Partial Differential Equations

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Partial Differential Equations

Correlation between two quasilinear elliptic problems with a source term involving the function or its gradient

Haydar Abdel Hamid, Marie Françoise Bidaut-Véron

Laboratoire de mathématiques et physique théorique, CNRS UMR 6083, faculté des sciences, 37200 Tours, France Received and accepted 5 October 2008

Available online 31 October 2008 Presented by Haïm Brezis

Abstract

Thanks to a change of unknown we compare two elliptic quasilinear problems with Dirichlet data in a bounded domain ofR^N. The first one, of the form−_pu=β(u)|∇u|^p+λf (x), whereβis nonnegative, involves a gradient term with natural growth.

The second one, of the form−pv=λf (x)(1+g(v))^p⁻¹wheregis nondecreasing, presents a source term of order 0. The correlation gives new results of existence, nonexistence and multiplicity for the two problems.To cite this article: H.A. Hamid, M.F. Bidaut-Véron, C. R. Acad. Sci. Paris, Ser. I 346 (2008).

Résumé

Corrélation entre deux problèmes quasilinéaires elliptiques avec terme de source relatif à la fonction ou à son gradient.

A l’aide d’un changement d’inconnue nous comparons deux problèmes elliptiques quasilinéaires avec conditions de Dirichlet dans un domaine bornéΩ deR^N. Le premier, de la forme−pu=β(u)|∇u|^p+λf (x), oùβ est positif, comporte un terme de gradient à croissance critique. Le second, de la forme−_pv=λf (x)(1+g(v))^p⁻¹oùgest croissante, contient un terme de source d’ordre 0. La comparaison donne des résultats nouveaux d’existence, nonexistence et multiplicité pour les deux problèmes.

Pour citer cet article : H.A. Hamid, M.F. Bidaut-Véron, C. R. Acad. Sci. Paris, Ser. I 346 (2008).

Version française abrégée

SoitΩun domaine borné régulier deR^N(N2)et 1< p < N. Dans cette Note nous comparons deux problèmes quasilinéaires. Le premier comporte un terme de source d’ordre 1 :

−pu=β(u)|∇u|^p+λf (x) dansΩ, u=0 sur∂Ω, (1)

oùβ∈C⁰([0, L)), L∞,à valeurs0, λ >0 etf ∈L¹(Ω), f 0 p.p. dansΩ.Le second problème comporte un terme de source d’ordre 0 :

−_pv=λf (x)

1+g(v)p−1

dansΩ, v=0 sur∂Ω, (2)

E-mail addresses:abdelham@lmpt.univ-tours.fr (H.A. Hamid), veronmf@univ-tours.fr (M.F. Bidaut-Véron).

doi:10.1016/j.crma.2008.10.002

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oùg∈C¹([0, Λ)), Λ∞, g(0)=0 etgest croissante.

Le changement d’inconnue

v(x)=Ψ u(x)

= u(x)

0

e^{γ (θ )/(p}⁻¹⁾dθ, oùγ (t )= t

0

β(θ )dθ,

conduit formellement du problème (1) au problème (2), etβ etgsont liés par la relationβ(u)=(p−1)g(v).En particulierβ est croissant si et seulement sigest convexe. Le changement d’inconnue inverse formel, apparemment moins utilisé, est donné explicitement par

u(x)=H v(x)

=

v(x)

0

ds 1+g(s).

Toutefois dans la transformation peuvent s’introduire des mesures. NotonsM_b⁺(Ω)l’espace des mesures de Radon positives bornées surΩ, etM_s⁺(Ω)le sous-ensemble des mesures concentrées sur un ensemble dep-capacité 0. Nous établissons une correspondance précise entre les deux problèmes :

Théorème 1.Soituune solution renormalisée du problème

−pu=β(u)|∇u|^p+λf (x)+αs dansΩ, u=0 sur∂Ω, (3) oùα_s∈M_s⁺(Ω),et0u(x) < Lp.p. dansΩ. Alors il existeμ_s ∈M_b⁺(Ω),telle quev=Ψ (u)est solution attei- gnable du problème

−_pv=λf (x)

1+g(v)p−1

+μ_s dansΩ, v=0 sur∂Ω. (4) Réciproquement soitvune solution renormalisée de(4), telle0v(x) < Λp.p. dansΩ,oùμ_s∈M_s⁺(Ω). Alors il existeα_s∈M_s⁺(Ω), telle queu=H (v)est solution renormalisée de(3). De plus, siμ_s=0,alorsα_s=0.SiL= ∞ etβ ∈L¹((0,∞)),alorsμs=e^{γ (}^∞⁾αs. SiL <∞,ouL= ∞etβ /∈L¹((0,∞)),etαs =0 alors(3)n’a pas de solution. SiΛ <∞etμs=0,alors(4)n’a pas de solution.

Dans le casβ constant, les résultats suivants généralisent ceux de [1] relatifs au casp=2 : Théorème 2.On suppose queβ(u)≡p−1,doncv=Ψ (u)=e^u−1etg(v)=v,et que

λ₁(f )=inf

Ω

|∇w|^pdx

Ω

f|w|^pdx

: w∈W₀^1,p(Ω)\ {0} >0.

Siλ > λ1(f ),ouλ=λ1(f )etf ∈L^N/p(Ω),alors(1)et(2)n’ont pas de solution renormalisée.

Si 0< λ < λ₁(f )alors(2) a une solution unique v₀∈W₀^1,p(Ω), et (1) a une solution unique u₀∈W₀^1,p(Ω) telle quee^u⁰−1∈W₀^1,p(Ω). Si de plusf ∈L^r(Ω)avecr > N/p,alorsu₀etv₀∈L^∞(Ω);et pour toute mesure μ_s∈M_s⁺(Ω),(4)a une solution renormaliséev_s, et donc(1)a une infinité de solutionsu_s=H (v_s)∈W₀^1,p(Ω)moins régulières queu₀.

Le Théorème 1 et l’utilisation du problème (1) nous permettent de déduire un résultat important pour le pro- blème (2), étendant un résultat classique de [2] dans le casp=2 :

Théorème 3.On suppose queΛ= ∞,lims→∞g(s)/s= ∞, g est convexe à l’infini, etf ∈L^r(Ω)avecr > N/p.

Alors il existe λ^∗>0tel que pour toutλ∈(0, λ^∗)le problème(2)a une solution minimale bornéev_λ, et pour tout λ > λ^∗il n’a aucune solution renormalisée.

Nous étudions aussi les propriétés de la fonction extrémalev^∗=sup_λ_λ∗v_λétendant certains résultats de [3,9,11].

Dans le cas où gest à croissance limitée par l’exposant de Sobolev, nous obtenons des résultats d’existence d’une seconde solution variationnelle, nouveaux même dans le casp=2,étendant ceux de [1] et de [5].

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1. Introduction and main results

LetΩ be a smooth bounded domain inR^N (N2)and 1< p < N. In this Note we compare two quasilinear problems. The first one presents a source gradient term with a natural growth:

−_pu=β(u)|∇u|^p+λf (x) inΩ, u=0 on∂Ω, (1)

whereβ ∈C⁰([0, L)), L∞,andβ is nonnegative, andλ >0 is a given real, andf ∈L¹(Ω), f 0 a.e. in Ω.

Hereβcan have an asymptote at pointL,and is not supposed to be nondecreasing.

The second problem involves a source term of order 0,with the sameλandf:

−_pv=λf (x)

1+g(v)p−1

inΩ, v=0 on∂Ω, (2)

whereg∈C¹([0, Λ)), Λ∞, g(0)=0 andgis nondecreasing.

Problems of type (1) and (2) have been intensively studied the last twenty years. The main questions are existence, according to the regularity off and the value ofλ,regularity and multiplicity of the solutions, and the existence with possible measure data.

It is well known that the change of unknown in (1)

v(x)=Ψ u(x)

=

u(x)

0

e^{γ (θ )/(p}⁻¹⁾dθ, whereγ (t )= t

0

β(θ )dθ,

leads formally to problem (2), whereΛ=Ψ (L)andgis given byg(v)=e^{γ (u)/(p}⁻¹⁾−1.This is a way for studying problem (1) from the knowledge of problem (2). It is apparently less used the reverse correlation, even in casep=2:

foranyfunctiongnondecreasing on[0, Λ), the substitution in (2)

u(x)=H v(x)

= v(x)

0

ds 1+g(s)

leads formally to problem (1), whereβis defined on[0, L)withL=H (Λ); indeedH=Ψ⁻¹.Andβ is linked tog by relationβ(u)=(p−1)g(v).In particularβis nondecreasing if and only ifgis convex; andLis finite if and only if 1/(1+g) /∈L¹(0, Λ).

Some examples withp=2.

1. β(u)=1 and 1+g(v)=1+v.

2. β(u)=q/(1+(1−q)u), q∈(0,1),and 1+g(v)=(1+v)^q. 3. β(u)=1+e^uand 1+g(v)=(1+v)(1+ln(1+v)).

4. β(u)=q/(1−(q−1)u), q >1 and 1+g(v)=(1+v)^q. 5. β(u)=1/(1−u)and 1+g(v)=e^v.

6. β(u)=q/(1−(q+1)u), q >0 and 1+g(v)=1/(1−v)^q.

It had been observed in [6] that the correspondence betweenuandvis more complex, because some measures can appear, in particular in the equation inv. Our first main result is to make precise the link between the two problems.

We denote byM_b(Ω)the set of bounded Radon measures, and byM_s(Ω)the subset of measures concentrated on a set ofp-capacity 0. AndM_b⁺(Ω)andM_s⁺(Ω)are the positive cones ofM_b(Ω), M_s(Ω), andM₀(Ω)is the subset of measures absolutely continuous with respect to thep-capacity. Recall thatM_b(Ω)=M₀(Ω)+M_s(Ω).

We recall one definition of renormalized solutions among four of them given in [4]. LetU be measurable and finite a.e. inΩ, such thatT_k(U )belongs toW₀^1,p(Ω)for anyk >0.One still denotes byuthe capp-quasi-continuous representative. Letμ=μ0+μ⁺_s −μ⁻_s ∈M_b(Ω). ThenUis a renormalized solution of problem

−_pU=μ inΩ, U=0 on∂Ω, (3)

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if|∇U|^p⁻¹∈L^τ(Ω),for anyτ ∈ [1, N/(N−1)),and for anyk >0,there existα_k, β_k∈M₀(Ω)∩M_b⁺(Ω),concentrated on the sets{U=k}and{U= −k}respectively, converging in the narrow topology toμ⁺_s , μ⁻_s such that for any ψ∈W₀^1,p(Ω)∩L^∞(Ω),

Ω

∇T_k(U )^p⁻²∇T_k(U )· ∇ψdx=

{|U|<k}

ψdμ0+

Ω

ψdαk−

Ω

ψdβk.

Theorem 1.1.Letube any renormalized solution of problem

−_pu=β(u)|∇u|^p+λf (x)+α_s inΩ, u=0 on∂Ω, (4)

whereα_s ∈M_s⁺(Ω)and such that0u(x) < La.e. inΩ. Then there existsμ_s ∈M_b⁺(Ω),such thatv=Ψ (u)is a reachable solution of problem

−_pv=λf (x)

1+g(v)p−1

+μ_s inΩ, v=0 on∂Ω. (5)

Conversely letvbe any renormalized solution of (5), whereμs∈M_s⁺(Ω)and such that0v(x) < Λa.e. inΩ. Then there existsαs∈M_s⁺(Ω)such thatu=H (v)is a renormalized solution of (4).

Moreover ifμ_s=0,thenα_s=0.IfL= ∞andβ∈L¹((0,∞)),thenμ_s=e^{γ (}^∞⁾α_s.IfL <∞or ifL= ∞and β /∈L¹((0,∞)),andα_s=0,then(4)has no solution. IfΛ <∞, andμ_s=0,then(5)has no solution.

This theorem extends in particular the results of [1] where p=2 and L= ∞. The nonexistence result when β /∈L¹((0,∞)),andα_s=0,answers to an open question of [10].

First we apply to the caseβconstant, which meansglinear.

Theorem 1.2.Assume thatβ(u)≡p−1,thusv=Ψ (u)=e^u−1andg(v)=v.Suppose that λ₁(f )=inf

Ω

|∇w|^pdx

Ω

f|w|^pdx

: w∈W₀^1,p(Ω)\ {0} >0. (6)

Ifλ > λ₁(f ),orλ=λ₁(f )andf ∈L^N/p(Ω),then(1)and(2)admit no renormalized solution.

If0< λ < λ₁(f )there exists a unique solutionv₀∈W₀^1,p(Ω)to(2), thus auniquesolutionu₀∈W₀^1,p(Ω)to(1) such thate^u⁰−1∈W₀^1,p(Ω). Iff∈L^r(Ω)withr > N/p,thenu₀andv₀∈L^∞(Ω),and moreover for any measure μ_s ∈M_s⁺(Ω), there exists a renormalized solution v_s of (5);then there exists aninfinity of less regular solutions u_s=H (v_s)∈W₀^1,p(Ω)of(1).

Remark 1.3.Under the assumption (6), most of these existence results extend to general gsuch that Λ= ∞and lim sup_τ_−→∞g(τ )/τ <∞. They extend to the case

lim sup

τ→∞g(τ )/τ^q<∞ for someq∈

1, N/(N−p)

(7) if moreoverf ∈L^r(Ω)withqr< N/(N−p).

Next consider problem (2) with generalg,andf ∈L^r(Ω)withr > N/p.It is easy to prove that for smallλ >0 there exists a minimal solutionv_λ∈W₀^1,p(Ω)such thatv_λL^∞(Ω)< Λ. Our main result is an extension of a well known result of [2] forp=2,and [3] forp >1. It is noteworthy thatthe proof uses problem(1):

Theorem 1.4. Assume thatΛ= ∞,and lims→∞g(s)/s= ∞, andgis convex near infinity, andf ∈L^r(Ω)with r > N/p. There exists a realλ^∗>0such that:

(i) forλ∈(0, λ^∗)problem(2)has a minimalboundedsolutionv_λ, (ii) forλ > λ^∗there exists norenormalizedsolution.

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Whengis subcritical with respect to the Sobolev exponentp^∗=Np/(N−p), we obtainnew multiplicity results for problem (2), even in the casep=2,extending [1] and [5]:

Theorem 1.5.Under the assumptions of Theorem1.4, assume that lim sup

τ→∞g^p⁻¹(τ )/τ^Q<∞ for someQ∈(1, p^∗−1), (8)

and f ∈L^r(Ω)with(Q+1)r< p^∗.Then there existsλ₀>0such that for any λ < λ₀,there exists at leasttwo solutionsv∈W₀^1,p(Ω)∩L^∞(Ω)of (2). Moreover ifp=2, g is convex, org satisfies the Ambrosetti–Rabinowitz growth condition andf ∈L^∞(Ω), the same result holds withλ₀=λ^∗.

Concerning the extremal solution, we get the following, extending some results of [3,11]:

Theorem 1.6.Under the assumptions of Theorem1.4, the extremal functionv^∗=sup_λ_λ∗v_λ is a renormalized so- lution of (2) withλ=λ^∗. IfN < p(1+p)/(1+p/r) thenv^∗∈W₀^1,p(Ω).Moreoverv^∗ ∈L^∞(Ω)in any of the following conditions:

(i) N is arbitrary and(8)holds and(Q+1)r< p^∗, (ii) N is arbitrary and(7)holds andqr< N/(N−p), (iii) N < pp/(1+1/(p−1)r).

Remark 1.7.Using Theorems 1.1, 1.4 and 1.5, we deduce existence and nonexistence results for problem (1). In Theorem 1.1, function f can depend on u or v, which strongly extends the range of applications. For example, takingg(v)=v,andf =u^b, b >0,problem−_pu=(p−1)|∇u|^p+λu^brelative touleads to−_pv=λ(1+ v)^p⁻¹ln^b(1+v)relative tov.Then for smallλthe problem inuhas an infinity of solutionsu∈W₀^1,p(Ω),two of them being bounded.

Remark 1.8. A part of our results is based on a growth assumption on g. Returning to problem (1), this condi- tion is not always easy to verify. When L= ∞, all the “usual” functions β, even with a strong growth, satisfy lim sup_τ_→∞g(τ )/τ^q<∞for anyq >0,see [1]. However using the converse correlation betweengandβ,we prove that the conjecture that this condition always holds iswrong: letF∈C⁰([0,∞))be any strictly convex function, with lims→∞F (s)= ∞.Then there exists an increasing functionβ such that limt→∞β(t )= ∞and the correspondingg satisfies lim sup_τ_→∞g(τ )/F (τ )= ∞.

2. Sketch of the main proofs

In some proofs we use a regularity lemma:

Lemma 2.1. Let 1< p < N, and F ∈L^m(Ω), and m=Np/(Np−N +p) (thus 1< m < N/p). Let U be a renormalized solution of problem

−_pU=F inΩ, U=0 on∂Ω.

If1< m < N/p,thenU^(p⁻¹⁾∈L^k(Ω),withk=N m/(N−pm).Ifm=N/p,thenU^(p⁻¹⁾∈L^k(Ω)for anyk1.

Ifm > N/p,thenU∈L^∞(Ω).If1< m < m,then|∇U|^(p⁻¹⁾∈L^τ(Ω),withτ =N m/(N−m). Ifmm,then U∈W₀^1,p(Ω).

Proof of Theorem 1.1. Forp=2,we cannot use approximations of the equations because of the nonuniqueness of the solutions of −pU=μ withμ∈M_b⁺(Ω). The main idea is to use the equations satisfied in the sense of distributions by the truncated functionsT_K(u)=min(K, u)andT_k(v)=min(k, v)withk=Ψ (K), using definition (ii) of renormalized solution given above:

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−_pT_K(u)=β

T_K(u)∇T_K(u)^p+λf χ_{_u_K_}+α_K, inD(Ω),

−_pT_k(v)=λf

1+g(v)p−1

χ_{_v_k_}+μ_k, inD(Ω),

where μ_k and α_K are two measures concentrated on the same set: {u=K} = {v=k}, and explicitly related by μ_k=(1+g(k))^p⁻¹α_K,and respectively converging weakly^∗toμ_sandα_s.The nonexistence results are consequences of some properties of renormalized solutions, also called Inverse Maximum Principle. 2

Proof of Theorem 1.2. The nonexistence is first proved for (1), and then for (2) by Theorem 1.1. The existence is obtained by iteration and approximation, using [4]. Uniqueness follows from Picone’s identity, adapted to renormalized solutions. 2

Proof of Theorem 1.4. Formally, ifv is a solution of (2) for someλ,andu=H (v),thenu¯=(1−ε)uis a supersolution of (1) relative toλ¯=(1−ε)^p⁻¹λ,andv¯=Ψ (u)¯ is a supersolution of (2) relative toλ¯;then there exists a solutionv1v.¯ Using Theorem 1.1 we show that it is not formal, since actuallyno measure appears. In the (best) case H (∞) <∞, v¯ is bounded, then alsov1 is bounded. Otherwise a bootstrapp using Lemma 2.1 is needed for constructing a bounded solution. 2

Proof of Theorem 1.5. The Euler functionJ_λ is well defined, and for smallλit has the geometry of Mountain Path near 0, but the Palais–Smale sequences may be unbounded. From [8], (2) has a second solution for almost any smallλ, and then for a sequenceλ_n→λ,and we are lead to prove that the solutionsv_λ_nrelative toλ_nconverge to a solution to (2) relative toλ. The usual estimates for the casep=2,using an eigenfunction as test function, cannot be extended.

We get an estimate of−_pv_λ_ninL¹(Ω)in another way, using the convexity ofg. The estimate ofv_λ_ninW₀^1,p(Ω)is obtained by contradiction. For largerλ,ifp=2,J_λhas still the geometry of mountain path nearv_λ; the question is open whenp=2. Under Ambrosetti–Rabinowitz condition we apply some results of [7]. 2

Proof of Theorem 1.6. The estimates come from Lemma 2.1 and well known regularity results for quasilinear equations, and from an extension of techniques of [9]. 2

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