Remarks on the Maximum Principle for Nonlinear Elliptic PDEs with Quadratic Growth Conditions Guy Barles(1) , Alain-Philippe Blanc(1) , Christine Georgelin

(1)

Magdalena Kobylanski(1) First version

Introduction The aim of this work is to study the conditions under which equations like −div(a(x, u, Du)) + b(x, u, Du) = f

in Ω

(1)

satisfy the maximum principle in H 1 (Ω) or in H 1 (Ω) ∩ L∞ (Ω) where Ω is a bounded domain of IRN . Here a and b are Caratheodory functions satisfying suitable ellipticity and growth conditions while f belongs a priori to H −1 (Ω). By maximum principle, we mean the following type of property  if u1 , u2 ∈ H1 (Ω) or H1 (Ω) ∩ L∞ (Ω)) are respectively sub- and    

supersolutions of (1), then

   

(2)

u1 ≤ u2 on ∂Ω =⇒ u1 ≤ u2 in Ω.

We recall here that the precise meaning of “u1 ≤ u2 on ∂Ω” is (u1 − u2 )+ ∈ H01 (Ω) while the definition of sub- and supersolutions of (1) will be given in Section 1. It is a classical (and easy) remark that the maximum principle implies the uniqueness of the solution of (1) in H01 (Ω) or in H01 (Ω) ∩ L∞ (Ω). We assume throughout this article that N ≥ 3 since the main difficulties occur in this case. At the end of Section 1, we show how the statements of our results has to be modified in order to be valid in dimensions N = 1 or N = 2. Roughly speaking, there are two main different types of approaches for proving properties like (2) in the literature. The first one (cf. D. Gilbarg and N.S. Trudinger[7], Theorem 8.1, Chapter 8) uses in an essential way the Sobolev embedding of H 1 (Ω) into Lq (Ω) with 1

Universit´e de Tours — Facult´e des Sciences et Techniques. Parc de Grandmont. 37200 Tours. France.

1

q = 2N/(N − 2) and provides for linear equations (and in particular for the linearized equation coming from (1)) results with rather general assumptions on the coefficients. But despite of this generality, this approach was unable to take in account problems with quadratic growth conditions since, in this case, the coefficients of the linearized equations are not in the right Lp – spaces. To solve this difficulty, F. Murat and the first author[1] used another approach which, in some sense, is more elementary since the conclusion is obtained through a classical eigenvalue argument. In fact, the main idea in [1] consists in first finding a structure condition on a and b ensuring that a general equation like (1) satisfies the maximum principle and then, when considering a particular equation, to find a change of variable which allows to come down to an equation like (1) which satisfies such a structure condition. Of course this is possible only if the equation at hand has suitable properties but it is worth mentionning that this strategy allows to obtain rather general results, under natural conditions, for equations with quadratic growth conditions. In this second approach, the coefficients of the linearized equation are not necessarely in the right Lp – spaces to apply the result of [7] but the idea is that the “large good terms” are able to compensate the “too large bad terms”. However this second approach is not more general than the first one since for other types of problems the first approach appears as being more efficient. The main contribution of this article is to match together these two approaches and therefore to provide uniqueness and maximum principle type results for equations with quadratic growth conditions which are more general than the ones given in [1] and in [7]. In order to be more specific on our results, let us consider as an example the equation −∆u + H(x, u, Du) = f

in Ω ,

(3)

where H is a Caratheodory function in Ω × IR × IRN and f ∈ LN/2 (Ω). We are able to prove that the maximum principle holds for (3) in H 1 (Ω) ∩ L∞ (Ω) if the function H(x, u, p) is locally lipschitz in u and p for almost all x ∈ Ω and if we have, on one hand, ∂H (x, u, p) ≤ C0 (|u|)(|p| + b1 (x))

∂p

a.e. x ∈ Ω, u ∈ IR, p ∈ IRN ,

(4)

and |H(x, u, 0)| ≤ C1 (|u|)b2 (x) a.e. x ∈ Ω, u ∈ IR,

(5)

where C0 and C1 are continuous functions of |u|, b1 ∈ LN (Ω) and b2 ∈ LN/2 (Ω); the assumptions (4) and (5) have to be considered as the quadratic growth conditions on H. On an other hand, we impose ∂H (x, u, p) ≥ α0 > 0 a.e. x ∈ Ω, u ∈ IR, p ∈ IRN . ∂u 2

(6)

In [1], the analogous result was proved under the assumption that b1 , b2 , f ∈ L∞ (Ω). As in [1], we will refer to this type of examples as the bounded case since the solution belongs here to H 1 (Ω) ∩ L∞ (Ω). In this framework, existence results have been proved in Boccardo, Murat & Puel[3, 4] for the model case where H is a Caratheodory function which satisfies H(x, u, p) = α0 u + H(x, u, p) with α0 > 0, |H(x, u, p)| ≤ C 0 + C 1 |p|2 . In fact, the results proved here completes the existence results obtained for such types of elliptic pdes by V. Ferrone and M.R. Posteraro[6] .......... and by V. Ferrone and F. Murat[5].. ... insuring the existence of a solution of when the source term f is “small” : the condition they found on the norm of f insuring that (1) has a at least one solution is the one we obtain for the maximum principle to hold H 1 (Ω) ∩ L∞ (Ω). It is worth noticing that we will NOT consider the unbounded case where the solutions do not belong to L∞ (Ω) because it is wellknown that, since there are more than one solution u ∈ H01 (Ω) for the problem −∆u − C1 |Du|2 = 0 in Ω .

(7)

and because maximum principle leads to uniqueness, it is hopeless to get a very general maximum principle in the unbounded case even in a very simple case (see the section 2 and the remark inside). We nevertheless prove in Section 2 that (7) has a unique solution u ∈ H01 (Ω) such that exp(nku) − 1 ∈ H01 (Ω) for some integer n. In order to prove these results, we follow the general approach described in [1]. The main new feature of the present work is the way we obtain what we call below the “basic result” i.e. the result we use after the change of variable we perform on the equation we are interested in. By “mixing” ideas coming from the proofs of the related results in [7] and in [1], we improve the general structure conditions under which one can prove that the maximum principle (2) holds for the equation (1). The paper is organized as follows. The first Section is devoted to the statement and proof of three “basic results” while the second one is devoted to the model equation (3). Finally, in the third Section, we indicate a few extensions of the results of the second Section to quasilinear equations, to obstacle problems and to time-dependent problems.

1

The basic result and its consequences

In this section, we present a maximum principle type result for linear equations which will be the corner-stone of all the results for quasilinear equations of this article. Then we describe its first consequences.

3

In the sequel, the space H01 (Ω) is equipped with the norm ||u||H 1 (Ω) = 0

Z

2

|Du| dx

1/2

,

Ω

for u ∈ H01 (Ω), and we denote by K(N ) the best constant in the Sobolev’s embedding of H01 (Ω) equipped with this norm in Lq (Ω) with q = 2N/(N − 2) (recall that we assume N ≥ 3) where Lq (Ω) is the equipped with the usual norm. Our first (and main) result is the Theorem 1.1 : Let (wk )k∈IN be a sequence of functions of H 1 (Ω) ∩ L∞ (Ω) such that, for any k ∈ IN , wk+ ∈ H01 (Ω) ∩ L∞ (Ω) and one has 



−div αk (x)Dwk + β k (x)wk + (γ k (x), Dwk ) + δ k (x)wk ≤ 0 in D0 (Ω) ,

(8)

where the measurable functions αk , β k , γ k and δ k satisfy the following properties : there 1 exists n > 0, 0 < θ ≤ 1, θ1 , θ2 > 0 with (θ1 + θ2 ) = θ, 0 < η ≤ η and a function 2 ς ∈ LN/2 (Ω) such that, for all k, one has k (x))i,j is a N × N matrix such that (i) for almost every x ∈ Ω, αk (x) = (αi,j 2

η|ξ| ≤

N X

k αi,j (x)ξi ξj ≤ η|ξ|2

a.e. x ∈ Ω , ∀ξ ∈ IRN .

i,j=1

(ii) δ k ∈ L1 (Ω), β k , γ k ∈ [L2 (Ω)]N δ k (x) −

n 1 k k (m−1 (m−1 γ k (x), γ k (x)) ≥ ς(x) a.e. x ∈ Ω , k β (x), β (x)) − 2θ1 2θ2 n k

 1 k α (x) + [αk (x)]t where [αk (x)]t is the adjoint matrix of αk (x). 2   − 4n (iii) lim sup δ k < (1 − θ) η[K(N )]−2 . (n + 1)2 LN/2 (Ω) k If Dwk+ → 0 in L2 (Ω) and almost everywhere, then for k large enough, we have

with mk =

wk (x) ≤ 0 a.e. x ∈ Ω . It is natural that a result on linear equations plays a central role in the proof of a Maximum Principle type result for nonlinear equations since most of these proofs consists more or less in linearizing the equation. The justification of the rather strange statement of Theorem 1.1 – the formulation with sequences – is given in the proof of Theorem 2.1 below.

4

Remark 1.1 : Despite of its apparent generality, many variants of this results may be considered : we just want to point out here that one may take θ1 = 0 if β k ≡ 0 (respectively θ2 = 0 if γ k ≡ 0) and (ii) becomes δ k (x) −

1 (m−1 γ k (x), γ k (x)) ≥ ς(x) a.e. x ∈ Ω 4θn k

δ k (x) −

n (m−1 β k (x), β k (x)) ≥ ς(x) a.e. x ∈ Ω ) . 4θ k

(respectively

To state the results on quasilinear equations, following [1], we first define what are sub- and supersolutions for the equation (1). Definition 1.1 : The function w ∈ H 1 (Ω) is a subsolution of (1) if a(x, w, Dw) ∈ (L2 (Ω))N ,

(a)

(b) b(x, w, Dw) ∈ L1 (Ω), (c)

R

Ω [a(x, w, Dw)Dψ

+ b(x, w, Dw)ψ] dx ≤ < f, ψ >

∀ψ ∈ H01 (Ω) ∩ L∞ (Ω),

ψ ≥ 0 in Ω.

The function w ∈ H 1 (Ω) is a supersolution of (1) if (a), (b), (c) hold with the opposite inequality in (c). The main consequence of Theorem 1.1 is the Theorem 1.2 : We assume that, for any k ∈ IN , uk , vk ∈ H 1 (Ω)∩L∞ (Ω) are respectively sub and supersolution of (1), that (uk − vk )+ ∈ H01 (Ω) ∩ L∞ (Ω) and that when k → ∞, D(uk − vk )+ → 0 in L2 (Ω) and almost everywhere. We also assume that f ∈ H −1 (Ω) and that a, b are Caratheodory functions satisfying the following properties : for almost all x ∈ Ω, (u, p) 7→ a(x, u, p), b(x, u, p) are locally Lipschitz functions in IR × IRN and there exists n > 0, 0 < θ ≤ 1, θ1 , θ2 > 0 with 21 (θ1 + θ2 ) = θ, 0 < η ≤ η and function ζ ∈ LN/2 (Ω) for R > 0 such that, one has 1. The equation is uniformly elliptic i.e., for almost every x ∈ Ω, u ∈ IR and p ∈ IRn , one has N X ∂ai η|ξ|2 ≤ (x, u, p)ξi ξj ≤ η|ξ|2 ∀ξ ∈ IRN . ∂p j i,j=1 2. If Rk := max(||uk ||∞ , ||vk ||∞ ) and if KRk := {(u, p) ∈ IR × IRN ; |u| ≤ Rk , |p| ≤ R}, then for any k ∈ IN , R > 0 and for any 1 ≤ i, j ≤ N ∂a ∂a i i ∞ sup (., u, p) ∈ L (Ω) , sup (., u, p) ∈ L2 (Ω) , ∂pj ∂u Kk Kk R

R

5

∂b ∂b sup (., u, p) ∈ L2 (Ω), sup (., u, p) ∈ L1 (Ω) . ∂pj ∂u Kk Kk R

R

3. For any k ∈ IN and for almost every x ∈ Ω, |u| ≤ Rk and p ∈ IRN , one has ∂b n ∂a ∂a 1 ∂b ∂b − (m−1 , ) − (m−1 , ) ≥ ζ(x) a.e. x ∈ Ω , ∂u 2θ1 ∂u ∂u 2θ2 n ∂p ∂p 

#t 

"

1 ∂a ∂a  with m =  + . 2 ∂p ∂p 4. For any k ∈ IN , there exists a function δ2k ∈ LN/2 (Ω) such that, for almost every x ∈ Ω, |u| ≤ Rk and p ∈ IRN , one has ∂b (x, u, p) ≥ δ2k (x) , ∂u and

  − lim sup δ2k k

< (1 − θ) LN/2 (Ω)

4n η[K(N )]−2 . 2 (n + 1)

Then for k large enough, one has uk (x) ≤ vk (x) a.e. x ∈ Ω . It is worth mentionning that assumptions 1. and 2. in Theorem 1.2 are basic assumptions on the ellipticity of the equation and on integrability properties for the nonlinearities : these assumptions will be clearly satisfied by all the equations we will consider (before and after changes of variables). The hypothesis 3. and 4. are the structure conditions which are required for having a Maximum Principle type result. Theorem 1.2 is a generalization of Theorem I.2 in [1]: the main difference with [1] is that the quantity which appears in the left-hand side of the inequality in 3. does not need ∂b to be “essentially positive” since ζ may be as large as we want. Instead of that, the ∂u – term has to be “essentially positive” in a way described by assumption 4. involving a measurement in norm LN/2 and not in norm L∞ . As in [1] we will generally use Theorem 1.2 after some change of variables. The interesting feature of this result is that the checking of its assumptions just consists in an estimation of the different derivatives of the nonlinearities a and b. Proof of Theorem 1.2 : we sketch the proof since it is a straightforward consequence of arguments given in [1] and of the proof of Theorem 1.1 below. The first step consists in following the arguments of the section III.1 of [1] which allow to reduce the proof to the case of non-linearities which are C 1 in u and p for a.e. x ∈ Ω and also to perform the computations for a fixed k on the set KR for some R > 0 devoted 6

to tend to +∞. This is where the assumption 2. plays a role since it allows to justify these computations. Then after these computations, we are (essentially) left with a linearized inequality satisfied by wk := uk − vk . Applying the arguments of Theorem 1.1 (more than the result itself), we get an inequality analogous to (14) below but with a right-hand side which is a o(1) as R → +∞. Keeping k fixed, we let R tend to +∞. And the conclusion follows as in the proof of Theorem 1.1. We leave the details to the reader. 2 Now we turn to the first real maximum principle type result for (1). Theorem 1.3 : Assume that a and b satisfy the assumptions of Theorem 1.2 and assume in addition that a(x, u, p) does not depend on u and

∂b ≥ 0 a.e. x ∈ Ω, u ∈ IR, p ∈ IRN , ∂u

(9)

then the Maximum Principle (2) holds for sub and supersolutions in H 1 (Ω) ∩ L∞ (Ω). This result for subsolutions adn supersolutions may be seen as the real analogue of the Theorem I.2 in [1] but it is not since, on one hand, we have to impose (9) which was not the case in [1] and on the other hand, the role of Theorem 1.3 will not be the same as the one of Theorem I.2 in [1] i.e. the result to be used after some change of variable. This role will be played by Theorem 1.2. We have stated here Theorem 1.3 since its short proof that we provide now, justifies at least partially the admittedly strange statements of Theorem 1.1 and 1.2. A more complete justification is given in the next section where we will use Theorem 1.2 to provide results for the model case. Proof of Theorem 1.3 : Let u1 , u2 ∈ H 1 (Ω) ∩ L∞ (Ω) be respectively a sub and a supersolution of (1) such that u1 ≤ u2 on ∂Ω. We argue by contradiction assuming that M := ||(u1 − u2 )+ ||∞ > 0. For ε > 0 small enough, we introduce the functions uε := u1 − M + ε. Thanks to assumption (9), uε ∈ H 1 (Ω) ∩ L∞ (Ω) is still a subsolution of (1). Moreover, if we set wε = uε − u2 , the definition of M and Stampacchia’s Theorem imply that wε+ → 0 in H 1 (Ω) and we can extract a subsequence denoted by (wεk )εk such that Dwε+k → 0 a.e. in Ω. We apply Theorem 1.2 to wk := uεk − u2 . This yields uεk (x) ≤ u2 (x) a.e. x ∈ Ω , for k large enough and therefore u1 (x) − u2 (x) ≤ M − εk 7

a.e. x ∈ Ω .

This property is a contradiction with the definition of M and the proof is complete.

2

We conclude this section by the Proof of Theorem 1.1 : We first remark that, according to the assumptions we made on the data αk , β k , γ k and δ k , all the terms in (8) are actually in D0 (Ω) and therefore this inequality has a sense. Moreover, one can easily show that it implies the following : for any function ϕ ∈ H01 (Ω) ∩ L∞ (Ω) such that ϕ ≥ 0 a.e. in Ω, one has Z h Ω

i

(αk (x)Dwk , Dϕ) + (β k (x), Dϕ)wk + (γ k (x), Dwk )ϕ + δ k (x)wk ϕ dx ≤ 0 .

(10)

We consider a C 1 –function S : [0, +∞[→ IR such that S(0) = 0 and S 0 (t) > 0 for t > 0. Since wk+ belongs to H01 (Ω) ∩ L∞ (Ω), S(wk+ ) ∈ H01 (Ω) ∩ L∞ (Ω) and therefore it can be used as a test-function in the inequality (10). Following [1], we will choose S(t) = tn if the assumptions of Theorem 1.1 hold with n ≥ 1 and S(t) = (t2 + ε2 )n/2 − εn if on the contrary n < 1. In the sequel, in order to simplify the computations and to point out the main arguments of the proof, we will assume that n ≥ 1; the other case, a little bit more complicated, consists in performing the same computations with ε > 0 and then to let ε tend to 0. The choice ϕ = (wk+ )n in (10) yields Z

h

{wk >0}

nwkn−1 (αk (x)Dwk , Dwk ) + nwkn (β k (x), Dwk )+ i

wkn (γ k (x), Dwk ) + δ k (x)wkn+1 dx ≤ 0 .

(11)

We first follows the strategy of the proof of [1] and we apply Young’s inequality to the β k and γ k –terms. We obtain



nwkn (β k (x), Dwk ) ≤

n θ1 n−1 k nwk (α Dwk , Dwk ) + (m−1 β k (x), β k (x))11{|Dwk |>0} wkn+1 , 2 2θ1 k

and



wkn (γ k (x), Dwk ) ≤

θ2 n−1 k 1 nwk (α Dwk , Dwk ) + (m−1 γ k (x), γ k (x))11{|Dwk |>0} wkn+1 . 2 2θ2 n k

Plugging these estimates in (11), we get Z

h

{wk >0}

i

(1 − θ)nwkn−1 (αk (x)Dwk , Dwk ) + Qk (x)wkn+1 dx ≤ 0 ,

where n 1 k k Q (x) := δ (x) − (m−1 (m−1 γ k (x), γ k (x)) 11{|Dwk |>0} . k β (x), β (x)) + 2θ1 2θ2 n k k

k





8

(12)

From now on we switch to the strategy of [7]. To do so, we set χk := (wk+ )(n+1)/2 and 4n Λ = (1 − θ) . With these new notations and after few elementary computations, (n + 1)2 (12) becomes Z Z Λ

Ω

(αk (x)Dχk , Dχk )dx ≤

Ω

[Qk (x)]− χ2k dx .

(13)

To proceed, we use the embedding of H01 (Ω) in Lq (Ω) which gives for the left-hand side Z 1 ||χk ||2Lq (Ω) , (αk (x)Dχk , Dχk )dx ≥ η ||χk ||2H 1 (Ω) ≥ η 2 0 K(N ) Ω and the H¨older inequality for the right-hand side which gives Z Ω





[Qk (x)]− χ2k dx ≤ [Qk (x)]−

LN/2 (Ω)

||χk ||2Lq (Ω) .

Gathering all these informations, (13) reduces to !

Λ k − η − [Q (x)] N/2 ≤0. L (Ω) K(N )2

||χk ||2Lq (Ω)



Now we examine the term [Qk (x)]−

LN/2 (Ω)

(14)

. By assumption (ii), we know that

δk− , [Qk (x)]− (x) ≤ [ς(x)]−

a.e. x ∈ Ω .

Moreover, since Dwk+ → 0 a.e., we have 11{|Dχk |>0} → 0 a.e. and therefore [Qk (x)]− − δk− (x) → 0 a.e. x ∈ Ω . By using the Lebesgue’s dominated convergence theorem, we deduce from these properties and from (iii) that



lim sup [Qk (x)]− k

LN/2 (Ω)

0 and the space LN (Ω) simultaneously by L2(1+ε) (Ω). 9

Remark 1.2 : For N ≥ 3,the space LN/2 (Ω) appears naturally above in the H¨ olderSobolev’s inequality Z Ω

f w2 dx ≤ [K(N )]2 ||f ||LN/2 (Ω)

Z

|Dw|2 dx .

Ω

H01 (Ω)

It is well-known because of the injection of in the Lorentz space Lq,2 (Ω) where q = 2N/(N − 2) that an analogous inequality is true when f ∈ LN/2,∞ (Ω). It would be natural to think that all the results we prove above (and below) remain valid if the space LN/2 (Ω) is replaced by the space LN/2,∞ (Ω) with straightforward adaptations of our assumptions. Unfortunately, we are only able to show that this is actually true for the Lorentz spaces LN/2,m (Ω) for any m < +∞ but not for m = +∞. The problem for the extension to LN/2,∞ (Ω) is the use of Lebesgue’s dominated convergence theorem at the end of the proof of Theorem 1.1; this key argument is untracktable with LN/2,∞ (Ω) and we were unable to turn around it.

2

The maximum principle for the model equations

Our result is the following Theorem 2.1 : Assume that (4), (5) and (6) are satisfied with α0 > 0. Then (2) holds in H01 (Ω) ∩ L∞ (Ω). In particular, (3) has at most one solution in H01 (Ω) ∩ L∞ (Ω). Theorem 2.1 states the uniqueness of the solution of the equation (0.1) in H01 (Ω) ∩ L (Ω). It is worth noticing that this uniqueness property does not hold in the larger class H01 (Ω) : we refer the reader of the counter-examples given in Kazdan and Kramer[8] (See also [4] or [1] where the counter-example is described). An assumption of the type (6) with α0 > 0 is necessary for Theorem 2.1 to hold if no additional assumption on the dependence of H in p is made. One can however obtain a analogous result for α0 = 0 if one of the following hypotheses holds ∞

∂H p − H ≤ K 1 |p|2 ∂p

a.e. x ∈ Ω, u ∈ IR, p ∈ IRN ,

(15)

for some constants K 1 ≥ 0, or there exist k ∈ IR, n > 0, 0 < θ ≤ 2 and a function δ ∈ LN/2 (Ω) such that −k 2 |p|2 + k

h ∂H

∂p

i

p−H +

2 ∂H 1 ∂H − − 2kp ≥ δ(x) , ∂u 2θn ∂p

for almost all x ∈ Ω, u ∈ IR, p ∈ IRN and with − δ

LN/2 (Ω)

θ 4n [K(N )]−2 . < (1 − ) 2 (n + 1)2

Our result is 10

(16)

Theorem 2.2 : Assume that (4), (5) and (6) are satisfied with α0 = 0. Then (i) If we assume in addition that (15) holds for sub- and supersolutions of equation (3), (2) holds in H 1 (Ω)∩L∞ (Ω). In particular, (3) has at most one solution in H01 (Ω)∩L∞ (Ω). (ii) If we assume in addition that (16) holds, (2) holds for sub- and supersolutions of equation (3) such that u1 , u2 ∈ H 1 (Ω) with exp(nku1 ), exp(nku2 ) ∈ H 1 (Ω) and in particular if u1 , u2 ∈ H 1 (Ω) ∩ L∞ (Ω). In particular, (3) has at most one solution u in H01 (Ω) such that exp(nku) − 1 ∈ H01 (Ω). Before giving the proofs of the two above theorems, let us try to justify assumption (16) by considering the analogue in our context of the example of Kazdan-Kramer [8]; such argument was already used in [1] to justify an analogous hypothesis. For C1 6= 0 and f ∈ LN/2 (Ω), f ≥ 0 a.e. in Ω (f being not identically 0), we consider the equation (17) −∆u − C1 |Du|2 = f in Ω . If we test our condition (16) with k = −C1 and the optimal value n = 1, we obtain the condition on f 1 (18) C1 ||f ||LN/2 < K(N )2 in order to find a nonnegative θ fullfilling the second condition inside (16). The condition (18) is the precise condition found by V. Ferrone and F. Murat in [5] required for this equation to have a positive solution u ∈ H01 (Ω) such that eC1 u − 1 ∈ H01 (Ω). Now it is worth noticing that one may construct (see again [5]) for any ε > 0 a ball B(0, Rε ) and a constant C1 such that : 1 1 < C1 ||f0 ||LN/2 < +ε 2 K(N ) K(N )2 N (N − 2) such that the problem finding a u ∈ H01 (B(0, Rε )) such that (1 + |x|2 )2 eC1 u − 1 ∈ H01 (B(0, Rε )) satisfying (17) with right-hand side f0 has no solution at all. This recall a Fredlholm-like alternative and is related with an eigenvalue problem allready observed when f is constant in the paper [1] and in Kazdan - Kramer [8] : the change of function v = eC1 u − 1 in (17) leads to the transformed equation with f0 (x) =

−∆v = C1 (v + 1)f

in Ω ,

and v ∈ H01 (Ω). Consider the eigenvalue problem ˜ 1 = min λ

Z

2

|Dw| dx ; w ∈

Ω

11

H01 (Ω),

Z Ω

2



f w dx = 1

.

Since f ∈ LN/2 (Ω) and because of the embedding of H01 (Ω) into Lq (Ω) with q = 2N/(N − ˜ 1 > 0 and there exists a positive function e˜1 ∈ H 1 (Ω), 2), this problem is well-posed, λ 0 such that ˜ 1 f e˜1 in Ω . −∆˜ e1 = λ Choosing e˜1 as test-function in the v equation, we obtain ˜1 λ

Z Ω

f e˜1 vdx = C1

and therefore ˜ 1 − C1 ) (λ

Z Ω

Z Ω

f e˜1 v =

(v + 1)f e˜1 dx ,

Z Ω

f e˜1 dx ≥ 0 .

˜ 1 ≥ C1 . Since v ≥ 0 a.e. in Ω, this implies that λ R 1 But on an other hand, if w ∈ H0 (Ω) satisfies Ω f w2 dx = 1, we have by H¨older inequality and by the Sobolev embedding of H01 (Ω) into Lq (Ω) with q = 2N/(N − 2) 1=

Z Ω

f w2 dx ≤ ||f ||LN/2 (Ω) ||w||2Lq (Ω) ≤ [K(N )]2 ||f ||LN/2 (Ω)

Z

|Dw|2 dx .

Ω

We deduce from that ˜1 , 1 ≤ [K(N )]2 ||f ||LN/2 (Ω) λ ˜ 1 ≥ C1 . and the condition (18) insures automatically that λ This means that this condition is in some sense sharp for equation (17) .

2

Proof of Theorem 2.1 : the idea of the proof follows the one given in [1] and consists essentially in using a change of function u = ϕ(v), where ϕ is a C 3 function with ϕ0 > 0 in IR in order to get a transformed equation to which we can apply Theorem 1.2. But, and this is the main difference with [1], the transformed equation which is of the type (1) ∂b will not will not satisfy the assumption (9) of Theorem 1.3 and in particular the term ∂u be positive. To take care of this difficulty, we argue in the following way : if u1 , u2 ∈ H 1 (Ω)∩L∞ (Ω) are respectively sub and supersolution of (3) such that u1 ≤ u2 on ∂Ω, we argue by contradiction assuming that M := ||(u1 − u2 )+ ||∞ > 0. By the same arguments as in the proof of Theorem 1.3 and using (6) in an essential way, there exists a sequence (εk )k of non-negative real numbers, converging to 0 such that uk1 := u1 − M + εk is still a subsolutions of (3), uk1 ∈ H 1 (Ω) ∩ L∞ (Ω) and uk1 ≤ u2 on ∂Ω if k is large enough since M > 0. Finally, (uk1 − u2 )+ → 0 in H 1 (Ω) and we have D(uk1 − u2 )+ → 0 a.e. in Ω. The new point here is really to perform the change of variable on uk1 and u2 and not on u1 and u2 . If v1k and v2 are define through uk1 = ϕ(v1k ) and u2 = ϕ(v2 ) then one checks easily that (v1k − v2 )+ → 0 in H 1 (Ω) and we have D(v1k − v2 )+ → 0 a.e. in Ω.

12

In order to apply Theorem 1.2, it is therefore enough to check the assumptions on the non-linearities. Here the computations are analogous to the one of [1] but, of course, the requirements on these non-linearities are different. y Using 0 where y ∈ H01 (Ω) ∩ L∞ (Ω) as test function in the variational formulation ϕ (v) of (3) shows that the transformed equation is −∆v −

1 ϕ00 (v) |Dv|2 + 0 H(x, ϕ(v), ϕ0 (v)Dv) = 0 in Ω. 0 ϕ (v) ϕ (v)

(19)

Therefore we have a(x, v, ξ) = ξ

and b(x, v, ξ) = −

ϕ00 (v) 2 1 |ξ| + 0 H(x, ϕ(v), ϕ0 (v)ξ) . 0 ϕ (v) ϕ (v)

Thanks to Remark 1.1, we can take θ1 = 0. In order to check the assumptions 2., 3., ∂b ∂b 4. of Theorem 1.2, we have essentially to estimate (x, v, ξ) and (x, v, ξ). ∂v ∂ξ As in [1], we use the “old variable” u = ϕ(v) and p = ϕ0 (v)ξ to examine these quantities and the change of variable ϕ defined by ϕ(v) = −

 1 1 log e−KAv + A K

where we will first fix A > 0 and then choose K > 0 large enough. Recall that u1 and u2 are assumed to be bounded. We thus only need the range of ϕ ˜ , +M ˜ ] with M ˜ = 3 max(ku1 kL∞ (Ω) , ku2 kL∞ (Ω) ) (the constant “3” is here to to cover [−M take in account the fact that we deal with uk1 instead of u1 ) . This will be the case if K ˜ is large enough, and more precisely if K > eM A . If the function ω defined by ω = ϕ0 ◦ ϕ−1 , i.e. ω(u) = ω(ϕ(v)) = ϕ0 (v),

and p = w(u)ξ,

one has h ∂H i o ∂b 1 n 00 ∂H −ω (u)|p|2 + ω 0 (u) (x, v, ξ) = p − H (x, u, p) + (x, u, p). ∂v ω(u) ∂p ∂u

An analogous computation yields ∂b ∂H ω 0 (u) (x, v, ξ) = (x, u, p) − 2 p. ∂ξ ∂p ω(u) Since u = ϕ(v) = −

1 1 log(e−KAv + ) and ω(u) = ϕ0 (v), we have A K ω(u) = K − eAu . 13

˜ Since we want to compare uk1 and u2 which both belong to L∞ (Ω) with kui kL∞ (Ω) ≤ M for i = 1, 2, it is enough to prove the estimates in this range of values. Thanks to this fact, we can replace the functions C0 and C1 in (4) and (5) by some constants . We get the estimate h ∂H i p − H (x, u, p) ≤ K2 |p|2 + K1 ([b1 (x)]2 + [b2 (x)] + |f (x)|) ,

∂p

˜ , +3M ˜ ] and p ∈ IRN for some nonnegative constants K1 , K2 . a.e. for x ∈ Ω, u ∈ [−3M Because of (6), this implies o eAu n 2 ∂b 2 2 (x, v, ξ) ≥ (A − K A)|p| − AK ([b (x)] + [b (x)] + |f (x)|) + α0 . 2 1 1 2 ∂v K − eAu

We choose A = K2 + 1 in this expression. It is then clear that, for K large enough, we have  AK1 eAu  ∂b 2 (x, v, ξ) ≥ η(K)|p|2 + α0 − [b (x)] + [b (x)] + |f (x)|) , 1 2 ∂v K − eAu  AK1 eAu  2 ≥ α0 − [b (x)] + [b (x)] + |f (x)|) , 1 2 K − eAu

where η(K) :=

1 AeAu min . 2 [−3M˜ ,3M˜ ] K − eAu

We notice that as K → +∞, η(K) behaves like c1 K −1 . Then we set, for x ∈ Ω, δ2 (x) = α0 − `(K)χ(x) , where `(K) :=

max

˜ ,3M ˜] [−3M

AK1 eAu , K − eAu

and 



χ(x) := [b1 (x)]2 + [b2 (x)] + |f (x)|) . From the above computations, we have ∂b (x, v, ξ) ≥ δ2 (x) , ∂v ˜ , 3M ˜ ]) and ξ ∈ IRN . Moreover, because of the for almost all x ∈ Ω, v ∈ ϕ−1 ([−3M assumptions on b1 , b2 and f , δ2 ∈ LN/2 (Ω) for any choice of K large enough. Now we are going to choose simultaneously K and n in order to have assumptions 3. and 4. of Theorem 1.2 being satisfied. To do so, we remark that − N/2 (δ2 ) N/2 L

(Ω)

≤ [`(K)]N/2

Z Ω

14

11{χ≥[`(K)]−1 } |χ(x)|N/2 dx .

But since χ ∈ LN/2 (Ω) and since `(K) behaves like c2 K −1 for some constant c2 > 0 as K → +∞, it is clear that the right-hand side of this inequality is a o(1)K −1 as K → ∞. On an other hand, by easy computations, we obtain ∂b 2 2 (x, v, ξ) ≤ L1 |p|2 + L2 b1 (x) ,

∂ξ

for some constants L1 , L2 depending on A and K; but since A has already be chosen, these constants may be considered as independent of K provided it is taken large enough (here the use of ω(u) and ω 0 (u) instead of v is the keystone of the proof). We recall that we choose θ2 = 1 which yields θ = 1/2 and thanks to the above estimates we have 2 2 1 ∂b 1 ∂b 1 (x, v, ξ) − (x, v, ξ) ≥ (η(K) − L1 )|p|2 + δ2 (x) − L2 b1 (x) . ∂v 2n ∂ξ 2n 2n

In order to satisfy the third assumption of the theorem 1.2, we first choose n such that 1 η(K) − L1 = 0 i.e. 2n 1 n := L1 , 2η(K) we have

2 2 ∂b 1 ∂b 1 (x, v, ξ) − (x, v, ξ) ≥ δ2 (x) − L2 b1 (x) , ∂v 2n ∂ξ 2n

and since the right-hand side is in LN/2 (Ω), assumption 3. holds for any choice of the parameter n satisfying the above condition. It remains to check assumption 4. and this will be done by a suitable choice of K. Indeed, as shown above − (δ2 ) N/2 ≤ o(1)K −1 , L

(Ω)

and we need the property − (δ2 )

LN/2 (Ω)


0 as 2n K → ∞ and therefore [K(N )]−2 behaves like c4 K −1 for some constant c4 > 0 as 2 (n + 1) K → ∞. It is then clear that, for a choice of K large enough, the assumption 4. is fullfilled. Therefore Theorem 1.2 applies and the proof is complete. 2 Now we turn to the proof of Theorem 2.2.

15

In the case where (15) holds, we choose again ϕ(v) = −

1 1 log(e−KAv + ), A K

ω(u) = K − eAu ,

for some A > 0 and K large enough to be fixed later. Here we have  o ∂b eAu n 2 2 (x, v, ξ) ≥ A − K A |p| , 1 ∂v K − eAu

where K 1 ≥ 0 is given by (15). We choose A = K1 and since with this choice we have ∂b (x, v, ξ) ≥ 0 , ∂v we conclude easily as in the situation of the proof of Theorem 2.1. Consider now the case where (16) holds. If k = 0, there is nothing to do since the assumptions of Theorem 1.3 hold. If k 6= 0, we choose ω(u) = eku . The function ϕ is nothing but 1 1 ϕ(v) = − log(1 − kv) for v < , k k which is equivalent to v = eku − 1. According to the computations done in the proof of Theorem 2.1, we have 2 ∂b 1 ∂b ω 00 (u) 2 (x, v, ξ) − (x, v, ξ) = − |p| ∂v 2θn ∂ξ ω(u) i ω 0 (u) h ∂H ∂H + p − H (x, u, p) + (x, u, p) ω(u) ∂p ∂u 1 ∂H ω 0 (u) 2 − (x, u, p) − 2 p . 2θn ∂p ω(u)

Since ω(u) = eku , the right hand side of this equality is nothing but −k 2 |p|2 + k

h ∂H

∂p

i

p−H +

2 ∂H 1 ∂H − − 2kp , ∂u 2θn ∂p

which, in view of hypothesis (16), is greater than δ(x). And one easily completes the proof. 2

16

3

Extensions

3.1

Extensions to more general equations.

We consider here extensions of the results of Section 2 to more general quasilinear elliptic equations of the form − div(d(x, Du)) + h(x, u, Du) = 0 in Ω,

(20)

where di (1 ≤ i ≤ N ) and h are Caratheodory functions in Ω × IR × IRN which are locally Lipschitz in (u, p) for almost all x in Ω. To state our result, we introduce the following assumptions: for almost every x ∈ Ω, u ∈ IR and p in IRN , we assume that  N  X ∂di    (x, p)ηi ηj ≥ γ|η|2     i,j=1 ∂pj        

and

∂d (x, p) ≤ C0 ∂p

∀η ∈ IRN

(γ > 0) (21)

and d(x, 0) = 0

 ∂h   (x, u, p) ≥ γ0 , (γ0 > 0)     ∂u ∂h (x, u, p) ≤ C1 (|u|)(|p| + b1 (x))    ∂p   

(22)

|h(x, u, 0)| ≤ C2 (|u|)b2 (x)

where C0 is a constant, C1 , C2 are continuous functions of |u| and b1 ∈ LN (Ω), b2 ∈ LN/2 (Ω). In addition to these “natural” assumptions, we need, as this was the case in [1], the following condition    for each ε > 0, there exists C(ε) such that ∂d  (x, p)p − d(x, p) ≤ ε|p| + C(ε) a.e. x ∈ Ω, p ∈ IRN .  ∂p

(23)

Our result is the following Theorem 3.1 : Assume that (21) (22) and (23) hold then the maximum principle holds for (20) in H 1 (Ω)∩L∞ (Ω). In particular, (20) has at most one solution in H01 (Ω)∩L∞ (Ω). We leave the proof of this result to the reader since it is a routine adaptation of the arguments of the analogous result in [1] and of the proof of Theorem 2.1 above. 17

3.2

Some Results in the Case of Non Lipschitz Continuous Nonlinearities

The aim of this section is to provide several results for equations involving nonlinearities which are not assumed to be Lipschitz continuous. In order to simplify the exposure and to point out the main ideas, we will only focus on the case of the model equation (3). 3.2.1

The Case of Uniformly Continuous Nonlinearities

Again to simplify, we assume that H has the form ˜ H(x, t, p) := H(x, p) + α0 t a.e. x ∈ Ω, u ∈ IR, p ∈ IRN . ˜ with H(x, 0) = 0. We introduce the following assumption which is the analogue of (4) : There exists m : IR+ → IR+ with m(0+) = 0 and a function b1 ∈ LN (Ω) such that i  h ˜ ˜ H(x, p) − H(x, q) ≤ m |p − q| |p| + |q| + b1 (x) ,

(24)

for almost every x ∈ Ω and every p, q ∈ IRN . Our result is the Theorem 3.2 : Assume that α0 > 0, that f ∈ LN/2 (Ω) and that (24) holds then the conclusion of Theorem 2.1 remains valid. Proof of Theorem 3.2 : Assume that u, v ∈ H 1 (Ω) ∩ L∞ (Ω) are respectively sub and supersolution of (3) and assume that (u − v)+ ∈ H01 (Ω). ˜ ε which are defined for a.e. x ∈ Ω and all We introduce the sequence of functions H p ∈ IRN by   1 ˜ ε (x, p) := inf H(x, ˜ H q) + |p − q| (|p| + q + b (x)) . 1 q∈IRN ε This regularization procedure is called Inf-convolution (See Lasry and Lions[10]) and ˜ ε are Caratheodory because of assumption (24) one can easily prove that the functions H functions such that ˜ ε (x, p) ≤ H(x, ˜ ˜ ε (x, p) + ρ(ε) for almost all x ∈ Ω and all p ∈ IRN where (i) H p) ≤ H ρ(ε) → 0 when ε → 0. ˜ ε is Lipschitz continuous in p for almost every x ∈ Ω and (4) holds with C0 (|u|) (ii) H replaced by C0ε > 0.

18

˜ is replaced Because of these properties, u is still a subsolution of the equation where H ˜ ε while v + ρ(ε) is a supersolution of the new equation. Moreover because of property by H α0 ˜ ε – equation and therefore (ii), we may apply Theorem 2.1 to the H u≤v+

ρ(ε) α0

a.e. in Ω

and the conclusion follows by letting ε tend to 0. 3.2.2

2

A Result in the Case of Non Uniformly Continuous Nonlinearities

In this section we present a result in the case when one part of the nonlinearity is only continuous but with a strict subquadratic behavior. Again in order to simplify the exposure we consider only the case of (3) and we assume that H has the following form H(x, u, p) := H1 (x, u, p) + H2 (x, p) a.e. x ∈ Ω, u ∈ IR, p ∈ IRN . where H1 , H2 are Caratheodory functions and where H2 (x, 0) = 0 for x ∈ Ω. We introduce the following assumptions : we suppose that, for almost all x ∈ Ω, (u, p) 7→ H1 (x, u, p) is locally Lipschitz continuous in IR × IRN and there exists α0 > 0 such that ∂H1 (x, u, p) ≥ α0 (1 + |p|2 ) a.e. x ∈ Ω, u ∈ IR, p ∈ IRN . (25) ∂u For H2 , we first introduce the subquadratic assumption : H2 (x, p) → 0 as |p| → +∞ , 1 + |p|2

(26)

for almost all x ∈ Ω, uniformly with respect to x ∈ Ω and the continuity assumption For any R > 0, there exists a function mR : IR+ → IR+ with mR (0+) = 0 and satisfying mR (t + s) ≤ mR (t) + mR (s) for any t, s ≥ 0 such that |H2 (x, p) − H2 (x, q)| ≤ mR (|p − q|) a.e. x ∈ Ω , ∀ |p| , |q| ≤ R .

(27)

This assumption means that the function H2 is continuous in p on each compact subset of IRN uniformly with respect to x ∈ Ω. Our result is the Theorem 3.3 : Assume that f ∈ H −1 (Ω), that H1 satisfies (4), (5), (25) and that H2 satisfies (26), (27). Then the conclusion of Theorem 2.1 remains valid for (3). Before giving the proof of Theorem 3.3, we remark that the continuity properties we impose on H2 are rather weak : this assumption holds typically if H2 (x, p) = F (p) − g(x) for any continuous over IRN function F and any g ∈ L∞ . This is compensated in the 19

proof below on one hand by the subquadratic assumption on H2 and on another hand by (25) which is a rather strong requirement. It is worth mentioning that the assumptions we impose on H1 in this result are exactly the ones which are indeed satisfied by the nonlinearity we obtain after the change of variable we make in the proof of Theorem 2.1. Proof of Theorem 3.3 : Assume that u, v ∈ H 1 (Ω) ∩ L∞ (Ω) are respectively sub and supersolution of (3) and assume that (u − v)+ ∈ H01 (Ω). We argue by contradiction, following the proof of [7], by assuming that M := ||(u − v)+ ||L∞ (Ω) > 0 , we choose 0 < M/2 ≤ k < M and we introduce the functions wk := (u − v − k)+ . Subtracting the inequalities satisfied respectively by u and v and multiplying by wkn where n is a large integer, we obtain by linearizing equation (3) Z Ω

Z Ω

nwkn−1 |Dwk |2 dx +

α(x)wkn (u − v)dx +

Z Ω

Z Ω

wkn (β(x), Dwk )dx+

(H2 (x, Du) − H2 (x, Dv)) wkn dx ≤ 0 ,

(28)

where, by assumptions (4) and (25) on H1 , the functions α and β satisfy, for some constants η > 0 and C > 0 α(x) ≥ η(1 + |Du|2 + |Dv|2 ) a.e. x ∈ Ω, |β(x)| ≤ C(|Du| + |Dv| + b1 (x)) a.e. x ∈ Ω. Easy computations then lead to Z Ω

Z Ω

α(x)wkn+1 dx +

Z Ω



nwkn−1

2

|Dwk | dx +

Z Ω

wkn (β(x), Dwk )dx+ 

H2 (x, Du) − H2 (x, Dv) + ηk(1 + |Du|2 + |Dv|2 ) wkn dx ≤ 0 . (29)

The main point is to estimate the last integral of the left-hand side of this inequality. We claim that there exists R > 0 such that H2 (x, p) − H2 (x, q) + ηk(1 + |p|2 + |q|2 ) ≥ 0 a.e. x ∈ Ω, if |p| ≥ R or |q| ≥ R. Indeed to build R, we first use (26) : there exists R0 > 0 such that if |p| ≥ R0 then |H2 (x, p)| ≤ ηk/2 a.e. x ∈ Ω. 1 + |p|2 Then we choose R ≥ R0 such that for any |p| ≤ R0 , one has |H2 (x, p)| ≤ ηk/2 a.e. x ∈ Ω. 1 + R2 20

(30)

This is possible because of (27). Finally a case by case analysis (|p| ≥ R or R0 ≤ |p| ≤ R or |p| ≤ R0 ) shows that the claim (30) is true. The next step consists in proving that, for k ≥ M/2, there exists K = K(R) > 0 such that H2 (x, p) − H2 (x, q) + ηk(1 + |p|2 + |q|2 ) ≥ −K |p − q| a.e. x ∈ Ω, if |p| ≤ R and |q| ≤ R. This property is a consequence of (27) since the function mR clearly satisfies for some K > 0 mR (t) ≤ ηM/2 + Kt for all t > 0 , and by (27) we have H2 (x, p) − H2 (x, q) ≥ −mR (|p − q|) ≥ −ηM/2 − K |p − q|

a.e. x ∈ Ω.

We denote by ΩR the set {x ∈ Ω; |Du(x)| ≤ R and |Dv(x)| ≤ R}. Using the above properties in (29) yields Z Ω

Z Ω

nwkn−1 |Dwk |2 dx +

α(x)wkn+1 dx −

Z ΩR

Z Ω

wkn (β(x), Dwk )dx+

K |Du − Dv| wkn dx ≤ 0 .

(31)

This inequality remains valid if we replace ΩR by Ω and the remainder of the proof consists in following the computations and the arguments of the proof of Theorem 1.1, choosing in particular n large enough. We leave the details to the reader. 2

References [1] G. Barles and F. Murat : The Maximum Principle for quasilinear elliptic equations with quadratic growth conditions. Arch. Rational Mech. Anal. 133 (1995) 77–101. [2] A. Bensoussan, L. Boccardo and F. Murat: On a nonlinear partial differential equation having natural growth terms and unbounded solution. Ann. Inst. H. Poincar´e, Anal. non lin´eaire, 5, (1988), pp. 347-364. [3] L. Boccardo, F. Murat and J.-P. Puel: Existence de solutions non born´ees pour certaines ´equations quasi-lin´eaires. Portugaliae Math., 41, (1982), pp. 507-534. [4] L. Boccardo, F. Murat and J.-P. Puel: Existence de solutions faibles pour des ´equations elliptiques quasi-lin´eaires `a croissance quadratique. Nonlinear Partial Differential and their Applications, Coll`ege de France Seminar, volume IV, H. Brezis and J.-L. Lions editors, Research Notes in Mathematics, 84, Pitman, London, (1983), pp. 19-73. 21

[5] V. Ferrone and F. Murat : Non linear problems having natural growth in the gradient : an existence result when the source term is small.Private communication [6] V. Ferrone and M.R. Posteraro : On a class of quasilinear elliptic equations with quadratic growth in the gradient Nonlinear Analysis T.M.A. 20 (1993). [7] D. Gilbarg and N.S. Trudinger: Elliptic Partial Differential Equations of Second Order. Springer-Verlag, Berlin, (1977). [8] J.L. Kazdan and R.J. Kramer: Invariant criteria for existence of solutions of second order quasi-linear elliptic equations. Comm. Pure Appl. Math., 31, (1978), pp. 619645. ´ [9] O.A. Ladyˇzenskaja and N.N. Ural’ceva: Equations aux deriv´ees partielles de type elliptique. Dunod, Paris, (1968). [10] J.M Lasry et P.L Lions: A remark on regularization in Hilbert spaces. Isr. J. Math. 55 (1986) pp 257-266.

22