We are interested in the existence of solutions for Dirichlet problem associated to the degenerate quasilinear elliptic equations f(x), on Ω in the setting of the weighted Sobolev spaces .

1. Introduction

In this paper we prove the existence of (weak) solutions in the weighted Sobolev spaces

for the homogeneous Dirichlet problem:

( P )

where L is the partial differential operator:

(1.1)

where Ω is a bounded open set in ℝ^N (N ≥ 2), ω₁ and ω₂ are two weight functions, and the functions 𝒜 : Ω × ℝ × ℝ^N → ℝ^N, g : Ω × ℝ → ℝ, and H : Ω × ℝ × ℝ^N → ℝ are Carathéodory functions.

By a weight, we will mean a locally integrable function ω on ℝ^N such that ω(x) > 0 for a.e. x ∈ ℝ^N. Every weight ω_i (i = 1, 2) gives rise to a measure on the measurable subsets on ℝ^N through integration. This measure will be denoted by μ_i. Thus, μ_i(E) = ∫_Eω_i(x)dx (i = 1,2) for measurable sets E ⊂ ℝ^N.

In general, the Sobolev spaces W^k,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 A_p-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 A_p-weights is the fact that powers of distance to submanifolds of ℝ^N often belong to A_p (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 ω(x) ≡ 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 −div (a(x, u, ∇u)) = h, has at least one solution , and in [1] the author proved the existence of solution when the nonlinear term H(x, η, ξ) is equal to zero.

Firstly, we prove an L^∞ estimate for the bounded solutions of (P): we assume that f/ω₁ ∈ L^q(Ω, ω₁), with r₂/(r₂ − 1) < q < ∞ (where r₂ > 1 as in Theorem 2.5), and we prove that any that solves (P) satisfies , where C depends only on the data, that is, Ω, N, p, q, α₁, α₂, C₀, C₁ and . After that, we prove the existence of solution for problem (P) if f/ω₁ ∈ L^q(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 A_p-weights, 1 < p < ∞, with ω₁ ≤ ω₂. Suppose the following.

(H1)
x ↦ 𝒜(x, η, ξ) is measurable in Ω for all (η, ξ) ∈ ℝ × ℝ^N: (η, ξ) ↦ 𝒜(x, η, ξ) is continuous in ℝ × ℝ^N for almost all x ∈ Ω.
(H2)
[𝒜(x, η, ξ) − 𝒜(x, η′, ξ′)]·(ξ − ξ′) > 0, whenever ξ, ξ′ ∈ ℝ^N, ξ ≠ ξ′.
(H3)
𝒜(x, η, ξ) · ξ ≥ α₁ | ξ|^p, with 1 < p < ∞, where α₁ > 0.
(H4)
|𝒜(x, 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 ∈ L^p′(Ω, ω₂) (1/p + 1/p′ = 1).
(H5)
x ↦ g(x, η) is measurable in Ω for all η ∈ ℝ : η ↦ g(x, η) is continuous in ℝ for almost all x ∈ Ω.
(H6)
|g(x, η)| ≤ K₁(x) + h₃(x) | η|^p/p′, where K₁ and h₃ are positive functions, with h₃ ∈ L^∞(Ω) and K₁ ∈ L^p′(Ω, ω₁).
(H7)
g(x, η) η ≥ α₀ | η|^p, for all η ∈ ℝ, where α₀ > 0.
(H8)
x ↦ H(x, η, ξ) is measurable in Ω for all (η, ξ) ∈ ℝ × ℝ^N: (η, ξ) ↦ H(x, η, ξ) is continuous in ℝ × ℝ^N for almost all x ∈ Ω.
(H9)
|H(x, η, ξ)| ≤ C₀ + C₁ | ξ|^p, where C₀ and C₁ are positive constants.
(H10)
f/ω₁ ∈ L^q(Ω, ω₁), with r₂/(r₂ − 1) < q < ∞ (where r₂ > 1 as in Theorem 2.5) and ω₂/ω₁ ∈ L^q(Ω, ω₁).

Let 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 , such that .

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 ∈ L^q(Ω, ω₁), with p′r₂/(r₂ − 1) < q < ∞;
(H12)
H(x, η, ξ) η ≥ 0, for all η ∈ ℝ.

Then there exists at least one solution of the problem P.

Theorem 1.2 will be proved by approximating problem (P) with the following problems:

( P m )

where H_m(x, η, ξ) = H(x, η, ξ)/1 + (1/m)|H(x, η, ε)|, for m ∈ ℕ. Note that |H_m | ≤|H| and that |H_m | ≤ m.

2. Definitions and Basic Results

Let ω be a locally integrable nonnegative function in ℝ^N and assume that 0 < ω(x) < ∞ almost everywhere. We say that ω belongs to the Muckenhoupt class A_p, 1 < p < ∞, or that ω is an A_p-weight, if there is a constant C = C_p,ω such that

(2.1)

for all balls B ⊂ ℝ^N, where |·| denotes the N-dimensional Lebesgue measure in ℝ^N. If 1 < q ≤ p, then A_q ⊂ A_p (see [4, 7, 12] or [13] for more information about A_p-weights). The weight ω satisfies the doubling condition if μ(2B) ≤ Cμ(B), for all balls B ⊂ ℝⁿ, where μ(B) = ∫_Bω(x)dx and 2B denotes the ball with the same center as B which is twice as large. If ω ∈ A_p, then ω is doubling (see Corollary 15.7 in [4]).

As an example of A_p-weight, the function ω(x) = |x|^α, x ∈ ℝ^N, is in A_p if and only if −N < α < N(p − 1) (see Corollary 4.4, Chapter IX in [7]). If φ ∈ BMO(ℝ^N), then ω(x) = e^αφ(x) ∈ A₂ for some α > 0 (see [6]).

Definition 2.1. Let ω be a weight, and let Ω ⊂ ℝ^N be open. For 0 < p < ∞, we define L^p(Ω, ω) as the set of measurable functions f on Ω such that

(2.2)

Remark 2.2. If ω ∈ A_p, 1 < p < ∞, then since ω^−1/(p−1) is locally integrable, we have for every open set Ω (see Remark 1.2.4 in [13]). It thus makes sense to talk about weak derivatives of functions in L^p(Ω, ω).

Definition 2.3. Let Ω ⊂ ℝ^N be open, 1 < p < ∞, and let ω₁ and ω₂ be A_p-weights, 1 < p < ∞. We define the weighted Sobolev space W^1,p(Ω, ω₁, ω₂) as the set of functions u ∈ L^p(Ω, ω₁) with weak derivatives D_ju ∈ L^p(Ω, ω₂), for j = 1, …, N. The norm of u in W^1,p(Ω, ω₁, ω₂) is given by

(2.3)

The space

is the closure of

with respect to the norm

. The dual space of

is the space

(2.4)

Remark 2.4. (a) If ω ∈ A_p, 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

(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 ℝ^N(N ≥ 2) and ω₂ ∈ A_p(1 < p < ∞). There exist constants C_Ω and δ positive such that for all and all r₂ satisfying 1 ≤ r₂ ≤ N/(N − 1) + δ,

(2.6)

where p^* = p r₂.

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

(2.7)

for all l > k ≥ k₀. Then φ(k₀ + d) = 0, where

Lemma 2.7. If ω ∈ A_p, then (|E|/|B|) ^p ≤ C_p,ω(μ(E)/μ(B)), whenever B is a ball in ℝ^N and E is a measurable subset of B.

Proof. See Theorem 15.5 Strong doubling of A_p-weights in [4].

By Lemma 2.7, if μ(E) = 0, then |E | = 0.

Lemma 2.8. Let ω₁ and ω₂ be A_p-weights, 1 < p < ∞, ω₁ ≤ ω₂, and a sequence {u_n}, satisfies the following:

(i)
u_n⇀u in and μ₂-a.e. in Ω;
(ii)
∫_Ω〈𝒜(x, u_n, ∇u_n) − 𝒜(x, u_n, ∇u), ∇(u_n − u)〉 ω₂ dx → 0 with n → ∞.

Then u_n → u in .

Proof. The proof of this lemma follows the line of Lemma 5 in [10].

Definition 2.9. We say that is a (weak) solution of problem (P) if

(2.8)

for all

3. Proof of Theorem 1.1

Set λ = (C₁/α₁) + 1 and define for k > 0 the functions ϕ ∈ C¹(ℝ) and G_k ∈ W^1,∞(ℝ) by

(3.1)

is a solution of problem (P), define the set A(k) = {x ∈ Ω:|u(x)| > k}. We will use the test functions v(x) = ϕ(G_k(u(x))). Since

and Ω is a bounded set in ℝ^N, then

and

(3.2)

where χ_A(k) is the characteristic function of the set A(k) (by sign (u(x)) we mean the function equal to +1 for u(x) > 0 and to −1 for u(x) < 0).

Since , we have that .

Using the function v in (2.8) we obtain

(3.3)

We have the following estimates.

(i)
By (H3) we obtain

(3.4)

(ii)
By (H7) we obtain

(3.5)

(iii)
Using (H9) we obtain

(3.6)

And we also have

(3.7)

Hence in (3.3) we obtain

(3.8)

Since λ = (C₁/α₁) + 1, we have for s ≥ 0

(3.9)

Hence in (3.8) we obtain

(3.10)

Using (3.9) and k<|u(x)| if x ∈ A(k), we obtain

(3.11)

Let us define the function ψ_k by ψ_k(x) = ϕ((|u(x)| − k) ⁺/p). We have that

and

(3.12)

We have the following.

(a)
For all s ≥ 0, e^λs − 1 ≥ (e^λs/p − 1) ^p.
(b)
There exists, a constant C₂ > 0 (C₂ = C₂(λ, p)) such that for all s ≥ 1

(3.13)

This implies the following.

(I1)
a.e. on Ω.
(I2)
If x ∈ A(k + 1), then

(3.14)

Combining (I1) and (I2) with (3.11) and (3.12) we obtain

(3.15)

Define the function h = |f | + C₀ ω₂. Since f/ω₁ ∈ L^q(Ω, ω₁) and ω₂/ω₁ ∈ L^q(Ω, ω₁), we have that h/ω₁ ∈ L^q(Ω, ω₁). Hence

(3.16)

If x ∈ A(k) − A(k + 1), we have k<|u | < k + 1. Hence

(3.17)

and we obtain

(3.18)

By Theorem 2.5, (3.15), and (3.18) we have

(3.19)

Therefore, there exist positive constants C₃ and C₄ (depending only on Ω, α₁, p, λ, C₂, and C_Ω) such that

(3.20)

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

(3.21)

Hence in (3.20) we obtain

(3.22)

Now, we can choose γ in order to have (1 − θ) γ^1/(1−θ) = C₃/2 and k₀ such that

(3.23)

We obtain, from (3.22), that for every k ≥ k₀ it results

(3.24)

where

. Hence for all k ≥ k₀ we have

(3.25)

Let us now take l > k ≥ k₀. Then we have

(3.26)

Therefore for all l > k ≥ k₀ we obtain (by (3.25) and (3.26))

(3.27)

that is,

Let φ(k) = μ₁(A(k)). Since β = p^*/p q′ > 1, by Lemma 2.6 there exists a constant C₇ > 0 such that

(3.28)

Using Lemma 2.7 we have |A(k)| = 0 for all k ≥ C₇. Therefore any solution u of problem (P) satisfies the estimate .

4. Proof of Theorem 1.2

Step 1. Let us define for m ∈ ℕ the approximation

(4.1)

( P m )

We say that

is (weak) solution of problem (Pm) if

(4.2)

for all

. We will prove that there exists at least one solution u_m of the problem (Pm). For

we define

(4.3)

Then

is a (weak) solution of problem (Pm) if

(4.4)

Let a(u, v, φ) = B(u, v, φ) + B_m(u, φ).

(i)
Using (H4) we obtain

(4.5)

(ii)
Using (H6) and |H_m(x, η, ξ)| ≤ m, we obtain

(4.6)

Hence,

(4.7)

For each

we have that a(u, v.) is linear and continuous. Hence, there exists a linear and continuous operator

(4.8)

such that 〈A(u, v), φ〉 = a(u, v, φ). We set

(4.9)

The operator

is semimonotone; that is, by similar arguments as in the proof of Theorem 2 in [11] we have the following.

(i)
〈A(u, u) − A(u, v), u − v〉≥0 for all .
(ii)
For each , the operator v ↦ A(u, v) is hemicontinuous and bounded from to and,

for each

the operator u ↦ A(u, v) is hemicontinuous and bounded from

(iii)
If u_n⇀u in and 〈A(u_n, u_n) − A(u_n, u), u_n − u〉→ 0, then A(u_n, u)⇀A(u, v) in as n → ∞ for all .
(iv)
If , u_n⇀ u in , and in , then as n → ∞.
(v)
The operator is bounded.

Hence the operator

is pseudomonotone (see [15]).

(vi)
By (H3), (H7), and (H12) we have

(4.10)

where α = min {α₀, α₁}. Since p > 1, we have

(4.11)

that is, the operator

is coercive. Then, by Theorem 27.B in [15], for each

, the equation

(4.12)

has a solution. Therefore, the problem (Pm), has a solution

Step 2. We will show that u_m ∈ L^∞(Ω) and , where C is independent of m. If is a solution of problem (P_m), we define

(4.13)

We have D_iu_n = D_iu if |u(x)| ≤ n. For k > 0, let us define the function ψ_n(x) = sign (u_n(x)) max {|u_n(x)| − k, 0}. We have

Now consider the function

(4.14)

Since Φ is a Lipschitz function and Φ(0) = 0, then

. Moreover, D_iΦ(ψ_n) = Φ′(ψ_n)D_iψ and

(4.15)

We also have, for all measurable subset E ⊂ Ω,

(4.16)

By applying Vitali’s Convergence Theorem, with ψ = Φ(u), we obtain

(4.17)

Since ω₁ ≤ ω₂, we obtain

(4.18)

Hence ψ_n → ψ in L^p(Ω, ω₁). Since

is a solution of problem (Pm) and

, we have

(4.19)

Using (H4), (H6), |H_m(x, η, ξ) | ≤ m, (4.17), and (4.18), we obtain in (4.19) as n → ∞

(4.20)

Using φ = ψχ_A(k) in (4.2) (where A(k) = {x ∈ Ω : | u(x)| > k}) we obtain

(4.21)

Since

(4.22)

we obtain the following.

(i)
By (H7) we have g(x, η) η ≥ 0 for all η ∈ ℝ, and

(4.23)

(ii)
Using (H12) we have H_m(x, η, ξ) η ≥ 0 for all η ∈ ℝ, and

(4.24)

We have ∇ψ = ∇u in A(k). Using (H3), (i), and (ii) we obtain in (4.21)

(4.25)

By Theorem 2.1.14 in [13] there is a positive constant C such that

(4.26)

Then we obtain

(4.27)

Using (4.27) and Young’s inequality we obtain in (4.25) (for all ε > 0)

(4.28)

where C(ε) = (εp) ^−p′/p/p′. We can choose ε > 0 so that Cε = α₁/2, and there exists a constant C₈ such that

(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^′)

(4.30)

Let us now take l > k > 0 and observe that A(l) ⊂ A(k). Then, from the previous inequality, it follows that

(4.31)

Hence we obtain

(4.32)

Since (1 − (p^′/q))(p^*/p) > 1, by Lemma 2.6 there exists a constant C₁₀ > 0 such that μ₁(A(k)) = 0 for all k ≥ C₁₀, and using Lemma 2.7 we obtain |A(k)| = 0. Therefore if u_m is a solution of problem (P_m), we have

and C₁₀ is independent of m.

Step 3. Since and , then the sequence {u_m} is relative compact in the strong topology of (by apply the analogous results of [10] and Lemma 2.8). Then, by extracting a subsequence which strongly converges in (i.e., there exists such that in ), we have for all

(4.33)

Therefore

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

(4.34)

where

. Let us consider the partial differential operator

(4.35)

and f(x, y) = (x² + y²) ^3/2q cos (1/(x² + y²)), with 2r₂/(r₂ − 1) < q < 6. Therefore, by Theorem 1.2, the problem

( P )

has a solution

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L^∞-Solutions for Some Nonlinear Degenerate Elliptic Equations

Abstract

1. Introduction

2. Definitions and Basic Results

3. Proof of Theorem 1.1

4. Proof of Theorem 1.2

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L∞-Solutions for Some Nonlinear Degenerate Elliptic Equations

Abstract

1. Introduction

2. Definitions and Basic Results

3. Proof of Theorem 1.1

4. Proof of Theorem 1.2

References

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L^∞-Solutions for Some Nonlinear Degenerate Elliptic Equations