Breaking of resonance and regularizing effect of a first order quasi-linear term in some elliptic equations

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www.elsevier.com/locate/anihpc

Breaking of resonance and regularizing effect of a first order quasi-linear term in some elliptic equations

^✩

Boumediene Abdellaoui

^a

, Ireneo Peral

^b,^∗

, Ana Primo

^b

aDépartement de Mathématiques, Université Aboubekr Belkaïd, Tlemcen, Tlemcen 13000, Algeria bDepartamento de Matemáticas, U. Autonoma de Madrid, 28049 Madrid, Spain Received 9 November 2006; received in revised form 14 June 2007; accepted 14 June 2007

Available online 17 October 2007

Abstract

In this article we study the problem

(P)

⎧⎨

⎩

−u+ |∇u|^q=λg(x)u+f (x) inΩ,

u >0 inΩ,

u=0 on∂Ω,

with 1q2 andf, gare positive measurable functions. We give assumptions ongwith respect toqfor which for allλ >0 and allf∈L¹,f 0, problem (P) has a positive solution. In particular we focus our attention ong(x)=1/|x|²to prove that the assumptions ongare optimal.

Résumé

Dans cet article nous étudions le problème

(P)

⎧⎨

⎩

−u+ |∇u|^q=λg(x)u+f (x) inΩ,

u >0 inΩ,

u=0 on∂Ω,

où 1q2 etf, gsont des fonctions mesurables positives. Nous donnons des hypothèses surgdépendant deqtelles que pour toutλ >0 et pour toutf ∈L¹,f 0, le problème (P) a une solution positive. Nous portons une attention particulière au cas g(x)=1/|x|²pour montrer que les hypothèses surgsont optimales.

MSC:35D05; 35D10; 35J20; 35J25; 35J70; 46E30; 46E35

Keywords:Quasilinear elliptic equations; Existence and nonexistence; Regularization; Resonance

✩ Work partially supported by projects MTM2004-02223, MEC, Spain and CCG06-UAM/ESP-0340, CAM-UAM.

* Corresponding author.

E-mail addresses:boumediene.abdellaoui@uam.es (B. Abdellaoui), ireneo.peral@uam.es (I. Peral), ana.primo@uam.es (A. Primo).

doi:10.1016/j.anihpc.2007.06.003

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1. Introduction

This paper is devoted to obtain existence and nonexistence results for nonlinear elliptic equations of the form −u+ |∇u|^q=λg(x)u+f (x) inΩ,

u >0 inΩ,

u=0 on∂Ω,

(1.1) whereΩ ⊂R^N is an open bounded domain, 1q 2 and λ∈R. In the whole of this work, we suppose thatf andgare positive measurable functions with some summability assumptions that we will specify in each case. For λ≡0, equations of the form (1.1) have been widely studied in the literature. We refer to [1,4,5,9,7,8,17,19] and the references therein. The caseq >2 andf Lipschitz has been studied in [22] with quite different methods. The case λ >0 and the presence of the term|∇u|^qin the right-hand side of the equation has been recently studied in [2]. In [10]

it is proved that for allλ >0 the equation

−u=λ u

|x|²+f (x) inΩ⊂R^N, N3 and 0∈Ω, (1.2)

has in general no solution for a positivef ∈L¹(Ω). However, in [3], by adding the gradient term|∇u|²on the left- hand side of Eq. (1.2), we find a positive solution for all λ >0 and for all positivef ∈L¹(Ω). This means that if q=2 andg(x)= |x|⁻²in problem (1.1), the term|∇u|²produces a strong regularizing effect. The new feature in this paper is to find the assumptions ongin terms ofqin order to solve problem (1.1) for allλ >0 and for allf ∈L¹(Ω), f 0. We can reformulate the main objective of the paper as follows: fixedgfind theoptimalq for which for all λ >0 and for all positive functionf ∈L¹(Ω), problem (1.1) has a positive solution. Precisely we prove the existence of solutionfor allλ >0and for allf ∈L¹(Ω),f 0, ifg0 is anadmissible weightin the sense of (2.1) below.

This condition, in general, also becomes optimal. The previous result also proves that the absorption term |∇u|^q is sufficient to break down any resonant effect of the linear zero order term. Forf ∈L¹(Ω), we say thatuis a weak solution to problem (1.1) ifu∈W₀^1,q(Ω),g(x)u∈L¹(Ω)and

Ω

u(−φ) dx+

Ω

|∇u|^qφ dx=λ

Ω

g(x)uφ dx+

Ω

f (x)φ dx, ∀φ∈C₀^∞(Ω).

Since we consider solutions to (1.1) with data inL¹(Ω), we refer to [16] and [15] for a complete discussion about this framework. The paper is organized as follows. Section 2 is devoted to prove existence results. In Subsection 2.1, fixed 1< q2 we prove the main result, that is, ifgis an admissible weight then for allλ >0 and for allf ∈L¹(Ω), f 0, there existsu∈W₀^1,q(Ω)positive solution to problem (1.1). This is the result of Theorem 2.3.

In Subsection 2.2, we obtain conditions ongandλsuch that for all 1< q2, there exists solution to problem (1.1) provided that f is in a suitable class associated explicitly to g. Due to the presence of the Laplace operator and the linear termλg(x)u, then the natural condition is to assume (2.12). In Subsection 2.3, we give some results on uniqueness of solution. In Section 3, we consider the Hardy potential to prove nonexistence results that show the optimality of the condition required for existence in Theorems 2.3 and 2.9. These nonexistence results are related to the classical Hardy inequality,

Ω

|∇u|²dxΛ_N

Ω

u²

|x|²dx, for allu∈C^∞₀ (Ω)whereΛ_N=

N−2 2

2

(see for instance [18]). More precisely by settingg(x)= |x|⁻², we prove nonexistence results for a very weak solution if 1q_N^N₋₁ andλ > Λ_N, the Hardy constant. Next we prove the optimality of the summability assumption onf in Theorem 2.9. Finally in Subsection 3.1 we study the potentialg(x)= |x|⁻^α, 1< α < ^N⁺₂². Givenua measurable function we consider thek-truncation ofudefined by

Tk(u)= u, |u|k, k_|^u_u_|, |u|> k.

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2. Existence of weak positive solutions

In this section we prove the existence of positive solution to problem (1.1) according to a relation betweenq, λ, g andf. In the first part, we prove that for allλ >0 and for allf ∈L¹(Ω),f 0, there exists a positive solution to (1.1), provided thatgsatisfies some assumptions related toq. In the second part, we study the class of dataf for which, for allqin[1,2]and under some conditions ong, there exists positive solution for small values ofλ.

2.1. Admissible weights: Global existence result

Let consider in (1.1) a fixedq, 1< q2, then we say thatgis anadmissible weightin the sense of (2.1) below, g0, g∈L¹(Ω) and C(g, q)= inf

φ∈W₀^1,q(Ω)\{0}

(

Ω|∇φ|^qdx)^q¹

Ωg|φ|dx >0. (2.1)

Remark 2.1.It is clear that ifgsatisfies (2.1), theng∈W⁻^1,q(Ω)∩L¹(Ω),q =_q₋^q₁. Moreover (2.1) implies that (a)

Ωg|u|dx <∞for allu∈W₀^1,q(Ω).

(b) DefiningT:W₀^1,q(Ω)→Rby T , u ≡

Ω

gu dx, (2.2)

thenT is a linear continuous form onW₀^1,q(Ω). That means that there existsF=(f₁, f₂, . . . , f_N)∈(L^q(Ω))^N such thatg= −div(F ) and then

T , u ≡

Ω

gu dx=

Ω

F ,∇udx.

As a consequence, the duality product is equivalent to the first integral and T_W−1,q (Ω)= F_(L^q_(Ω))N.

Proposition 2.2.Assume thatgsatisfies(2.1)and letT be defined by(2.2). Considergn(x)=min{g(x), n}and the corresponding linear continuous form

T_n:W₀^1,q(Ω)→R, u→T_n(u)=

Ω

g_nu dx.

The following statements hold.

(i) T_n→T strongly inW⁻^1,q(Ω).

(ii) Assume thatu_n uweakly inW₀^1,q(Ω),u_n0andu_n→ua.e., theng_nu_n→gustrongly inL¹(Ω).

Proof. (i) As in Remark 2.1 we also have that T_n, u =

Ω

g_nu dx,

that is, the duality is realized by the integral and moreover C(g_n, q)= inf

φ∈W₀^1,q(Ω)\{0}

(

Ω|∇φ|^qdx)¹^q

Ωg_n|φ|dx >0.

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Notice that

Tn_W^−1,q = sup

u

W1,q 0 (Ω)1

Tn, u sup

u

W1,q 0 (Ω)1

Ω

gnu

sup

u

W1,q 0 (Ω)1

Ω

g|u|T_W^−1,q.

Then {Tn}n∈N, up to a subsequence, converges weakly in W⁻^1,q(Ω). Since for all u∈W₀^1,q(Ω), the Lebesgue theorem proves that

g_n|u| →g|u| strongly inL¹(Ω), then{T_n}n∈N T therefore,

T_W−1,q lim inf

n→∞ T_n_W−1,q lim sup

n→∞ T_n_W−1,q T_W−1,q , hence,

T_n→T , strongly inW⁻^1,q (Ω).

(ii) According to the strong convergence proved in (i), we have that ifu_n uweakly inW₀^1,q(Ω), then, T_n, u_n =

Ω

g_nu_ndx→

Ω

gu dx= T , u asn→ ∞.

If we assume that, moreover, u_n 0 and u_n→u a.e., then by using a well-known result in [12] we obtain that g_nu_n→gustrongly inL¹(Ω). (See also [21], Theorem 1.9, page 21.) 2

The main result of this section is the following one.

Theorem 2.3.Assume that1< q2and letgbe a positive function such that(2.1)holds, then for allλ0and for allf ∈L¹(Ω),f0, there existsu∈W₀^1,q(Ω),u0, that solves problem(1.1)in the sense of distributions.

To prove Theorem 2.3 we start by proving the result in some particular cases and then we proceed by approximation ofgandf. Notice that since 1< q2, then^N₂ ^N_q, therefore the following first approach is quite natural.

Theorem 2.4. Assume that f, g∈L^r(Ω) are positive functions withr > ^N_q, then for all λ0, there exists u∈ W₀^1,2(Ω)∩L^∞(Ω)a weak positive solution to problem(1.1).

Proof. Step 1: For every fixedk >0, considerv∈W₀^1,2(Ω)∩L^∞(Ω)such that−v=λkg(x)+f (x)inΩ and denote M_k = vL^∞. Notice thatv is bounded by the assumptions on gandf and by standard elliptic estimates (see [24]). It follows that zero is a subsolution andvis a supersolution to problems

−w_n+₁₊^|∇1^wⁿ^|^q

n|∇w_n|^q =λg(x)T_kw_n+f (x), w_n∈W₀^1,2(Ω), w_n0

(2.3) for alln∈N. By a simple variation of the arguments used in [8] and [4], there exists a sequence of nonnegative minimal solutions {w_n}to problems (2.3). It follows that−w_nλkg(x)+f (x)= −v, hence by the weak comparison principle we conclude that 0wnvM, uniformly inn, in particular,wn∈W₀^1,2(Ω)∩L^∞(Ω). Takingwnas test function in (2.3) we obtain,

Ω

|∇w_n|²dx+

Ω

H_n(∇w_n)w_ndx=λ

Ω

g(x)T_kw_nw_ndx+

Ω

f (x)w_ndx,

whereH_n(s)= |s|^q/(1+¹_n|s|^q). It is easy to check that there exists a positive constantCsuch that

Ω

|∇w_n|²dxC(f, g, Ω, k) uniformly inn,

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therefore, up to subsequencesw_n u_k weakly inW₀^1,2(Ω). By weak-* convergence inL^∞(Ω)we also have that uk∈W₀^1,2∩L^∞(Ω)andukMk. Next we investigate the equation satisfied byuk. To do that we prove the following claim.

Convergence claim.w_n→u_k strongly inW₀^1,2(Ω).

Proof of the convergence claim. We follow closely the argument used in [5]. Considerφ(s)=se¹⁴^s², which satisfies φ(s)− |φ(s)|¹₂. Takingφ(w_n−u_k)as test function in (2.3),

Ω

∇wnφ(wn−uk)∇(wn−uk) dx+

Ω

Hn(∇wn)φ(wn−uk) dx

=λ

Ω

g(x)Tkwnφ(wn−uk) dx+

Ω

f (x)φ(wn−uk) dx. (2.4)

Asw_n u_kweakly inW₀^1,2(Ω), then a direct computation shows that

Ω

∇w_nφ(w_n−u_k)∇(w_n−u_k) dx=

Ω

∇(w_n−u_k)²φ(w_n−u_k) dx+o(1).

Sinceq2, it is well known that∀ε >0 there exists a nonnegative constantC_εsuch that

s^qεs²+C_ε, s0. (2.5)

Hence the second term in the left-hand side can be estimated in the following way,

Ω

H_n(∇w_n)φ(w_n−u_k) dxε

Ω

|∇w_n|²φ(w_n−u_k)dx+C(ε)

Ω

φ(w_n−u_k)dx

=ε

Ω

|∇w_n− ∇u_k|²φ(w_n−u_k)dx−ε

Ω

|∇u_k|²φ(w_n−u_k)dx +2ε

Ω

∇w_n∇u_kφ(w_n−u_k)dx+C(ε)

Ω

φ(w_n−u_k)dx.

Sincewn ukweakly inW₀^1,2(Ω)and by the fact that|φ(wn−uk)| →0 almost everywhere (and inL²(Ω)), then it follows

(i)

Ω|∇u_k|²|φ(w_n−u_k)|dx→0 asn→ ∞, (ii)

Ω∇w_n∇u_kφ(w_n−u_k) dx→0 asn→ ∞. Therefore, passing to the limit asntends to∞, we have

Ω

H_n(∇w_n)φ(w_n−u_k) dxε

Ω

|∇w_n− ∇u_k|²φ(w_n−u_k)dx+o(1).

Moreover, it is clear that the right-hand side in (2.4) goes to zero asn→ ∞. Sinceφ(s)− |φ(s)|>¹₂, choosingε1 we conclude that

1 2

Ω

|∇w_n− ∇u_k|²dx

Ω

φ(w_n−u_k)−εφ(w_n−u_k)|∇w_n− ∇u_k|²dxo(1), whencewn→uk inW₀^1,2(Ω)and the claim is proved. Moreover, from (2.5) it follows that

H_n(∇w_n)c₁|∇w_n|²+c₂.

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By the claim, we have in particular the almost everywhere convergence of the gradients and therefore we conclude that

H_n(∇w_n)→ |∇u_k|^q inL¹(Ω).

Hence we find thatu_ksolves problem

−u_k+ |∇u_k|^q=λg(x)T_ku_k+f (x) inΩ, u_k∈W₀^1,2(Ω). (2.6) Step 2. We claim that there existsM >0 such thatu_kL^∞(Ω)Mfor allk >0.

First of all we prove that{u_k}is uniformly bounded inL^a(Ω)for alla >1.

Consider

λ₁(θ, g)= inf

φ∈C₀^∞(Ω), φ=0

Ω|∇φ|^θdx

Ωg|φ|^θdx.

Sinceg∈L^r(Ω)withr >^N_q, it follows thatλ1(θ ) >0 for allθq.

Sinceuk∈L^∞(Ω), then usingu^a_k, witha2, as a test function in (2.6), we get a

Ω

u^a_k⁻¹|∇uk|²dx+

Ω

u^a_k|∇uk|^qdxλ

Ω

gu^a_k⁺¹dx+

Ω

f u^adx.

Thus 4a (2+a)²

Ω

|∇u

a 2+1 k |²dx+

q a+q

q

Ω

|∇u

a q+1

k |^qdxλ

Ω

gu^a_k⁺¹dx+

Ω

f u^adx.

Using Hölder, Young and Poincaré inequalities there results that

Ω

gu^a⁺¹dx=

Ω

g^2(aâ⁺^q)uâ²g²¹uâ²⁺¹g

q 2(a+q)dx

Ω

gu^a⁺^qdx ^a

2(a+q)

Ω

gu^a⁺²dx ¹

2

Ω

g dx ^q

2(a+q)

ε

Ω

gu^a⁺^qdx+ε

Ω

gu^a⁺²dx+C(ε)

Ω

g dx

ε

λ1(q, g)

Ω

|∇u

a q+1

k |^qdx+ ε λ1(2, g)

Ω

|∇u

a 2+1

k |²dx+C(ε, g), whereεis a positive constant that will be chosen later. On the other hand we have

Ω

f u^adx

Ω

u²^∗⁽^a²⁺¹⁾dx _2∗(a/2^a

+1)

Ω

f

N (a+2) 2(N+a)dx

^2(N_{N (a+2)}⁺^a)

C(f, S)

Ω

|∇u^a²⁺¹⁾|²dx _a₊^a₂

ε

Ω

|∇u^a²⁺¹⁾|²dx+C(f, S, ε),

whereSis the Sobolev constant and 2^∗=_N^2N₋₂. Therefore, putting together the above inequalities it follows that 4a

(1+a)² − ελ λ1(2, g)−ε

Ω

|∇u

a 2+1 k |²dx+

q a+q

q

− ελ λ1(q, g)

Ω

|∇u

a q+1

k |^qdxC(f, g, S, ε).

Choosingεsmall enough we conclude that

Ω

|∇u

a 2+1 k |²dx+

Ω

|∇u

a q+1

k |^qdxC,

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whereC is independent ofk. Hence using Sobolev inequality we obtain thatu_kL^a(Ω)C(a, λ, f, g)for alla >1.

Whence{gu_k+f}L^r(Ω)is uniformly bounded ink, for somer > ^N₂. The uniform boundedness is a consequence of classical results about elliptic regularity, see [24].

Therefore ifk > M,u≡ukand is a solution to problem (1.1). 2

Remark 2.5.Notice that ifq2, then the passage to the limit in theconvergence claimabove can be performed in a different way; indeed using a compactness result by Boccardo–Murat in [6], the gradients converge almost everywhere, therefore using Vitali theorem we get the strong convergence of the gradient.

However, we prove theconvergence claimwith arguments valid in a more general setting, which are needed to obtain strong convergence in the next theorems.

In the following result, we continue considering a weight with extra summability, but a generalf∈L¹(Ω).

Theorem 2.6.Assume thatf, g are positive functions,f ∈L¹(Ω),f 0and g∈L^r(Ω)withr > ^N_q, then for all λ0, problem(1.1)has a positive solutionu∈W₀^1,q(Ω).

Proof. Consider a sequencef_n∈L^∞(Ω)such thatf_n↑f inL¹(Ω). Thanks to Theorem 2.4, there exists a sequence of positive bounded functions{u_n}, solutions to problems

⎧⎨

⎩

−u_n+ |∇u_n|^q=λg(x)u_n+f_n(x) inΩ, u_n>0 inΩ, u_n=0 on∂Ω,

un∈W₀^1,2(Ω)∩L^∞(Ω).

(2.7)

Notice that, in particularu_nsolves a problem of the form−u_n=F_n∈L¹(Ω), thenu_nis the unique entropy solution to this problem. As a consequence we can useT_k(u_n)as a test function. See for instance [24] or [16].

TakingT_ku_nas test function in (2.7), it follows that

Ω

|∇Tkun|²dx+

Ω

|∇un|^qTkundx=λ

Ω

g(x)unTkundx+

Ω

fn(x)Tkundx inΩ.

DefineΨ_k(s)=s

0T_k(t )¹^qdt, that explicitly is,

Ψk(s)=

⎧⎨

⎩

q q+1s

q+1

q ifs < k,

q q+1k

q+1

q +(s−k)k¹^q ifs > k,

then by using the assumptions ongandf, it follows that for allε >0, there existsC_ε>0 such that

Ω

|∇T_ku_n|²dx+

Ω

|∇Ψ_ku_n|^qdxkελ

Ω

g(x)u_ndx q

+λkC(ε)+kf_nL¹

εkλ

C(q, g)

Ω

|∇un|^qdx+λkC(ε)+kfnL¹. (2.8)

Hence, as in the second step of the proof of Theorem 2.4,

Ω

|∇u_n|^qdx

Ω

|∇T_ku_n|²dx+k

{unk}

|∇u_n|^qdx+C_q|Ω|

εkλ

C(q, g)

Ω

|∇u_n|^qdx+λkC(ε)+C_q|Ω| +kfL¹,

(8)

then u_n u weakly inW₀^1,q(Ω) and T_ku_n T_ku weakly in W₀^1,2(Ω). It is clear by the assumption on g that gu_n→gustrongly in L¹(Ω). LetG_k(s)=s−T_k(s)and defineψ_k₋₁(s)=T₁(G_k₋₁(s)), thenψ_k₋₁(u_n)|∇u_n|^q

|∇u_n|^qχ_{_u_n_k_}. As in [5], usingψ_k₋₁(u_n)as a test function in (2.7) there results that

Ω

∇ψ_k₋₁(u_n)²dx+

Ω

ψ_k₋₁(u_n)|∇u_n|^qdx=

Ω

λg(x)u_n+f_n(x)

ψ_k₋₁(u_n) dx.

Since{u_n}is uniformly bounded inL^p(Ω),∀pq^∗,it follows that

x∈Ω, such thatk−1< u_n(x) < k→0, x∈Ω, such thatu_n(x) > k→0 ask→ ∞, uniformly inn. Thus we conclude

klim→∞

{unk}

|∇u_n|^qdx=0, uniformly inn. (2.9)

We follow the same arguments as in the proof of theconvergence claimin Theorem 2.4. Takeφ(Tk(un)−Tk(u))as a test function in (2.7), then

T_ku_n→T_ku strongly inW₀^1,2(Ω). (2.10)

To finish the proof, it is sufficient to show that

|∇u_n|^q→ |∇u|^q strongly inL¹(Ω).

Since the sequence of gradients converges a.e. inΩ, we only have to show that is equi-integrable and to apply Vitali’s theorem. LetE⊂Ω be a measurable set. Then,

E

|∇u_n|^qdx

E

|∇T_ku_n|^qdx+

{unk}∩E

|∇u_n|^qdx.

From (2.10) it follows that for allk >0,T_k(u_n)→T_k(u)strongly inW₀^1,p(Ω)for allp2. In particular we obtain the strong convergence forp=q. Hence the integral

E|∇T_k(u_n)|^qdxis uniformly small if|E|is small enough. On the other hand, thanks to (2.9) we obtain that

{unk}∩E

|∇u_n|^qdx

{unk}

|∇u_n|^qdx→0 ask→ ∞uniformly inn.

The equi-integrability of|∇u_n|^qfollows immediately, and the proof is completed. 2

Proof of Theorem 2.3. We consider the truncationgn(x)=min{g(x), n} ∈L^∞(Ω). Due to Theorem 2.6, there exists a sequence of positive functions{un}such thatunsolves

⎧⎨

⎩

−u_n+ |∇u_n|^q=λg_n(x)u_n+f (x) inΩ, un>0 inΩ, un=0 on∂Ω,

un∈W₀^1,q(Ω).

(2.11)

SinceT_ku_n∈W₀^1,q(Ω)∩L^∞(Ω), then we can useT_ku_nas a test function in (2.11), it follows that

Ω

|∇T_ku_n|²dx+

Ω

|∇Ψ_ku_n|^qdxλ

Ω

g_n(x)T_ku_nu_ndx+

Ω

f (x)T_ku_ndx

kλ

Ω

g_n(x)u_ndx+k

Ω

f (x) dx.

Since

Ω

|∇Ψkun|^qdx

{u_nk}

|∇Ψkun|^qdxk

{u_nk}

|∇u|^qdx,

(9)

then as above

Ω

∇T_k(u_n)²dx+k

{unk}

|∇u_n|^qdxkελ

Ω

g_n(x)u_ndx q

+k

Ω

f (x) dx+λkC(ε, Ω).

Therefore by (2.1) we have that

Ω

|∇u_n|^qdx kελ C(g, q)

Ω

|∇u_n|^qdx+k

Ω

f (x) dx+λkC(ε, q, Ω),

henceu_n uweakly inW₀^1,q(Ω). We have thatu_n0 and by Sobolev and Rellich theorems, up to a subsequence, we obtain thatu_n→ua.e. Then we apply Proposition 2.2 to obtain that

gnun→gu strongly inL¹(Ω).

Therefore to finish the proof it is sufficient to show thatu_n→ustrongly inW₀^1,q(Ω). This fact follows by proving thatTkun→Tkustrongly inW₀^1,2(Ω)and using Vitali’s theorem as in the previous steps. 2

Remarks 2.7.

(1) The solution of problem (1.1) obtained in Theorem 2.3 is also an entropy solution in the sense that we can take in problem (1.1) test functions of the formT_k(u−v)withv∈W^1,2(Ω)∩L^∞(Ω).

(2) The same existence result holds iff is a bounded positive Radon measure such thatf ∈L¹(Ω)+W⁻^1,2(Ω)(fis absolutely continuous with respect to the classical capacity). These results are obtained with some minor technical changes, i.e. the result follows using the same approximation arguments. See [15] to find the precise meaning of solution in this framework and equivalent definitions. By the regularity theory of renormalized solutions we easily get that ifuis a positive solution to problem (1.1), thenu∈W₀^1,q(Ω)∩W₀^1,p(Ω)for all 1p <_N^N₋₁.

(3) The existence result obtained in Theorem 2.3 in particular means that resonance phenomenon doesn’t occur if we add|∇u|^q as an absorption term. Without the gradient term, positive solution exists just by assuming thatλis less than the infimum of the spectrum of the operator−with the corresponding weightgand under a suitable condition onf.

As a direct application of Theorem 2.3 we obtain the following well known result.

Corollary 2.8.Assume thatλ=0, then problem(1.1)has an entropy positive solution for all0f ∈L¹(Ω)and for all1q2.

In fact we can considerg∈L^∞(Ω)in Theorem 2.3 and then pass to the limit using a priori estimates asλgoes to 0.

A direct proof of this particular existence result can be obtained using truncation argument. See [5]. We will analyze in particular the Hardy potential in section 3 in order to prove the optimality of the assumptions in Theorem 2.3.

2.2. Existence of solution for allq∈ [1,2]and smallλ

In this section we will find general conditions ongandf that assure the existence of solution for allq2. The presence of the linear termλg(x)umotivates the following assumption ong,

g0, g≡0 andg∈L¹(Ω)withλ₁(g,2)= inf

φ∈W₀^1,2(Ω)\{0}

Ω|∇φ|²dx

Ωg|φ|²dx >0. (2.12)

It is easy to check that under assumption (2.12), for everyλ < λ¯ ₁(g,2)there exists a uniqueϕ∈W₀^1,2(Ω), positive weak solution to the problem

−ϕ= ¯λg(x)ϕ+g(x) inΩ, ϕ=0 on∂Ω. (2.13)

The main result in this section is the following one.

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Theorem 2.9. Assume that 0< λ < λ₁(g,2)and let ϕ be the solution to problem (2.13) with λ <λ < λ¯ ₁(g,2).

Suppose thatf is a positive function such that

Ωf ϕ dx <∞, then there existsu∈W₀^1,q(Ω)positive solution to problem(1.1)and moreover,

Ω|∇u|^pdx <∞,∀pqifq_N^N₋₁andp <_N^N₋₁, on the contrary case.

Proof. By using Theorem 2.4, we find a sequence{u_n}of positive solutions to the approximated problems

−u_n+ |∇u_n|^q=λg_n(x)u_n+f_n(x) inΩ,

u_n∈W₀^1,q(Ω), (2.14)

withg_n(x)=min{g(x), n} ∈L^∞(Ω)andf_n(x)=min{f (x), n} ∈L^∞(Ω). It is clear that

Ωf_n(x)ϕ dx < C uniformly inn. By a duality argument we obtain

Ω

un(−ϕ) dx+

Ω

|∇un|^qϕ dx=λ

Ω

gn(x)unϕ dx+

Ω

fn(x)ϕ dx.

Therefore, (¯λ−λ)

Ω

g(x)u_nϕ dx+

Ω

g(x)u_ndx+

Ω

|∇u_n|^qϕ dx

Ω

f_n(x)ϕ dx

Ω

f (x)ϕ dxC,

and hence we conclude that

Ω

g(x)unϕ dx C

λ¯−λ uniformly inn,

Ω

g(x)undxC uniformly inn,

Ω

|∇un|^qϕ dxC uniformly inn.

TakingT_ku_nas a test function in (2.14), we have that

Ω

|∇T_ku_n|²dx+

Ω

|∇u_n|^qT_ku_ndxλk

Ω

g_n(x)u_ndx+k

Ω

f (x) dx.

Thus ¹_k

Ω|∇T_ku_n|²dxCandT_ku_n T_kuweakly inW₀^1,2(Ω). In particular fork=1 we obtain that

Ω

|∇u_n|^q

Ω

|∇T₁u_n|²dx+C(q, Ω)+

Ω

|∇u_n|^qT₁u_ndxC,

thereforeu_n uweakly inW₀^1,q(Ω). Ifq >_N^N₋₁ andg∈L^m(Ω)withm >^N₂ org(x)= |x|⁻²the existence result is a consequence of the previous theorem. On the contrary, if q_N^N₋₁, then using the regularity result for entropy solutions obtained in [16] (see also [15] for the case of Radon measures), we obtain an extra regularity, that is,u_n u inW₀^1,p(Ω)for allp <_N^N₋₁. We claim thatgn(x)un→g(x)ustrongly inL¹(Ω). To prove the claim we consider the sequence{w_n} ∈W₀^1,2(Ω)of solutions to the problem,

−w_n=λg_n(x)w_n+f_n(x) inΩ,

wn=0 on∂Ω. (2.15)

Since λ < λ₁(g,2), then using the assumption on f we obtain that w_nw everywhere and w_n w weakly in W₀^1,p(Ω), for allp < _N^N₋₁, withwthe unique entropy solution to problem−w=λg(x)w+f. Notice that solutions to (2.14) are subsolutions to (2.15), henceg_nu_ng_nw_ngw. Thanks to the dominated convergence theorem, g_n(x)u_n→g(x)uinL¹(Ω)and we conclude. To finish we have just to prove the strong convergence of|∇u_n|^q to

|∇u|^qinL¹(Ω). The proof is done in two steps. First we start by proving the strong convergence ofT_k(u_n)toT_k(u) inW₀^1,2(Ω), which follows by applying toT_k(u_n)andT_k(u)the techniques used in the proof ofconvergence claimin

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Theorem 2.4. Then to get the main convergence we use Vitali’s Lemma and a suitable test function as in the last step in the proof of Theorem 2.6. 2

Remarks 2.10.

(1) Notice that ifg(x)= |x|⁻², then the regularity required onf in Theorem 2.9, depends onλ. See [10].

(2) In Section 3 we will show that the condition on the integrability off in Theorem 2.9 is, in general, optimal.

2.3. Some remarks on the uniqueness

We consider the caseq=2 for which a change of variables allows us to find a comparison principle and uniqueness.

Lemma 2.11.Letu∈W₀^1,2(Ω)be such thatu=0and−u+ |∇u|²0, then there exist constantsC, R >0such thatuCinB_R(0)⊂Ω.

Proof. The change of variablesv=1−e⁻^utransforms−u+ |∇u|²0 into−v0 withv=0. It is sufficient to apply the classical maximum principle of the Laplacian operator and we conclude. 2

Lemma 2.12.Assume thatf ∈L¹(Ω)is a positive function andgsatisfies(2.1). Then for allλ >0the problem

⎧⎨

⎩

−u+ |∇u|²=λg(x)u+f (x) inΩ,

u >0 inΩ,

u=0 on∂Ω,

(2.16) has at most a positive solutionu∈W₀^1,2(Ω).

Proof. Ifu∈W₀^1,2(Ω)is a solution to (2.16) thenv=1−e⁻^usatisfies 0v1 inΩ and it is a solution to

−v=λg(x)(1−v)log(₁₋¹_v)+(1−v)f (x) inΩ,

v=0 on∂Ω. (2.17)

Define

H (x, v)= λg(x)(1−v)log(₁₋¹_v)+(1−v)f (x), if 0v <1,

0, ifv1.

By a direct computation we find that ^{H (v,x)}_v is a nonincreasing function invforv0, then by similar arguments as in [11] we conclude thatvis the unique solution to (2.17). Therefore,uis the unique solution to (2.16). 2

Remark 2.13.Using the same ideas of Lemma 2.12 we get a comparison principle for a sub- and super-solution to problem (2.16), namely ifu₁is a nonnegative subsolution andu₂is a nonnegative supersolution such thatu₁, u₂∈ W^1,2(Ω)andu₂u₁on∂Ω, thenu₂u₁inΩ.

We will prove the following comparison principle, that we will use below, without change of variables.

Theorem 2.14. Assume that1q < _N^N₋₁ and 0f ∈L¹(Ω). Letv, u be two nonnegative functions such that u, v∈W^1,q(Ω),u, v∈L¹(Ω)and

−u+ |∇u|^qf (x) inΩ,

−v+ |∇v|^qf (x) inΩ,

vu on∂Ω,

(2.18) thenvuinΩ.

Proof. Considerw=v−u, then it is clear thatw∈W^1,q(Ω),w0 on∂Ωandw∈L¹(Ω). In order to conclude, it is sufficient to prove thatw₊=0. It follows that

−w+ |∇v|^q− |∇u|^q0.