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Uniqueness for unbounded solutions to stationary viscous Hamilton–Jacobi equations
Guy Barles, Alessio Porretta
To cite this version:
Guy Barles, Alessio Porretta. Uniqueness for unbounded solutions to stationary viscous Hamilton–
Jacobi equations. Annali della Scuola Normale Superiore di Pisa, Classe di Scienze, Scuola Normale Superiore 2006, 5 (1), pp.107–136. �hal-00017876�
ccsd-00017876, version 1 - 26 Jan 2006
Uniqueness for unbounded solutions to stationary viscous Hamilton–Jacobi equations
Guy Barles, Alessio Porretta
AbstractWe consider a class of stationary viscous Hamilton–Jacobi equations as λ u−div(A(x)∇u) =H(x,∇u) in Ω,
u= 0 on∂Ω
whereλ≥0,A(x) is a bounded and uniformly elliptic matrix andH(x, ξ) is convex inξand grows at most like |ξ|q+f(x), with 1< q <2 andf ∈LNq′(Ω). Under such growth conditions solutions are in general unbounded, and there is not uniqueness of usual weak solutions. We prove that uniqueness holds in the restricted class of solutions satisfying a suitable energy–type estimate, i.e. (1 +|u|)q−¯ 1u∈H01(Ω), for a certain (optimal) exponent ¯q. This completes the recent results in [14], where the existence of at least one solution in this class has been proved.
MSC:35J60(35R05, 35Dxx).
Running head: Viscous Hamilton–Jacobi equations
1 Introduction
In this paper we consider a class of elliptic equations in a bounded domain Ω⊂IRN,N >2
λu−div(A(x)∇u) =H(x,∇u) in Ω,
u= 0 on∂Ω (1.1)
where the functionH(x, ξ) is convex and superlinear with respect toξ.
Equations of this type are sometimes referred to as stationary viscous Hamilton–
Jacobi equations and appear in connection to stochastic optimal control prob- lems. In that context, the convexity ofH is a natural assumption.
The model example which we are going to treat is the following λu−div(A(x)∇u) =γ|∇u|q+f(x) in Ω,
u= 0 on∂Ω (1.2)
whereq >1,λ≥0 andA(x) = (ai,j(x)) is a matrix ofL∞(Ω) functionsai,j(x) satisfying uniform ellipticity and boundedness conditions
α|ξ|2≤A(x)ξ·ξ≤β|ξ|2 ∀ξ∈RN, a.e. x∈Ω. (1.3)
Without loss of generality, we let γ >0. We draw our attention to the “sub- critical” case, namelyq <2, and, more precisely, to the question of uniqueness ofunbounded solutions.
Let us first recall that some regularity condition is needed onf in order that problem (1.2) admits a solution. In the class of Lebesgue spaces, this condition amounts to ask that
f ∈LNq′(Ω), (1.4)
where q′ is the conjugate exponent ofq, i.e. q′ = q
q−1. When q < 2, (1.4) implies that f ∈ Lm(Ω) with m < N2, hence solutions are expected to be unbounded. Moreover since, by Sobolev embedding theorem, one has
LNq′(Ω)⊂H−1(Ω) ⇐⇒ q≥1 + 2
N , (1.5)
the value q = 1 + N2 is a critical one. Indeed, the solutions belong toH01(Ω) only ifq≥1 +N2, whenqis below this value solutions are not only unbounded but have not even finite energy and should be defined in a suitable generalized sense.
The fact that (1.4) is a necessary condition for having solutions can be easily justified by a heuristic argument: ifA(x) =I, i.e. in case of the Laplace operator, the Calderon–Zygmund regularity implies that
−∆u∈Lm(Ω)→u∈W2,m(Ω)→ |∇u| ∈Lm∗(Ω),
wherem∗ is associated to mthrough the Sobolev embedding, i.e., for N > m, m∗= N m
N−m. In order to be consistent with (1.2), this means thatf ∈Lm(Ω) and |∇u| ∈ Lqm(Ω) so that one needs qm = m∗, i.e. m = Nq′. We refer the reader to [1], [15] for rigorous and sharper necessary conditions onf in order to have weak solutions. It is important to recall that ifλ= 0 the dataf, γ, α must also satisfy a size condition in order that a solution exists.
Pioneering results for such kind of equations were given by P.L. Lions ([16], [17]), mainly in case of Lipschitz solutions and includingq >2. Existence results for the caseq = 2 can be found in several works, among which we recall the series of papers by L. Boccardo, F. Murat, J.P. Puel (see e.g. [8], [9]) and more recently, assumingf inLN2(Ω), in [13], [10].
Under assumption (1.4) with 1 < q < 2, the existence of a solution for problems as (1.2) has been recently proved in [14] if eitherλ >0 orλ= 0 and a size condition is satisfied
γq−11kfk
L
N q′
(Ω)< αq′C∗, (1.6) whereC∗only depends on qandN.
In this paper we deal with the problem of uniqueness of solutions. Up to now, uniqueness results for problems like (1.2) have been proved in the Lipschitz case ([16]) and in [2], [3] if either solutions are bounded or q = 2. Note that
these two cases share a common feature, which is that f is required to be in Lm(Ω), m ≥ N2: for less summable f as we consider, the approach of these previous papers seems not to work. Further results when q ≤ 1 + N2 can be found in [4], [5].
When dealing with the question of uniqueness, one has to consider the fol- lowing well–known counterexample (see also [16], [1]) forq > N−1N
u(x) =Cα(|x|α−1) (α=−2−qq−1,Cα= (N+α−2)
1 q−1
|α| ) solves −∆u=|∇u|q inD′(B1(0)),
u∈W01,q(Ω)
(1.7)
This shows that uniqueness does not hold in the class of weak solutions u in W01,q(Ω), and, ifq≥1 +N2, not even inH01(Ω) (one can check thatu∈H01(Ω) in this case). It is then natural to look for a suitable class of solutions in which problem (1.2) is well–posed. A linearization argument would suggest that there is uniqueness in the class
usol. of (1.2): |∇u| ∈LN(q−1)(Ω). (1.8) On the other hand, ifq >1 + N2 (which givesN(q−1)>2), the existence of such kind of solutions can not be obtained unless the Calderon–Zygmund regu- larity theorem applies; thus, in order to deal with general (bounded measurable) coefficientsai,j, this approach is not reasonable.
Our main purpose here is to prove the uniqueness of solutions of (1.2) in a regularity class which is consistent with the existence results available from [14]. In this latter paper it has been proved that a natural class of solutions for which botha priori estimatesandexistencehold is given through the extra energy condition
usol. of (1.2): (1 +|u|)q−1¯ u∈H01(Ω), with ¯q= (N−2)(q−1)2(2−q) . (1.9) We are going to prove that this regularity is precisely what is needed to select a unique solution, so that problem (1.2) is actually well–posed in this class.
Our main result concerns the caseq≥1 +N2, which corresponds toH01(Ω) solutions (see (1.5)).
Theorem 1.1 Let 1 +N2 ≤q <2. Assume (1.3), (1.4) and that (i) either λ >0
(ii) orλ= 0and (1.6) is satisfied.
Then problem (1.2) has only one (distributional) solution u such that (1 +
|u|)q−1¯ u∈H01(Ω), withq¯= (N−2)(q−1)2(2−q) .
Note that the functionuin the counterexample (1.7) satisfies (1+|u|)r−1u∈ H01(Ω) for anyr <q¯but not forr= ¯q, which proves the optimality of our result.
Observe also that ¯qtends to infinity asq→2, which is consistent with the case f ∈LN2(Ω), for which existence and uniqueness have been proved (see [13], [3]
respectively) in the class of solutionsu such that exp(µ u)−1 ∈H01(Ω) for a suitable constantµ.
We leave to Section 2 the proof of Theorem 1.1, actually in a generalized version which includes problem (1.1) where H is convex and satisfies similar growth conditions. Some extensions to Neumann boundary conditions as well as to the case of unbounded domains is also discussed.
In Section 3 we deal with the case N−1N < q < 1 + N2, which corresponds tof ∈Lm(Ω) with 1< m < N+22N . A similar result as Theorem 1.1 is proved, but since, in this case, solutions do not belong to H01(Ω), we use a slightly stronger formulation than the distributional one, namely uniqueness is proved for so–called renormalized solutions (still in the class (1.9)). This notion (see Definition 3.1), first introduced in [12] for transport equations, is now currently used in several different contexts when dealing with solutions of infinite energy.
Still in Section 3, we prove in fact a more general uniqueness result whenq is below the critical value 1 +N2. Indeed, we will see that if q ≤1 + N2 then the regularity (1.9) implies (1.8). This fact allows to prove uniqueness through a simpler linearization principle, which does not need any convexity argument and which can be applied to more general situations like, for instance, nonlinear operators (see Theorem 3.2). Note that the limiting value q = 1 + N2 is also admitted here; actually, (1.9) and (1.8) coincide in that case with u∈H01(Ω).
On the other hand, as mentioned before, this argument was not possible for q >1 +N2 since (1.8) will no more be true in general.
Finally, some further remarks will be discussed at the end of Section 3, including the caseq < N−1N , where uniqueness holds simply inW01,q(Ω).
2 The case q ≥ 1 +
N2: finite energy solutions
We consider a natural generalization of (1.2), namely the following equation λu−div(A(x)∇u) =H(x,∇u) in Ω,
u= 0 on∂Ω (2.1)
We still assume that λ ≥ 0, that A(x) satisfies (1.3) and that H(x, ξ) is a Carath´eodory function satisfying
ξ7→H(x, ξ) is convex, for a.e. x∈Ω, (2.2) and the growth condition
∃q∈(N−1N ,2): |H(x, ξ)| ≤γ|ξ|q+f(x), γ >0, f(x)∈LNq′(Ω). (2.3) Note that this assumptions include the possibility that the equation contains transport terms; indeed, the basic choice forH is
H(x,∇u) =b(x)· ∇u+γ(x)|∇u|p+f(x)
whereb ∈
LN(Ω)N
, γ∈Lr(Ω) and f ∈LN r(ppr−−1)N (Ω), with r ∈(N,+∞] and
N(r−1)
r(N−1)< p <2−Nr.
In virtue of (2.3) and (1.5), assumingq≥1 +N2 corresponds to having data inH−1(Ω), so that we can reasonably talk ofH01(Ω) weak solutions.
Definition 2.1 We say thatu∈H01(Ω)is a weak subsolution of (2.1) ifH(x,∇u)∈ L1(Ω) and
λR
Ωu ξ dx+R
ΩA(x)∇u∇ξ dx≤R
ΩH(x,∇u)ξ dx
∀ξ∈H01(Ω)∩L∞(Ω), ξ≥0. (2.4) A super-solution of (2.1) is defined if the opposite inequality holds. A function ubeing both a sub and a super-solution is said to be a weak solution of (2.1).
Our proof of the comparison principle for sub and super-solutions of (2.1) relies on two basic ideas: the first one is that if
−div(A(x)∇w)≤ |∇w|q
(w+)q¯∈H01(Ω), (2.5)
then w ≤ 0; in other words, the homogeneous problem has only the trivial solution in this regularity class. Secondly, we aim at applying inequality (2.5) to (a small perturbation of) the difference of two solutionsu−v. In order to obtain this inequation, we use a convexity argument, which gives account of assumption (2.2). A further technical tool will be required in order to justify some regularity claimed onu−v: here we apply a truncation argument.
In order to do that we introduce the following truncation function Tn(s) =
Z s 0
θn(ξ)dξ , θn(ξ) =
1 if|ξ|< n
2n−|ξ|
n ifn <|ξ|<2n, 0 if|ξ|>2n
(2.6) and we start by giving a sort of renormalization principlefor the “truncated”
equation.
Lemma 2.1 Let u∈ H01(Ω) be a weak subsolution of (2.1). Then u satisfies, for any nonnegativeξ∈H01(Ω)∩L∞(Ω) and for everyn
λ Z
Ω
Tn(u)ξ dx+ Z
Ω
A(x)∇Tn(u)∇ξ dx
≤ Z
Ω
H(x,∇Tn(u))ξ dx+hIn, ξi,
(2.7)
whereIn is defined as hIn, ξi= 1
n Z
{n<u<2n}
A(x)∇u∇u ξ dx−1 n
Z
{−2n<u<−n}
A(x)∇u∇u ξ dx
+λ Z
Ω
(Tn(u)−uθn(u))ξ dx+ Z
Ω
(H(x,∇u)θn(u)−H(x,∇Tn(u)))ξ dx . (2.8)
If moreover|u|q−1¯ u∈H01(Ω), whereq¯=(N2(2−q)−2)(q−1), we have
n→+∞lim n2¯q−1kInkL1(Ω)= 0. (2.9) Proof. Letξ ∈H01(Ω)∩L∞(Ω), ξ≥0, and letn > 0. Multiplying equation (2.1) byθn(u)ξwe obtain
λ Z
Ω
u θn(u)ξ dx+ Z
Ω
A(x)∇u∇ξθn(u)dx= Z
Ω
H(x,∇u)θn(u)ξ dx + 1
n Z
{n<u<2n}
A(x)∇u∇u ξ dx−1 n
Z
{−2n<u<−n}
A(x)∇u∇u ξ dx (2.10)
Recalling thatθn(u) =Tn′(u) and definingInas in (2.8) we have obtained (2.7).
Now letu be such that|u|q−1¯ u∈H01(Ω), where ¯q= (N2(2−q)−2)(q−1). We have, by definition ofIn
n2¯q−1kInkL1(Ω)≤ n2¯q−1
1 n
Z
{n<u<2n}
A(x)∇u∇u dx+ 1 n
Z
{−2n<u<−n}
A(x)∇u∇u dx
+λ n2¯q−1 Z
Ω
|Tn(u)−uθn(u)|dx+n2¯q−1 Z
Ω
|H(x,∇u)θn(u)−H(x,∇Tn(u))|dx (2.11) Observe that
n2¯q−1 1 n
Z
{n<u<2n}
A(x)∇u∇u dx≤ Z
{n<u<2n}
A(x)∇u∇u|u|2¯q−2dx
≤ β
¯ q2
Z
{n<u<2n}
|∇|u|q¯|2dxn→+∞→ 0,
(2.12)
and similarly
n2¯q−2 Z
{−2n<u<−n}
A(x)∇u∇u dxn→+∞→ 0 (2.13)
We also have, using Young’s inequality, and by definition of ¯q,
|∇u|q|u|2¯q−1≤ q
2|∇u|2|u|2(¯q−1)+2−q
2 |u|2¯q+2(q2−−1)q
= q
2¯q2|∇(|u|q−1¯ u)|2+2−q
2 (|u|q¯)2∗. Since|u|q−1¯ u∈H01(Ω), we conclude that
|∇u|q|u|2¯q−1∈L1(Ω) (2.14)
Similarly, since (2¯q−1)(Nq′)′ = 2∗q¯ we have that |u|2¯q−1 ∈ L(Nq′)
′
(Ω); since f ∈LNq′(Ω) we deduce that
f|u|2¯q−1∈L1(Ω) (2.15)
Now we have, by definition ofθn andTn,
|H(x,∇u)θn(u)−H(x,∇Tn(u))| ≤[|H(x,∇u)|+|H(x,∇Tn(u))|]χ{|u|>n}, hence, using the growth assumption (2.3),
n2¯q−1|H(x,∇u)θn(u)−H(x,∇Tn(u))| ≤2(γ|∇u|q+f(x))|u|2¯q−1χ{|u|>n}. Thanks to (2.14)–(2.15) we conclude
n2¯q−1 Z
Ω
|H(x,∇u)θn(u)−H(x,∇Tn(u))|dxn→+∞→ 0. (2.16) Finally, sinceu∈L2∗q¯(Ω),
n2¯q−1 Z
Ω
|Tn(u)−uθn(u)|dx≤2 Z
Ω
|u|2¯qχ{|u|>n}dxn→+∞→ 0. (2.17) From (2.11), (2.12), (2.13), (2.16), (2.17) we get (2.9).
Remark 2.1 Clearly ifu∈H01(Ω) is a super-solution of (2.1) then (2.7) holds with the opposite sign. In particular, ifuis a weak solution of (2.1), then
λ Z
Ω
Tn(u)ξ dx+ Z
Ω
A(x)∇Tn(u)∇ξ dx
= Z
Ω
H(x,∇Tn(u))ξ dx+hIn, ξi, withn2¯q−1kInkL1(Ω)→0 provided|u|q−1¯ u∈H01(Ω).
We come to the main comparison result.
Theorem 2.1 Assume (1.3), (2.2), (2.3) with q ≥ 1 + N2. Let λ > 0. If u and v are respectively a subsolution and a super-solution of (2.1) such that (1 +|u|)q−1¯ u∈H01(Ω) and(1 +|v|)q−1¯ v∈H01(Ω), then we have u≤v in Ω.
In particular, problem (2.1) has a unique weak solution u such that (1 +
|u|)q−1¯ u∈H01(Ω).
Proof. From Lemma 2.1 we obtain that λ
Z
Ω
Tn(u)ξ dx+ Z
Ω
A(x)∇Tn(u)∇ξ dx≤ Z
Ω
H(x,∇Tn(u))ξ dx+hInu, ξi, (2.18)
and λ
Z
Ω
Tn(v)ξ dx+ Z
Ω
A(x)∇Tn(v)∇ξ dx≥ Z
Ω
H(x,∇Tn(v))ξ dx+hInv, ξi, (2.19) for everyξ∈H01(Ω)∩L∞(Ω),ξ≥0.
Let nowε >0 be fixed. Subtracting (2.18) and (2.19), we obtain λ
Z
Ω
[Tn(u)−(1−ε)Tn(v)]ξ dx+ Z
Ω
A(x)∇(Tn(u)−(1−ε)Tn(v))∇ξ dx
≤ Z
Ω
[H(x,∇Tn(u))−(1−ε)H(x,∇Tn(v))]ξ dx+hInu, ξi −(1−ε)hInv, ξi. Now we use the convexity assumption onH, which gives
H(x, p)≤(1−ε)H(x, η) +εH(x,p−(1−ε)η
ε ), ∀p , η∈RN. Withp=∇Tn(u),η =∇Tn(v) we obtain
H(x,∇Tn(u))−(1−ε)H(x,∇Tn(v))≤εH
x,∇Tn(u)−(1−ε)∇Tn(v) ε
, hence we have
λ Z
Ω
[Tn(u)−(1−ε)Tn(v)]ξ dx+ Z
Ω
A(x)∇[Tn(u)−(1−ε)Tn(v)]∇ξ dx
≤ε Z
Ω
H
x,∇[Tn(u)−(1−ε)Tn(v)]
ε
ξ dx+hInu, ξi −(1−ε)hInv, ξi. (2.20) We define now
wn=Tn(u)−(1−ε)Tn(v)−εϕ , (2.21) whereϕis a positive function that belongs toH01(Ω) and will be chosen later.
From (2.20) we obtain λ
Z
Ω
wnξ dx+ Z
Ω
A(x)∇wn∇ξ dx≤ε Z
Ω
H
x,∇wn
ε +∇ϕ
ξ dx
−ε Z
Ω
λϕ ξ+A(x)∇ϕ∇ξ dx
+hInu, ξi −(1−ε)hInv, ξi.
(2.22)
Using assumption (2.3) we have H
x,∇wn
ε +∇ϕ
≤γ
∇wn
ε +∇ϕ
q
+f(x)
≤(γ+δ)|∇ϕ|q+Cδ
∇wn
ε
q
+f(x),
(2.23)
whereδis any positive constant.
Using (2.23) in (2.22) we obtain λ
Z
Ω
wnξ dx+ Z
Ω
A(x)∇wn∇ξ dx≤ Cδ
εq−1 Z
Ω
|∇wn|q ξ dx
−ε Z
Ω
λϕ ξ+A(x)∇ϕ∇ξ dx−(γ+δ)|∇ϕ|qξ−f(x)ξ
+hInu, ξi −(1−ε)hInv, ξi.
(2.24)
We choose nowϕas a solution of
λ ϕ−div(A(x)∇ϕ) = (γ+δ)|∇ϕ|q+f(x) in Ω,
(1 +|ϕ|)¯q−1ϕ∈H01(Ω). (2.25)
The existence of such a function ϕis proved in [14]. Moreover, we have that ϕ≥0 (sincef ≥0 from (2.3)). Thanks to (2.25) we obtain from (2.24)
λ Z
Ω
wnξ dx+ Z
Ω
A(x)∇wn∇ξ dx≤ Cδ
εq−1 Z
Ω
|∇wn|q ξ dx
+hInu, ξi −(1−ε)hInv, ξi.
(2.26)
Forl >0, we choose in (2.26)ξ=ξn defined as ξn= [(wn−l)+]2¯q−1
Note thatξnis a positive function, and it belongs toH01(Ω)∩L∞(Ω). Moreover, by the definition ofwn in (2.21), we have
kξnkL∞(Ω)≤(2n)2¯q−1, so that we can apply Lemma 2.1 foruandv and get
|hInu, ξni| ≤(2n)2¯q−1kInukL1(Ω)n→+∞→ 0
|hInv, ξni| ≤(2n)2¯q−1kInvkL1(Ω)
n→+∞→ 0.
Thus (2.26) implies λ
Z
Ω
wn[(wn−l)+]2¯q−1dx+ (2¯q−1) Z
Ω
A(x)∇wn∇wn[(wn−l)+]2¯q−2dx
≤ Cδ εq−1
Z
Ω
|∇wn|q [(wn−l)+]2¯q−1dx+o(1)n,
whereo(1)n goes to zero asntends to infinity. Neglecting the zero order term which is positive, and using thatA(x)≥αI, we have
α(2¯q−1) Z
Ω
|∇wn|2[(wn−l)+]2¯q−2dx≤ Cδ
εq−1 Z
Ω
|∇wn|q [(wn−l)+]2¯q−1dx+o(1)n. (2.27)
Young’s inequality implies Cδ
εq−1 Z
Ω
|∇wn|q [(wn−l)+]2¯q−1dx≤α
2(2¯q−1) Z
Ω
|∇wn|2[(wn−l)+]2¯q−2dx +Cε,δ
Z
Ω
[(wn−l)+]2¯q+2(q
−1) 2−q dx , hence, using that 2¯q+2(q−1)2−q = ¯q2∗we get
α
2(2¯q−1) Z
Ω
|∇wn|2[(wn−l)+]2¯q−2dx≤Cε,δ Z
Ω
[(wn−l)+]q2¯∗dx+o(1)n. Using Sobolev inequality in the left hand side we obtain
C Z
Ω
[(wn−l)+]q2¯∗dx 1−N2
≤Cε,δ
Z
Ω
[(wn−l)+]q2¯∗dx+o(1)n
We let nowntend to infinity; sinceu,v and ϕall belong to Lq2¯∗(Ω), we have that
wn→w := u−(1−ε)v−εϕ strongly inLq2¯∗(Ω) asntends to infinity, hence we get
C Z
Ω
[(w−l)+]q2¯∗dx 1−N2
≤Cε,δ Z
Ω
[(w−l)+]q2¯∗dx .
Since 1−N2 <1, last inequality implies thatw≤0; indeed, if supw >0 (even possibly infinite), one gets a contradiction by letting l converge to supw and using that [(w−l)+]q2¯∗ would tend to zero inL1(Ω).
The conclusion is then thatw≤0, i.e.
u≤(1−ε)v+εϕ , and, lettingε→0,u≤v in Ω.
Let us now deal with the caseλ= 0. Indeed, the same proof can be applied, provided there exists a solution of
−div(A(x)∇ϕ) = (γ+δ)|∇ϕ|q+f(x) in Ω,
ϕ= 0 on∂Ω, (2.28)
for some δ > 0. This requires a further assumption, which is a sort of size condition on the data.
Indeed, it is known from [14] that there exists a constantC∗, only depending onq andN, such that if
bq−11kfk
L
N q′
(Ω)< αq
′
C∗ (2.29)
then the problem
n−div(A(x)∇z) =b|∇z|q+f(x) in Ω,
z= 0 on∂Ω (2.30)
admits a solutionz such that (1 +|z|)q−1¯ z∈H01(Ω). In particular, if we fix α (the coercivity constant ofA(x)) andf, the set
Bf : ={b >0 : problem (2.30) has a solutionz : (1 +|z|)q−1¯ z∈H01(Ω)}
is non empty. Indeed, it is not difficult to see that Bf is even an interval. In order to assure that (2.28) has a solution for a certainδ >0, we are then led to assume that (2.3) holds withγ <supBf.
Theorem 2.2 Let λ= 0. Assume (1.3), (2.2) and (2.3) with q≥1 +N2 and γ <supBf, which is defined above. Ifuandv are respectively a subsolution and a super-solution of (2.1) such that (1 +|u|)q−1¯ u∈H01(Ω) and(1 +|v|)q−1¯ v ∈ H01(Ω), then we have u ≤ v in Ω. In particular problem (2.1) has a unique solutionusuch that (1 +|u|)q−1¯ u∈H01(Ω).
The result of Theorem 2.2 may be rephrased more explicitly in terms of a size condition on the norm off. Indeed, letC∗ be the maximal possible choice in (2.29), i.e.
C∗= sup{C >0 : ifα−q′bq−11 kfk
L
N q′
(Ω)< C then problem (2.30) has a solutionz: (1 +|z|)q−1¯ z∈H01(Ω)}
Then we have
Corollary 2.1 Letλ= 0. Assume (1.3), (2.2) and (2.3) with q≥1 + N2 and α−q′γq−11kfk
L
N q′
(Ω)< C∗ (2.31)
Then problem (2.1) has a unique solutionusuch that (1 +|u|)q−1¯ u∈H01(Ω).
Remark 2.2 Applying Theorem 2.1 and Corollary 2.1 to the model problem λu−div(A(x)Du) =γ|Du|q+f(x) in Ω,
u= 0 on∂Ω (2.32)
we obtain the results stated in the Introduction. Observe that if γ > 0 and f ∈LNq′(Ω), one can easily prove thatanyweak solution satisfies (u−)q¯∈H01(Ω);
in particular, in that case uniqueness holds in the class of solutionsu∈H01(Ω) such that (u+)q¯ ∈ H01(Ω). Clearly, when γ is negative we should apply the result to the equation satisfied by−u.
Remark 2.3 When considering the caseλ= 0, a more careful look at the proof of Theorem 2.1 shows that in the inequality (2.23) one could replace f with f˜= sup
ξ
(H(x, ξ)−γ|ξ|q)+. The size condition of Theorem 2.2 and Corollary 2.1 would then concern ˜f instead of f. If one looks at the model problem (2.32), this simply means that if λ = 0 and γ > 0, the required size condition only concernsf+, as it is expected.
Remark 2.4 Whenq→2, the exponent ¯q→+∞. In fact, ifq= 2 uniqueness for problems like (2.32) holds in the class of solutions u ∈ H01(Ω) such that eµu−1 ∈ H01(Ω) for some suitable µ > 0. This result is proved (in a more general framework) in [3]: the idea is to use the change of unknown function v=eγu−1, so that the standard choice is to takeµ=γand to prove uniqueness whenv ∈H01(Ω). Otherwise one should takeµ =nγ for some n >1; in that case one proves uniqueness for solutions such that |v|n−1v ∈H01(Ω). However, we point out that this requires to apply to the equation ofva similar truncation argument as in Lemma 2.1.
2.1 Comments and extensions
1. Data in W−1,r.
The results of this section still hold if the right hand side in (2.1) is replaced byH(x,∇u) + div(g(x)) withg(x)∈LN(q−1)(Ω).
2. Neumann boundary conditions.
Our method easily extends to prove a comparison principle for the homo- geneous Neumann problem which can be written in a strong form as
λu−div(A(x)∇u) =H(x,∇u) in Ω,
A(x)∇u·ν(x) = 0 on∂Ω, (2.33)
whereν(x) is the outward, unit normal vector to∂Ω at x. Of course, we use the classical weak formulation which says thatu∈ H1(Ω) is a weak solution of (2.33) if
λ Z
Ω
uϕ dx+
Z
Ω
A(x)∇u∇ϕ dx= Z
Ω
H(x,∇u)ϕ dx ∀ϕ∈H1(Ω)∩L∞(Ω).
Then one has
Theorem 2.3 Assume (1.3), (2.2), (2.3) and thatλ >0. Letq≥1 +N2. If u and v are respectively a subsolution and a super-solution of (2.33) such that(1 +|u|)q−1¯ u∈H1(Ω)and(1 +|v|)q−1¯ v∈H1(Ω), then we have u≤v inΩ.
In particular, problem (2.33) has a unique weak solutionusuch that(1 +
|u|)q−1¯ u∈H1(Ω).
