Ammari H., Benkirane A., Touzani A. (editors) Recent Developments in Nonlinear Analysis

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December 28, 2009 12:15 WSPC - Proceedings Trim Size: 9in x 6in recent

Here we present a survey on some recent existence and L

∞

-regularity

results for the Dirichlet problem associated to nonlinear elliptic equations

on bounded open subset Ω of IR

, N ≥ 2, involving the following non-

everywhere nonlinear diﬀerential operator of Leray-Lions type

A(u) := −div a(x, u, ∇u).

The function a may have nonstandard growth and satisﬁes the condition:

a(x, s, ξ)·ξ ≥

−1

(M(h(|s|)))M(|ξ|), (9)

where M is an N-function (take as example M(t) = t

, p > 1) and h :

→ ]0, +∞[ is a continuous decreasing function such that : 0 < h(0)≤1

and its primitive

H(s) =

h(t)dt (10)

is unbounded (take for instance h(t) =

(1+t)

, 0 ≤ θ ≤ 1.)

It is worth recalling that in the L

setting, a priori bounds for solutions

of nonlinear elliptic Dirichlet problems can be derived by using either the

method of Stampacchia

or by means of the rearrangements techniques

widely developed in Talenti.

17–20

Observe that because of the lack of coercivity generated by the as-

sumption (9), the operator A degenerates when its second argument has

large values. Therefore, the classical existence results in Leray-Lions

and

Gossez-Mustonen

used to prove the existence of a solution for the prob-

lem (4) cannot be applied even if the datum f is very regular. To get rid

of this diﬃculty, we will consider approximate equations in which we intro-

duce a truncation, in order to get non degenerate approximated problems

which thus have solution. Once this done, some a priori estimates on the

solutions of these problems are proved. To complete the ingredients which

allow us to pass to the limit, we prove the almost everywhere convergence

of the gradients of solutions.

2. Standard growth

One stands in the L

setting. Let M(t) = t

with 1 < p < N. It is known

that the Dirichlet problem



A(u) = f in Ω,

u = 0 on ∂Ω,

(11)

has bounded solutions provided that the datum f satisﬁes (6). Even in the

case when h is the constant function, the limit case f ∈ L

yields a solution

December 28, 2009 12:15 WSPC - Proceedings Trim Size: 9in x 6in recent

u in a suitable Orlicz space of exponential type and it is (in general) not

bounded. Precisely, H(u) ∈ L

(Ω) with φ(t) = exp(t) −t − 1 and H is the

function deﬁned in (10).

We consider a subclass of L

(Ω) larger than (6) which guaranties the

boundedness of solutions of the problem (11).

In addition of (9), assume that the Carath´eodory function a : Ω×IR×IR

→

satisﬁes for almost x in Ω, for every s ∈ IR and for every ξ, η ∈ IR

with ξ 6= η

|a(x, s, ξ)| ≤ a

(x) + |s|

p−1

+ |ξ|

p−1

(12)

where a

is a non negative function in L

(Ω) with p

p−1

, and

(a(x, s, ξ) − a(x, s, η)).(ξ − η) > 0. (13)

Let us denote by L

log

L the Orlicz space generated by the N -function

Θ(t) = t

log

(e + t). Assume that

f ∈ L

log

L with α >

N(p −1)

. (14)

Our ﬁrst result is the following

Theorem 2.1 (Benkirane-Youssﬁ-Meskine,

2008).

Suppose that (9), (12), (13) and (14) are ﬁlled. Then, the problem (11)

has at least one solution u in W

1,p

(Ω) ∩ L

∞

(Ω) in the sense that

Ω

a(x, u, ∇u) · ∇vdx =

Ω

fvdx (15)

for all v in D(Ω).

Proof. We only give a sketch. Denote by T

, the truncation at level k > 0,

deﬁned as T

(s) = max(−k, min(s, k)) and set G

(s) = s−T

(s). To get rid

of the lack of coerciveness of the operator A, we introduce the truncation

operator. Let {f

} be a sequence of L

∞

-functions such that

→ f in L

(Ω) and |f

| ≤ |f|. (16)

The approximate problem











∈ W

1,p

(Ω) such that

Ω

a(x, T

), ∇u

) · ∇φ dx =

Ω

φ dx

holds for every φ in W

1,p

(Ω)

(17)

December 28, 2009 12:15 WSPC - Proceedings Trim Size: 9in x 6in recent

admits at least one solution thanks to the Leray-Lions existence theorem

(see Leray-Lions

). For t > 0 and  > 0, we use T



)) as test function

in (17). Being h decreasing, letting  tends to 0

we arrive at

p−1

(t)

−

{|u

|>t}

|∇u

≤

{|u

|>t}

|f|dx. (18)

On the other hand, H¨older’s inequality enables us to get

−

{|u

|>t}

|∇u

|dx ≤ (−µ

(t))

−

{|u

|>t}

|∇u

, (19)

where µ

denotes the distribution function of u

, that is

(t) = |{x ∈ Ω : |u

(x)| > t}|.

Combining (18), (19) and the following well known inequality (see Talenti

)

(µ

(t))

1−

≤ −

{|u

|>t}

|∇u

|dx,

we obtain

h(t) ≤

−µ

(t)

(µ

(t))

(1−

)

{|u

|>t}

|f|dx

In the above inequality, we use H¨older’s inequality in Orlicz spaces, then

we integrate both sides between 0 and τ obtaining by the deﬁnition of the

rearrangement

H(u

∗

(σ)) ≤

kfk

log

|Ω|

s log

(e + s)

Thus, by (14), the sequence {u

} is uniformly bounded in L

∞

(Ω). It is

now easy to get an estimation of {u

} in the energy space W

1,p

(Ω) which

implies that there is a function u such that

→ u weakly in W

1,p

(Ω) and a.e in Ω. (20)

Then, we prove that

∇u

→ ∇u a.e in Ω.

Which enables us to pass to the limit in (17) to get (15).

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3. Nonstandard growth

Here, we do not assume the ∆

-condition on the N-function M. In partic-

ular, the corresponding Orlicz-Sobolev spaces are not reﬂexive. We extend

the class of functions satisfying (6) to the Orlicz-Sobolev setting by assum-

ing one of the following two assumptions: Either

f ∈ L

(Ω), (21)

f ∈ L

(Ω) and

+∞



M(t)



N −m

dt < +∞.

(22)

In addition of (9) and (13), we make the following assumption:

|a(x, s, ξ)| ≤ a

(x) + k

−1

M(k

|s|) + k

−1

M(k

|ξ|) (23)

where a

(x) belongs to E

(Ω), P is an N-function such that P M and

, k

are constants in IR

∗

. We prove the following

Theorem 3.1 (Youssﬁ,

2007). Suppose that (9), (13) and (23) are

ﬁlled. Under either (21) or (22), the problem (11) has at least one solu-

tion u in W

(Ω) ∩ L

∞

(Ω) in the sense that

Ω

a(x, u, ∇u) · ∇vdx =

Ω

fvdx (24)

for all v in D(Ω).

Remark 3.1. The previous theorem is surprising, since the function h

does not inﬂuence the result. This seems to be natural, since if one looks

for bounded solutions the degeneracy of the operator A ”disappears”. Our

result includes classical similar results, notably the G. Stampacchia’s one.

We extend the result of the previous theorem 3.1 to the strongly nonlinear

problem

(

A(u) + H(x, u, ∇u) = f in Ω,

u = 0 on ∂Ω,

(25)

where H is a Carath´eodory function which doesn’t satisfy necessarily the

famous sign condition H(x, s, ξ)s ≥ 0, but only the following growth con-

dition

|H(x, s, ξ)| ≤ β(s)M(|ξ|), (26)

December 28, 2009 12:15 WSPC - Proceedings Trim Size: 9in x 6in recent

where β : IR → IR

is a continuous function such that the function

t →

β(t)

−1

(M(h(|t|)))

belongs to L

(IR). So by deﬁning

γ(s) =

β(t)

−1

(M(h(|t|)))

for all s ∈ IR, the function γ is bounded.

Theorem 3.2 (Benkirane-Youssﬁ,

2008). Suppose that (9), (13),

(23) and (26) are ﬁlled. Under either (21) or (22), the problem (25) has at

least one solution u in W

(Ω) ∩ L

∞

(Ω) in the sense that

Ω

a(x, u, ∇u) · ∇vdx +

Ω

H(x, u, ∇u)vdx =

Ω

fvdx (27)

for all v ∈ W

(Ω) ∩ L

∞

(Ω).

Remark 3.2. In addition of the degeneration of the operator A, the fact

that H does not satisfy the sign condition creates another problem of getting

the a priori estimates. To overcome this hindrance, we use test functions of

exponential type containing the function γ. Note that the result of Theorem

3.2 extends those obtained in Boccardo et al.

and Trombetti.

Proof. Summarized.

Step 1: A priori estimates: We deﬁne A

and B

(u) := −div a(x, T

(u), ∇u)

and

(x, u, ∇u) = T

(B(x, u, ∇u)).

Denote by m

∗

either N or m according as we assume (21) or (22), and let

} ⊂ W

−1

(Ω) be a sequence of smooth functions such that

→ f strongly in L

∗

(Ω)

and

∗

≤ kf k

∗

By Proposition 2 of Gossez-Mustonen,

there exists at least one solution

∈ D(A

+ B

) ⊂ W

(Ω) to the approximate equation

−div a(x, T

), ∇u

) + B

(x, u

, ∇u

) = f

December 28, 2009 12:15 WSPC - Proceedings Trim Size: 9in x 6in recent

in the sense that

Ω

a(x, T

), ∇u

) · ∇vdx +

Ω

(x, u

, ∇

)vdx =

Ω

vdx (28)

for all v ∈ W

(Ω).

Lemma 3.1. Let u

be a solution of (28). For all t,  in IR

∗

, one has the

following inequalities:

{t<u

≤t+}

a(x, T

), ∇u

) · ∇u

γ(u

)

≤

>t}

γ(u

)



))dx.

(29)

{−t−<u

≤−t}

a(x, T

), ∇u

) · ∇u

γ(u

−

)

≤

<−t}

−

γ(u

−

)



−

))dx.

(30)

Proof. To prove (29) we use the test function

γ(T

))



)))

that belongs to W

(Ω) for all k > 0 and we use the function

−e

γ(T

−

))



−

)))

which belongs to W

(Ω) to prove (30).

Lemma 3.2. There exists a constant c

, not depending on n, such that for

almost every t > 0

−

{|u

|>t}

−1

(M(h(|u

|)))M(|∇u

|)dx ≤ c

{|u

|>t}

|dx. (31)

Proof. The two functions e

γ(u

)

and e

γ(u

−

)

are bounded in L

∞

(Ω), we

sum up both inequalities (29) and (30) obtaining (31).

The following comparison lemma constitutes the crucial step in the proof.

Lemma 3.3. Let K(t) =

M(t)

and µ

(t) = |{x ∈ Ω : |u

(x)| > t}|, for all

t > 0. We have for almost every t > 0:

h(t) ≤

2M(1)(−µ

(t))

−1

(M(1))NC

(t)

1−

−1







{|u

|>t}

|dx

−1

(M(1))NC

(t)

1−







(32)

December 28, 2009 12:15 WSPC - Proceedings Trim Size: 9in x 6in recent

where C

stands for the measure of the unit ball in IR

and c

is the

constant which appears in (31).

Proof. The function C(t) =

−1

(t)

is decreasing and convex (see Tal-

enti

). Hence, Jensen’s inequality yields







{t<|u

|≤t+k}

−1

(M(h(|u

|)))M(|∇u

|)dx

{t<|u

|≤t+k}

−1

(M(h(|u

|)))|∇u

|dx







= C







{t<|u

|≤t+k}

K(|∇u

|)M

−1

(M(h(|u

|)))|∇u

|dx

{t<|u

|≤t+k}

−1

(M(h(|u

|)))|∇u

|dx







≤

{t<|u

|≤t+k}

−1

(M(h(|u

|)))dx

{t<|u

|≤t+k}

−1

(M(h(|u

|)))|∇u

|dx

≤

−1

(M(h(t)))(−µ

(t + k) + µ

(t))

−1

(M(h(t + k)))

{t<|u

|≤t+k}

|∇u

|dx

Taking into account that M

−1

(M(h(t))) ≤ M

−1

(M(1)), using the convex-

ity of C and then letting k → 0

, we obtain for almost every t > 0

−1

(M(1))

−1

(M(h(t)))







−

{|u

|>t}

−1

(M(h(|u

|)))M(|∇u

|)dx

−1

(M(1))(−

{|u

|>t}

|∇u

|dx)







≤

−µ

(t)

−

{|u

|>t}

|∇u

|dx

Recall the following inequality, (see Talenti

−

{|u

|>t}

|∇u

|dx ≥ NC

(t)

1−

for almost every t > 0. (33)

December 28, 2009 12:15 WSPC - Proceedings Trim Size: 9in x 6in recent

The monotonicity of the function C, (31) and (33) yield

−1

(M(h(t)))

≤

−µ

(t)

−1

(M(1))NC

(t)

1−

−1







{|u

|>t}

|dx

−1

(M(1))NC

(t)

1−







Using the inequality

M(t) ≤ t

−1

(M(t)) for all t ≥ 0 (34)

and the fact that 0 < h(t) ≤ 1, we obtain (32).

Step 2: L

∞

-bound: If we are under (21) we get

∞

≤ H

−1

2M(1)

−1

(M(1))NC

−1

kfk

−1

(M(1))NC

N|Ω|

(35)

and if we are under (22) we get

∞

≤

−1

2M(1)c

kfk

−1

(M(1)))

r+1

−1

(λ)

+∞

−1

(λ)



M(s)



(36)

This, implies that

∞

≤ c. (37)

Step 3: Estimation in W

(Ω). An estimation in W

(Ω), is obtained

by proving the following

Lemma 3.4.

(1) −

{0≤u

}

a(x, T

), ∇u

) · ∇u

γ(u

)

dx ≤

Ω

γ(u

)

dx.

(2) −

≤0}

a(x, T

), ∇u

) · ∇u

γ(u

−

)

dx ≤

Ω

−

γ(u

−

)

−

dx.

Proof. We use

γ(u

)

∈ W

(Ω)∩L

∞

(Ω)

December 28, 2009 12:15 WSPC - Proceedings Trim Size: 9in x 6in recent

to prove the ﬁrst inequality and use

−e

γ(u

−

)

−

∈ W

(Ω)∩L

∞

(Ω)

for the second.

Summing up both inequalities (1) and (2) we get

Ω

M(|∇u

|)dx ≤

kfk

∗

|Ω|

1−

∗

−1

M(h(c))

. (38)

Hence, there is a function u ∈ W

(Ω) such that

* u weakly in W

(Ω) for σ(ΠL

, ΠE

), (39)

and

→ u in E

(Ω) strongly and a.e. in Ω. (40)

Step 4: Almost everywhere convergence of the gradients. We prove the

following

Lemma 3.5. The sequence {a(x, T

), ∇u

)} is uniformly bounded in

(Ω))

Then, we prove that

∇u

→ ∇u a.e. in Ω. (41)

and

a(x, T

), ∇u

) * a(x, u, ∇u) weakly in (L

(Ω))

for σ(ΠL

, ΠE

(42)

Step 5: Modular convergence of the gradients. We prove that

→ u in W

(Ω)

for the modular convergence.

Step 6: Equi-integrability of the non-linearities. We prove that

(x, u

, ∇u

) → B(x, u, ∇u) strongly in L

(Ω). (43)

Step 7: Passage to the limit. Testing by v ∈ W

(Ω)∩L

∞

(Ω), we have

all ingredients to pass to the limit in (28) and obtain that u is a solution

of (25) in the sense of (27).

December 28, 2009 12:15 WSPC - Proceedings Trim Size: 9in x 6in recent

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