RESEARCH ARTICLE A Caffarelli-Kohn-Nirenberg inequality in Orlicz ...

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Applicable Analysis Vol. 00, No. 00, January 2008, 1–11

RESEARCH ARTICLE A Caffarelli-Kohn-Nirenberg inequality in Orlicz-Sobolev spaces and applications Marian Boceaa and Mihai Mih˘ailescub∗ a

Department of Mathematics, North Dakota State University, NDSU Dept. # 2750, P.O. Box 6050, Fargo, ND 58108-6050 U.S.A. b Department of Mathematics, University of Craiova, 200585 Craiova, Romania (v3.3 released January 2011) A generalization of the classical Caffarelli-Kohn-Nirenberg inequality is obtained in the setting of Orlicz-Sobolev spaces. As applications, we prove a compact embedding result, and we establish the existence of weak solutions of the Dirichlet problem for a non-homogeneous and degenerate/singular elliptic PDE.

Keywords: Caffarelli-Kohn-Nirenberg inequality; critical point; degenerate elliptic equations; Orlicz spaces AMS Subject Classification: 35D30; 35J70; 35J75; 49J45

1.

Introduction

In the context of more general interpolation inequalities, L. Caffarelli, R. Kohn & L. Nirenberg proved in [1] the existence of a positive constant C(p, q, α, β) such that (∫

−βq

|x|

)p/q ∫ |u| dx ≤ C(p, q, α, β) |x|−αp |∇u|p dx, ∀ u ∈ Cc∞ (Ω) , q

Ω

(1)

Ω

where N ≥ 1 is a positive integer, Ω ⊆ RN is an arbitrary open domain, and the constants p, q, α and β satisfy 1 < p < N,

−∞ < α
0 Φ(t) t>0 Φ(t)

ϕ0 := inf we will assume that 1 < ϕ0 ≤

tϕ(t) ≤ ϕ0 < ∞, Φ(t)

for all t > 0 .

(5)

Before stating our main result we indicate several examples of functions ϕ : R → R which are odd increasing homeomorphisms from R onto R, and for which (5) holds. For more details the reader is referred to [11, Examples 1-3, p. 243]. 1) ϕ(t) = |t|p−2 t, with p > 1. It can be showed that ϕ0 = ϕ0 = p; 2) ϕ(t) = log(1 + |t|r )|t|p−2 t, with p, r > 1. In this case ϕ0 = p, and ϕ0 = p + r;

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|t| t 3) ϕ(t) = log(1+|t|) , if t ̸= 0, ϕ0 = p − 1 and ϕ0 = p. p−2

3

ϕ(0) = 0, with p > 2. In this case it turns out that

The main result of this paper is the following theorem. Let Φ be defined by (4), N ≥ 2 be an integer, and α ∈ R be such

Theorem 1.1 : that

N − ϕ0 + 1 + α > 0 .

(6)

If (5) holds, then for any open domain Ω ⊂ RN with smooth boundary we have ∫ ∫ 1 α |x| Φ(|u(x)|) dx ≤ |x|α Φ(|x||∇u(x)|) dx , (7) N − ϕ0 + 1 + α Ω Ω for all u ∈ Cc∞ (Ω). In view of Lemma 2.2 in the next section of the paper we obtain the following Corollary 1.2: Under the same hypotheses as in Theorem 1.1 the following inequality holds: ∫ ∫ 1 α |x| Φ(|u(x)|) dx ≤ γ(|x|)Φ(|∇u(x)|) dx , (8) N − ϕ0 + 1 + α Ω Ω for all u ∈ Cc∞ (Ω), where { γ(t) :=

tϕ0 +α if t ∈ (0, 1] 0 tϕ +α if t ∈ (1, ∞) .

Remark. In the particular case where ϕ(t) = |t|p−2 t we have Φ(t) = |t|p and ϕ0 = ϕ0 = p. Thus, in this case, in Corollary 1.2 we have γ(t) = tp+α , and we recover the result of [12, Theorem 20.7] but with a different constant on the right hand side. This is due to the fact that the estimates used to prove these two results are different. Precisely, the proof of [12, Theorem 20.7] uses Hölder’s inequality while in the proof of Corollary 1.2, mainly due to the lack of homogeneity, we need to make use of Young’s inequality and of the estimates given by Lemmas 2.1 and 2.2 below. p

2.

Auxiliary results

Lemma 2.1:

Assume that (5) holds. Then Φ⋆ (ϕ(s)) ≤ (ϕ0 − 1)Φ(s),

for all s ≥ 0 .

Proof : We have ∫

t

Φ⋆ (t) =

ϕ−1 (s) ds .

0

It follows that ∫ ⋆

Φ (ϕ(s)) = 0

ϕ(s)

ϕ−1 (σ) dσ .

(9)

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Changing variables, σ = ϕ(r), we have ∫ ⋆

Φ (ϕ(s)) =

s

∫

′

rϕ (r) dr = sϕ(s) −

0

s

ϕ(r) dr = sϕ(s) − Φ(s),

for all s ≥ 0 .

0

Taking into account (5), we deduce that (9) holds. Lemma 2.2:

Assume that (5) holds, and define β : (0, ∞) → R by { β(σ) :=

σ ϕ0 if σ ∈ (0, 1] 0 σ ϕ if σ ∈ (1, ∞) .

Then Φ(σt) ≤ β(σ)Φ(t),

for all t > 0 and σ > 0 .

Proof : We first consider the case where σ > 1. Since deduce that ∫ log(Φ(σt)) − log(Φ(t)) =

σt

t

ϕ(s) ds ≤ Φ(s)

∫

σt

t

tϕ(t) Φ(t)

(10)

≤ ϕ0 for all t > 0 we

ϕ0 0 ds = log(σ ϕ ) , s

for all t > 0. Thus, 0

Φ(σt) ≤ σ ϕ Φ(t),

for all t > 0 and σ > 1 .

Next, assume that σ ∈ (0, 1). Using the fact that that ∫

t

log(Φ(t))−log(Φ(σt)) = σt

ϕ(s) ds ≥ Φ(s)

∫

t

σt

tϕ(t) Φ(t)

(11)

≥ ϕ0 for all t > 0, we deduce

ϕ0 ds = ϕ0 [log(t)−log(σt)] = − log(σ ϕ0 ) , s

for all t > 0. Hence, Φ(σt) ≤ σ ϕ0 Φ(t),

for all t > 0 and σ ∈ (0, 1) .

In view of (11) and (12), we deduce that (10) holds.

3.

(12)

Proof of Theorem 1.1

Following ideas in [12, Theorem 20.7], we start by noting that after approximating |x|α by (|x|2 + ϵ)α/2 we find div(|x|α x) = (α + N )|x|α , in the sense of distributions.

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Let u ∈ Cc∞ (Ω) be arbitrary. We have div(Φ(|u(x)|)|x|α x) =

N ∑ ∂ (Φ(|u(x)|)|x|α xi ) ∂xi i=1

= N |x| Φ(|u(x)|) + α

N ∑

[ xi ϕ(|u(x)|)

i=1 α

∂ (|u(x)|)|x|α + Φ(|u(x)|)α|x|α−2 xi ∂xi α

= (N + α)|x| Φ(|u(x)|) + ϕ(|u(x)|)|x|

N ∑ i=1

xi

]

∂ (|u(x)|), ∂xi

for a.e. x ∈ Ω. After integrating this over ω, an open and sufficiently smooth subset of Ω such that supp u ⊂ ω, we obtain ∫

∫

∫ N ∑ ∂ div(Φ(|u(x)|)|x| x) dx = (N +α) |x| Φ(|u(x)|) dx+ ϕ(|u(x)|)|x|α xi (|u(x)|) dx. ∂xi ω ω ω α

α

i=1

Thus, by the Divergence Theorem, ∫

∫ |x| Φ(|u(x)|) dx = − α

(N + α) ω

α

ϕ(|u(x)|)|x| ω

xi

i=1

∫ =−

N ∑

α

ϕ(|u(x)|)|x| ω

N ∑ i=1

∂ (|u(x)|) dx ∂xi

∂u xi (x) dx. ∂xi

Since ϕ(|u(x)|) ≥ 0 for x ∈ ω, we deduce that ∫ ∫ α (N + α) |x| Φ(|u(x)|) dx ≤ ϕ(|u(x)|)|x|α |x| |∇u(x)| dx. ω

ω

Thus, using the fact that supp u ⊂ ω, and taking into account the properties of Φ, we obtain ∫ ∫ α (N + α) |x| Φ(|u(x)|) dx ≤ ϕ(|u(x)|)|x|α |x| |∇u(x)| dx. Ω

Ω

Next, using Young’s inequality in the right hand side of the above inequality, we have ∫ ∫ ∫ α α (N + α) |x| Φ(|u(x)|) dx ≤ |x| Φ(|x||∇u(x)|) dx + |x|α Φ⋆ (ϕ(|u(x)|)) dx . Ω

Ω

Ω

Finally, taking into account (9), we find ∫ ∫ ∫ α α 0 (N +α) |x| Φ(|u(x)|) dx ≤ |x| Φ(|x||∇u(x)|) dx+(ϕ −1) |x|α Φ(|u(x)|) dx Ω

Ω

Ω

or, equivalently, ∫ (N − ϕ + 1 + α)

∫ |x| Φ(|u(x)|) dx ≤

0

|x|α Φ(|x||∇u(x)|) dx .

α

Ω

Ω

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We conclude that (7) holds.

4.

Applications

In this section we discuss some applications of Theorem 1.1 to the study of a degenerate boundary value problem. We will assume from now on that Ω = B1 (0) is the ball centered in the origin of radius 1 in RN (N ≥ 2), and we take α = 0. Hence, inequality (7) becomes ∫

1 Φ(|u(x)|) dx ≤ N − ϕ0 + 1 Ω

∫ Φ(|x||∇u(x)|) dx ,

(13)

Ω

for all u ∈ Cc∞ (Ω). Before we can proceed further, we need to recall some basic facts about Orlicz spaces. For more details we refer to the books by D. R. Adams & L. L. Hedberg [10], R. Adams [9], and M. M. Rao & Z. D. Ren [13], and to the papers by Ph. Clément et al. [11, 14], M. Garc´ıa-Huidobro et al. [15] and J. P. Gossez [16]. With ϕ, Φ, and Φ⋆ as defined in the Introduction, the Orlicz space LΦ (Ω) is the space of measurable functions u : Ω → R such that {∫

∫

∥u∥LΦ := sup

uv dx : Ω

} Φ⋆ (|v|) dx ≤ 1 < ∞ .

Ω

Endowed with the norm ∥ · ∥LΦ , LΦ (Ω) is a Banach space. We note that ∥ · ∥LΦ is equivalent to the Luxemburg norm, defined by { ( ) } ∫ u(x) ∥u∥Φ := inf µ > 0 : Φ dx ≤ 1 . µ Ω In the context of Orlicz spaces Hölder’s inequality reads as follows (see [13, Inequality 4, p. 79]): ∫ uvdx ≤ 2 ∥u∥LΦ ∥v∥LΦ⋆

for all u ∈ LΦ (Ω) and v ∈ LΦ (Ω). ⋆

Ω

The Orlicz-Sobolev space W 1,Φ (Ω) is defined by W

1,Φ

{ } ∂u Φ Φ (Ω) := u ∈ L (Ω) : ∈ L (Ω), i = 1, · · · , N , ∂xi

and it is a Banach space with respect to the norm ∥u∥1,Φ := ∥u∥Φ + ∥|∇u|∥Φ . In what follows W01,Φ (Ω) stands for the closure of Cc∞ (Ω) in W 1,Φ (Ω). By [16, Lemma 5.7], ∥u∥0,Φ := ∥|∇u|∥Φ . is an equivalent norm on W01,Φ (Ω). Let ϕ0 and ϕ0 be defined as in the Introduction, and assume that (5) holds. We note that this implies that Φ satisfies the ∆2 -

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condition: Φ(2t) ≤ KΦ(t), ∀ t ≥ 0 ,

(14)

where K is a positive constant (see [18, Proposition 2.3]). On the other hand (see, e.g. [17, Lemma 2.1] or [18, Proposition 2.1]), we have 0 ∥u∥ϕΦ

∫ ≤ Ω

and

Φ(|u(x)|) dx ≤ ∥u∥ϕΦ0 , ∀ u ∈ LΦ (Ω), ∥u∥Φ < 1,

∫ ∥u∥ϕΦ0 ≤

Ω

0

Φ(|u(x)|) dx ≤ ∥u∥ϕΦ , ∀ u ∈ LΦ (Ω), ∥u∥Φ > 1.

(15)

(16)

√ Finally, we assume that Φ is such that the map [0, ∞) ∋ t → Φ( t) is convex. We note that this, together with (14), implies that the Orlicz space LΦ (Ω) is an uniformly convex (and hence reflexive) Banach space (see [18, Proposition 2.2]). Remark. Let ϕ(t) = |t|p−2 t,

∀ t ∈ R,

with p > 1. As we already mentioned in example 1) in the Introduction, it can be shown that in this case we have ϕ0 = ϕ0 = p. Moreover, in this particular case the corresponding Orlicz space LΦ (Ω) reduces to the classical Lebesgue space Lp (Ω) while the Orlicz-Sobolev space W 1,Φ (Ω) becomes the classical Sobolev space W 1,p (Ω). In what follows D01,Φ (Ω) stands for the closure of Cc∞ (Ω) under the norm ∥u∥ := ∥ |x||∇u(x)| ∥Φ . √ The convexity of the map [0, ∞) ∋ t → Φ( t) ensures that D01,Φ (Ω) is a reflexive Banach space.

4.1.

A compactness result

In this section we prove the following compact embedding theorem: Theorem 4.1 : B1 (0), and ϕ0 > ) ( 2N ϕ0 q ∈ 1, 2N +ϕ0 .

Assume that the hypotheses of Theorem 1.1 are satisfied, Ω = Then D01,Φ (Ω) is compactly embedded into Lq (Ω) for every

2N 2N −1 .

Proof : First, note that under our assumptions LΦ (Ω) is continuously embedded into Lϕ0 (Ω), and that W 1,Φ (Ω) is continuously embedded in W 1,ϕ0 (Ω). Indeed, by [9, Lemma 8.12(b)] it is enough to show that Φ dominates Ψ, defined by Ψ(t) := |t|ϕ0 near infinity, i.e. there exists k > 0 and t0 > 0 such that Ψ(t) ≤ Φ(k · t), ∀ t ≥ t0 . That is a simple consequence of the definition of ϕ0 (see, e.g., the proof of [19, Lemma 2] for more details).

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Let {un } be a bounded sequence in D01,Φ (Ω). In view of our Theorem 1.1, {un } is bounded in LΦ (Ω). Let ε0 ∈ (0, 1) be such that B ε0 (0) ⊂ Ω, and let ε ∈ (0, ε0 ) be arbitrary. We deduce that {un } is bounded in W 1,Φ (Ω \ B ε (0)), and since W 1,Φ (Ω \ B ε (0)) ⊂ W 1,ϕ0 (Ω \ B ε (0)), the sequence {un } is bounded in W 1,ϕ0 (Ω \ B ε (0)). The classical compact embedding theorem implies that there exists a subsequence of {un }, still denoted by {un }, which converges in Lq (Ω \ B ε (0)). Thus, for sufficiently large n, m ∈ N, we have ∫ |un − um |q dx < ε.

(17)

Ω\B ε (0)

On the other hand, Hölder’s inequality implies that ∫

∫

|x|− 2 |x| 2 |un − um |q dx

Bε (0)

q

q

|un − um | dx = q

Bε (0)

q

≤ |x|− 2 χBε (0)

L

ϕ0 ϕ0 −q

q

2

|x| |un − um |q

L

(Ω)

ϕ0 q

(Ω)

.

Taking into account (13), (15), and (16), we obtain (∫

q

2 q |x| |u − u |

n m

L

ϕ0 q

(Ω)

|x|

=

)q/ϕ0 |un − um |ϕ0 dx

ϕ0 2

Ω

≤ ∥un − um ∥qLϕ0 (Ω) ≤ D2 ∥un − um ∥qΦ [( ∫ )q (∫ ) q0 ] ϕ0 ϕ Φ(|un − um |) dx ≤ D2 + Φ(|un − um |) dx Ω

Ω

[( ∫

) Φ(|x||∇(un − um )|) dx

≤ D3

q ϕ0

(∫ +

Ω

) q0 ] ϕ Φ(|x||∇(un − um )|) dx ,

Ω

where D2 and D3 are positive constants. Hence, there exists a constant M > 0 such that ∫

q

|un − um |q dx ≤ M |x|− 2 χBε (0)

ϕ0

L ϕ0 −q (Ω)

Bε (0)

.

Next, since

− q2

|x| χBε (0)

(∫ ϕ0

L ϕ0 −q (Ω)

|x|

=

= =

ε

ωN r (

qϕ0 2(ϕ0 −q)

)(ϕ0 −q)/ϕ0 dx

Bε (0)

(∫

where α = N −

−qϕ0 2(ϕ0 −q)

0

ωN εα α

N −1

r

−qϕ0 2(ϕ0 −q)

)(ϕ0 −q)/ϕ0 dr

)(ϕ0 −q)/ϕ0 ,

> 0 and ωN is the area of the unit ball in RN , we deduce

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that there exist two constants, β > 0 and M1 > 0, such that ∫ |un − um |q dx ≤ M1 εβ . Bε (0)

Combining this with (17) gives ∫ |un − um |q dx ≤ M2 (ε + εβ ) , Ω

where M2 is a positive constant. Since ε ∈ (0, ε0 ) was arbitrary, it follows that {un } is a Cauchy (and hence, convergent) sequence in Lq (Ω). This concludes the proof.

4.2.

Existence of solutions for a singular PDE

Under the hypotheses of Theorems 1.1 and 4.1, we investigate the existence of weak solutions for the problem {

−div(|x|2 a(|x||∇u(x)|)∇u(x)) = λ|u(x)|q−2 u(x) for x ∈ Ω, u(x) = 0 for x ∈ ∂Ω ,

(18)

where Ω = B1 (0) ⊂ RN (N ≥ 2), λ is a positive constant, q ≥ 2, and a : (0, ∞) → R is a function such that the mapping ϕ : R → R defined by { ϕ(t) =

a(|t|)t if t ̸= 0, 0 if t = 0 ,

is an odd, increasing homeomorphism from R onto R satisfying (5) and such that √ [0, ∞) ∋ t → Φ( t) is convex. One particular choice of a map a which satisfies the above requirements is a(x) = xp−2 , with p ≥ 2. We say that u ∈ D01,Φ (Ω) is a weak solution of problem (18) if ∫

∫ |x| a(|x||∇u(x)|)∇u(x)∇v(x) dx−λ 2

Ω

Ω

|u(x)|q−2 u(x)v(x) dx = 0, ∀ v ∈ D01,Φ (Ω).

Theorem 4.2 : Assume that the hypotheses of Theorems 1.1 and 4.1 are satisfied. Then, for each λ > 0, the problem (18) has a nontrivial weak solution. Proof : For λ > 0, the energy functional associated to problem (18) is Jλ : D01,Φ (Ω) → R, defined by ∫

λ Jλ (u) = Φ(|x||∇u(x)|) dx − q Ω

∫ |u(x)|q dx. Ω

Indeed, standard ( ) arguments (see, e.g. [18, Proposition 4.1]) show that Jλ ∈ 1,Φ C 1 D0 (Ω), R and that ⟨Jλ′ (u), v⟩

∫

∫ |x| a(|x||∇u(x)|)∇u(x)∇v(x) dx − λ

|u(x)|q−2 u(x)v(x) dx ,

2

= Ω

Ω

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for u, v ∈ D01,Φ (Ω). Therefore, u is a weak solution of (18) if and only if it is a critical point of Jλ . Consequently, it suffices to show that Jλ possesses a nontrivial critical point. To this aim, we claim that the following assertions hold: (a) Jλ is weakly lower semi-continuous; (b) Jλ is bounded from below and coercive; (c) there exists θ ∈ D01,Φ (Ω) \ {0} such that Jλ (θ) < 0. The arguments for proving (a) are very similar to those used in the proof of [18, Proposition 4.4], and we will not present them here. To prove (b), first note that if u ∈ D01,Φ (Ω) is such that ∥u∥ > 1, we deduce, by (16) and Theorem 4.1, that there exists a constant K > 0 such that Jλ (u) ≥ ∥u∥ϕ0 −

Kλ ∥u∥q . q

2N ϕ0 Taking into account the fact that 1 < q < 2N +ϕ0 < ϕ0 , we have Jλ (u) → ∞ as ∥u∥ → ∞. Thus, Jλ is coercive. On the other hand, making use again of (15) and (16), we obtain that

{ } Kλ 0 Jλ (u) ≥ min ∥u∥ϕ0 , ∥u∥ϕ − ∥u∥q q for any u ∈ D01,Φ (Ω). It follows that Jλ is bounded from below. It remains to prove (c). To this aim, let θ ∈ Cc∞ (Ω), θ ̸= 0. In view of Lemma 2.2, for each t ∈ (0, 1) we have ∫

∫ λ Jλ (tθ) = Φ(|x||∇(tθ)(x)|) dx − |(tθ)(x)|q dx q Ω Ω ∫ q ∫ λt ≤ tϕ0 Φ(|x||∇θ(x)|) dx − |θ(x)|q dx. q Ω Ω Thus, there exist two positive constants, L1 and L2 , such that Jλ (tθ) ≤ L1 tϕ0 − L2 tq , for each 1). { (t ∈) (0, } Since q < ϕ0 , this means that Jλ (tθ) < 0 for any 0 < t < 1 L2 ϕ0 −q min 1, L . 1 From (a) and (b) above we obtain, via the Direct Method of the Calculus of Variations (see, e.g. [20, Theorem 1.2 ]), that there exists uλ ∈ D01,Φ (Ω) a global minimum point of Jλ . Moreover, assertion (c) guarantees that uλ ̸= 0. Standard arguments relying on Theorem 4.1 show that uλ is in fact a critical point of Jλ and hence, a nontrivial weak solution of (18). This concludes the proof. Acknowledgements. The research of M. Bocea was partially supported by the U.S. National Science Foundation under Grant No. DMS-0806789. M. Mih˘ailescu has been partially supported by the Grant CNCSIS PD-117/2010 “Probleme neliniare modelate de operatori diferent¸iali neomogeni”.

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