gx, uϕω1 dx Ω
Ω
Ax, u, ∇u · ∇ϕω2
4.33
Hx, u, ∇uϕω2 dx.
1,p
Therefore u ∈ W0 Ω, ω1 , ω2 ∩ L∞ Ω is the solution of problem P .
Example 4.1. Let Ω {x, y ∈ R2 : x2 y2 < 1}, and consider the weights ω1 x, y x2 y2 1/2 and ω2 x2 y2 1/4 ω1 and ω2 ∈ A2 , and the functions A : Ω × R × R2 → R2 ,
g : Ω × R → R, and H : Ω × R × R2 → R defined by
A
H
x, y , η, ξ h2 x, y ξ,
g x, y , η η cos2 xy 1 ,
x, y , η, ξ |ξ|2 sin xy arctan η ,
4.34
International Journal of Differential Equations
where h2 x, y 2ex
2
y2
17
. Let us consider the partial differential operator
Lu x, y − div ω2 x, y A x, y , u, ∇u
ω1 x, y g x, y , u
ω2 x, y H x, y , η, ξ ,
4.35
and fx, y x2 y2 3/2q cos1/x2 y2 , with 2r2 /r2 − 1 < q < 6. Therefore, by
Theorem 1.2, the problem
Lu x, y f x, y ,
on Ω,
u x, y 0,
on ∂Ω
P has a solution u ∈ W01,2 Ω, ω1 , ω2 ∩ L∞ Ω.
References
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spaces,” Bulletin of the Belgian Mathematical Society. Simon Stevin, vol. 17, no. 1, pp. 141–153, 2010.
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Mathematics, Springer, Berlin, Germany, 2000.
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