It is clear from the above example that the differential operator is defined on W01,p(Ω, ν), but that it may not be coercive on the same space asunear to 1

ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu

NONLINEAR DEGENERATE ELLIPTIC EQUATIONS IN WEIGHTED SOBOLEV SPACES

AHARROUCH BENALI, BENNOUNA JAOUAD

Abstract. We study the existence of solutions for the nonlinear degenerated elliptic problem

−diva(x, u,∇u) =f in Ω, u= 0 on∂Ω,

where Ω is a bounded open set inR^N,N≥2,ais a Carath´eodory function having degenerate coercivitya(x, u,∇u)∇u ≥ν(x)b(|u|)|∇u|^p, 1< p < N, ν(·) is the weight function,bis continuous andf∈L^r(Ω).

1. Introduction

In this article we prove the existence of solutions for some nonlinear elliptic equations with principal part having degenerate coercivity. The model case is

−divν(·)|∇u|^p−2∇u (1− |u|)^α

=f in Ω, u= 0 on∂Ω,

(1.1) with Ω a bounded open subset of R^N, N ≥ 2, p > 1, α ≥ 0, ν(·) is weight function defined on Ω andf a measurable function on whose summability we will make different assumptions. It is clear from the above example that the differential operator is defined on W₀^1,p(Ω, ν), but that it may not be coercive on the same space asunear to 1. Because of this lack of coercivity, standard existence theorems for solutions of nonlinear elliptic equations cannot be applied. We consider the nonlinear degenerate elliptic problem

A(u) =−div(a(x, u,∇u)) =f in Ω, u= 0 on∂Ω,

where, Ω is a bounded open subset ofR^N,N ≥2, 1< p < N, anda: Ω×R×R^N → R^N is a Carath´eodory function, such that the following assumption holds

a(x, s, ξ).ξ≥ν(x)b(|s|)|ξ|^p, for almost everyxin Ω, for every (s, ξ)∈R×R^N, with

b(|s|) = 1/(1− |s|)^α, (1.2)

2010Mathematics Subject Classification. 35J70, 46E30, 35J85.

Key words and phrases. Nonlinear degenerated elliptic operators; weighted Sobolev space;

monotony and rearrangement methods.

c

2020 Texas State University.

Submitted December 4, 2019. Published October 12, 2020.

1

under various assumptions on f. As stated before, due to assumption (1.2), the operator A may not be coercive on W₀^1,p(Ω, ν), when the solutions approach the critical values±1. To overcome this difficulties, we will reason by approximation, cutting by means of truncatures the nonlinearitya(x, s, ξ) in order to get coercive differential operator on W₀^1,p(Ω, ν), and give a sense to the equation when the solutions near to±1 and to manage the set{x∈Ω : |u(x)|= 1}. For the caseν(·) being a constant, the existence of solutions to problem (1.1) is proved in [11], when f a measurable function on whose summability have make different assumptions, the analogous problems was treated by many other authors. See, for example, [3, 4, 9, 10, 8] where problems such as

−div 1

(1± |u|)^α|∇u|^p−2∇u

=f, are considered.

This article is organized as follows: In section 2, we recall some preliminaries on Weighted Sobolev spaces and properties of rearrangement. In section 3, we first prove the propositions that we will use to prove some a priori estimates of the solutions, then we prove the existence of weak and entropy solution with respect to the summability off.

2. Preliminaries

Assumptions. Letb : [0, l[→ (0,∞), with l > 0, be a continuous function such that

lim

s→l⁻b(s) = +∞. (2.1)

We define

A(s) = Z s

0

b(t)^p−11 dt, fors∈[0, l), A(l⁻) = lim

s→l⁻

Z s

0

b(t)^p−11 dt= +∞.

We study Dirichlet problems of the form

−diva(x, u,∇u) =f in Ω,

u= 0 on∂Ω, (2.2)

where Ω is a bounded open set inR^N,N ≥2, 1< p < N, anda: Ω×(−l, l)×R^N → R^N, is a Carath´eodory function andν : Ω→R+satisfies the following assumptions:

a(x, s, ξ)·ξ≥b(|s|)ν(x)|ξ|^p, ν∈L^r(Ω), r≥1, ν⁻¹∈L^t(Ω), t≥N, 1 + 1

t < p < N(1 +1

t). (2.3) for a.e.x∈Ω, for alls∈(−l, l) and allξ∈R^N;

|a(x, s, ξ)| ≤ν(x)[h(x) +b(|s|)|ξ|^p−1], (2.4) for a.e.x∈Ω, for alls∈(−l, l), for allξ∈R^N, andh∈L^p0(Ω, ν);

(a(x, s, ξ)−a(x, s, ξ⁰))·(ξ−ξ⁰)>0, (2.5) for a.e.x∈Ω, for alls∈(−l, l) and allξ∈R^N,ξ6=ξ⁰. Moreover,f is a measurable function on whose summability we will make several assumptions.

For stating existence results in the next section, we need some classes of solutions.

Definition 2.1. We say thatu∈W₀^1,p(Ω, ν) is a weak solution to problem (2.2) if Z

Ω

a(x, u,∇u)· ∇ϕ dx= Z

Ω

f ϕ dx, ∀ϕ∈W₀^1,p(Ω, ν). (2.6) Definition 2.2. A measurable function u∈ W₀^1,p(Ω, ν) is an entropy solution to problem (2.2) if

|u| ≤l a.e. in Ω (2.7)

and for all 0< k < l, Z

Ω

a(x, u,∇u)· ∇Tk(u−ϕ)dx≤ Z

Ω

f Tk(u−ϕ)dx, (2.8) for anyϕ∈W₀^1,p(Ω, ν)∩L^∞(Ω) such thatkϕk_L∞(Ω)< l−k.

Weighted Sobolev spaces. Let 1≤p < N, andν : Ω→Rbe a weight function, i.e. a function which is measurable and positive almost everywhere in Ω. The weighted Lebesgue spacesL^p(Ω, ν) is defined as

L^p(Ω, ν) =

u: measurable, real-valued function, Z

Ω

ν(x)|u(x)|^pdx <∞ . which is a Banach space (uniformly convex and hence reflexive ifp >1) equipped with the norm

kuk_Lp(Ω,ν)=Z

Ω

ν(x)|u(x)|^pdx^1/p .

By W^1,p(Ω, ν) we denote the completion of the spaceC¹(Ω) with respect to the norm

kukW^1,p(Ω,ν)=kukL^p(Ω,ν)+k|∇u|kL^p(Ω,ν).

Moreover we denote by W₀^1,p(Ω, ν) the closure of C¹(Ω) in W^1,p(Ω, ν) which is normed by

kuk_W^1,p

0 (Ω,ν)=k|∇u|kL^p(Ω,ν).

We denote by W^−1,p0(Ω,1/ν) the dual space ofW₀^1,p(Ω, ν); for more details see [16].

Rearrangement properties. We recall some definitions about decreasing re- arrangement of functions. Let Ω be a bounded open set of R^N and u : Ω → R a measurable function.

Definition 2.3. The distribution function ofuis defined as µu(t) =|{x∈Ω :|u(x)|> t}|, t≥0.

The functionµu is decreasing and right continuous.

Definition 2.4. The decreasing rearrangement of uis defined as u_∗(s) := sup{t≥0 :µ_u(t)> s}, s≥0.

The functionu_∗ is the generalized inverse ofµ_u. We recall that Z

Ω

|u|^pdx=p Z +∞

0

t^p−1µ_u(t)dt, forp≥1. (2.9) Then theL^p-norm, for 1 ≤p <+∞, is invariant with respect to rearrangement, that is,

kukL^p(Ω)=ku∗kL^p[0,|Ω|].

Moreover, ifu∈L^∞(Ω), by definitionu∗(0) = ess sup_Ω|u|. For more details about rearrangements we refer the reader to [6, 13, 18]. We recall that a measurable functionu: Ω→Rbelongs to the Marcinkiewicz spaceM^p(Ω) (or weak-L^p) if the distribution functionµ_u satisfies

µu(t)≤ c

t^r, ∀t >0,

for some constantc. We observe that the above condition is equivalent to u_∗(s)≤ c

s^1/r, ∀s >0, and we define

kuk_Mp(Ω)= sup

s>0

u_∗(s)s^1/r.

We observe that the Marcinkiewicz spaces are “intermediate” between Lebesgue spaces. Indeed, it is not difficult to show that

L^p(Ω)⊂M^p(Ω)⊂L^q(Ω),

for 1≤q < p. Now, we give a sense to the gradient of a functionu∈L¹(Ω) such that the truncates ofuare Sobolev functions.

Lemma 2.5([7]). For each measurable functionu: Ω→Rsuch that for everyk >

0 the truncated functionTk(u)belong to W_loc^1,1(Ω), there exists a unique measurable function v: Ω→R^N such that

∇Tk(u) =vχ_|u|<k a.e. inΩ. (2.10) Furthermore,u∈W₀^1,1(Ω)if and only ifv∈L¹_loc(Ω), and thenv=∇uin the usual weak sense.

Now we recall some Sobolev-type inequalities which will be used later.

Lemma 2.6 ([16]). Let ν be a nonnegative function on Ω such that ν ∈ L^r(Ω), r≥1,ν⁻¹∈L^t(Ω),t≥N. And letp, p^] be two real number that satisfyt≥N/p, 1 + ¹_t < p < N(1 + ¹_t),1/p^]= 1/p(1 +¹_t)−_N¹. Then

kuk_p] ≤c0k∇uk_Lp(ν), ∀u∈W₀^1,p(Ω, ν).

Lemma 2.7. Suppose thatλ >0 and1≤γ <+∞. Letψa non-negative measur- able function on (0,+∞). Then the

Z +∞

0

t^−λ

Z t

0

ψ(s)dsγdt t ≤c

Z +∞

0

(t^1−λψ(t))^γdt

t , , (2.11) Z +∞

0

t^λ

Z +∞

t

ψ(s)dsγdt t ≤c

Z +∞

0

(t^1+λψ(t))^γdt

t . (2.12)

Also we shall need the following proposition of weak approximation (see [5]). Let u∈W₀^1,p(Ω), and fors∈[0,|Ω|], letG(s) be a measurable subset of Ω such that

|G(s)|=s s1< s2⇒G(s1)⊂G(s2) G(s) ={x∈Ω :|u(x)|> t} ifs=µ(t).

For a given a functionϕ∈L¹(Ω), we set φ(s) = d

ds Z

G(s)

ϕ(x)dx.

Lemma 2.8 ([5]). If ϕ∈L^p(Ω) withp >1, then there exists a sequence (ϕ(s))n, such that ϕ^∗_n(s) =ϕ^∗(s) andϕn * φweakly inL^p(0,|Ω|).

3. Main result

The following Proposition gives a sufficient condition for the gradient of a func- tion to belong to some Marcinkiewicz space, These are the generalized results of [7]

in the Weighted Sobolev spacesW₀^1,p(Ω, ν).

Proposition 3.1. Let 1< p < N, andu∈ T₀^1,p(Ω, ν)be such that Z

{|u|<k}

|∇u|^pν(x)dx≤M k^λ

for every k >0. Then u∈ M^p1(Ω) wherep₁=p^](1−λ/p). More precisely, there exists ac such thatmeas{|u|> k}= meas{x∈Ω :|u(x)|> k} ≤ck^−p1.

Proof. Fork >0, from (2.3), we have

kTk(u)k_p] ≤c1k∇Tk(u)kL^p(ν)≤c1k^λ/p.

For 0< ε≤k, we have{x∈Ω :|u|> ε}={x∈Ω :|Tk(u)|> ε}. Hence meas{|u|> ε} ≤(kTk(u)k_p]

ε )^p] ≤c1k^λp]/pε^−p].

Settingε=k, we obtain meas{|u|> ε} ≤c₁k^−p1, wherep₁=p^](1−λ/p).

Proposition 3.2. Let 1< p < N, andu∈ T₀^1,p(Ω, ν)be such that Z

{|u|<k}

|∇u|^pν(x)dx≤M k^λ

for everyk >0. Thenν^1/p∇u∈ M^p2(Ω) wherep2=pp1/(λ+p1). More precisely, there exists ac such that meas{ν^1/p|∇u|> h} ≤ch^−p2.

Proof. Fork, h >0. Setφ(k, α) = meas{ν(x)|∇u|^p > α, |u|> k}. From Proposi- tion 3.1 we have

φ(k,0)≤c1k^−p1.

Using that the functionα7→φ(k, α) is non-increasing, fork, λ >0 we obtain φ(0, α)≤1

α Z α

0

φ(0, s)ds

=1 α

Z α

0

φ(0, s) +φ(k,0)−φ(k,0)ds

≤φ(k,0) + 1 α

Z α

0

φ(0, s)−φ(k,0)ds

≤φ(k,0) + 1 α

Z α

0

φ(0, s)−φ(k, s)ds.

(3.1)

Sinceφ(0, s)−φ(k, s) = meas{ν(x)|∇u|^p> s, |u|< k}we have 1

α Z α

0

φ(0, s)−φ(k, s)ds= 1 α

Z

|u|<k

ν(x)|∇u|^pdx≤ck^λ α, which by (3.1) gives

φ(0, α)≤c₁k^−p1+c₂k^λ

α . (3.2)

By minimizing (3.2) inkand settingα=h^pwe obtain meas{ν^1/p|∇u|> k} ≤ch^{−pp1/(λ+p1)}

3.1. A priori estimate. Letεbe positive and sufficiently small. We consider the problem

−diva_ε(x, u_ε,∇uε) =f_ε in Ω,

uε= 0 on∂Ω, (3.3)

where a_ε(x, s, ξ) = a(x, T_l−ε(s), ξ), with x ∈ Ω, s ∈ R and ξ ∈ R^N and f_ε ∈ L^∞(Ω). We use some classical results (see, for example [1, 2]) to assure that problem (3.3) has at least one solutionu_ε∈W₀^1,p(Ω, ν)∩L^∞(Ω). Then, we define b_ε(t) = b(T_l−ε(t)) for allt∈[0,+∞), and

Aε(s) = Z s

0

bε(r)^1/(p−1)dr.

First, we prove an integral inequality for weak solutions of problem (3.3).

Proposition 3.3. Let uε be a weak solution of (3.3). Then Aε(u^∗_ε(s))≤CN

Z |Ω|

s

r^−p0/N0[D(r)]^p0/pZ r 0

f_ε^∗(σ)dσp⁰/p

dr, s∈[0,|Ω|], (3.4) whereD: [0,|Ω|]→Ris a measurable function such that

Z

|uε|>y

ν^−t(x)dx= Z µ(y)

0

(D(r))^tdr.

Proof. Letφ=T_h(u_ε−T_θ(u_ε)) be a test function in (3.3). Then we have 1

h Z

θ<|uε|≤θ+h

b(|uε|)ν(x)|∇uε|^pdx≤ Z

|uε|>θ

|f|dx

Applying Hardy-Littlewood inequality and passing to the limit onhto 0, we obtain b(θ)

− d dθ

Z

|u_ε|>θ

ν(x)|∇u_ε|^pdx

≤

Z µ_uε(θ)

0

f_ε^∗(s)ds. (3.5) On the other hand by H¨older inequality, we obtain

−d dθ

Z

|uε|>θ

|∇uε|dx≤

− d dθ

Z

|uε|>θ

ν(x)|∇uε|^pdx^1/p

×

− d dθ

Z

|uε|>θ

ν^−p0/p(x)dx1/p⁰

≤

− d dθ

Z

|uε|>θ

ν(x)|∇uε|^pdx^1/p

×

− d dθ

Z

|uε|>θ

ν^−t(x)dx1/r1p⁰

(−µ⁰_uε(θ))^1/r2p0. (3.6)

where 1/r₁+ 1/r₂ = 1 andp⁰r₁/p =t. By Lemma 2.8, since ν⁻¹ ∈ L^t(Ω), t > 1 there existsD∈L^t([0,|Ω|]) such that

−d dθ

Z

|uε|>θ

ν^−t(x)dx=−µ⁰_u

ε(θ)[D(µ_uε(θ))]^t.

Then inequality (3.6), becomes

−d dθ

Z

|uε|>θ

|∇uε|dx≤

− d dθ

Z

|uε|>θ

ν(x)|∇uε|^pdx^1/p

× (−µ⁰_u

ε(θ))^1/p0[D(µ_uε(θ))]^t/r1p0 .

(3.7)

From isoperimetric inequality and Fleming-Rishel formula (see [15]), it follows that C_Nb(θ)^1/p(µ_uε(θ))^1/N0 ≤

− d dθ

Z

|u_ε|>θ

ν(x)|∇u_ε|^pdx1/p

×

(−µ⁰_uε(θ))^1/p0[D(µu_ε(θ))]^t/r1p0b(θ)^1/p ,

(3.8)

which by (3.5) gives

b(θ)^1/(p−1)≤CN(µu_ε(θ))^−p0/N0(−µ⁰_uε(θ))[D(µu_ε(θ))]^t/r1Z µ_uε(θ) 0

f_ε^∗(s)dsp⁰/p

integrating between 0 andu_∗(s) we obtain A(u_∗(s))≤CN

Z u∗(s)

0

h

(µu_ε(θ))^−p0/N0(−µ⁰_uε(θ))[D(µu_ε(θ))]^t/r1

×Z µ_uε(θ) 0

f_ε^∗(s)dsp⁰/pi dθ,

(3.9)

which gives the results.

Remark 3.4. Since 1 + ¹_t < p < N(1 + ¹_t), and t≥N/p, we have qp⁰/p≥1 and q/r₁⁰ ≥1, where r₁ =t(p−1), which allows us to apply the Proposition 2.11 and Proposition 2.12 to prove estimation (3.10) and (3.11), below.

Proposition 3.5. Let u_ε be a solution of (3.3).

(a) If 1< r < tN/(tp−N), then

k(Aε(|uε|))^qk_L1(Ω)≤ckfk^qp_Lr⁰(Ω)^/p; (3.10) whereq=rtN(p−1)/(t(N−rp) +rN).

(b) If r= 1, then

kAε(|uε|)k_MN t(p−1)/(N+t(N−p)) ≤ckfk^p_L⁰1^/p(Ω)kDk^p_L⁰t^/p[0,|Ω|]. (3.11) Proof. Case 1 < r < tN/(tp−N). Let us observe that Aε being monotone, by Proposition 3.3, properties of rearrangements, (2.12) and (2.11), we obtain

k(Aε(|uε|))^qk_L1(Ω)≤CN

Z +∞

0

hZ |Ω|

s

r^−p0/N0[D(r)]^p0/pZ r 0

f_∗(σ)dσp⁰/p

driq

ds

≤CN

Z +∞

0

hZ |Ω|

s

r⁻

p0r0 1 N0

Z r

0

f_∗(σ)dσ

p0r0 1 p dri_r^q0

1ds

≤CN

Z +∞

0

h s

r0 1 q

Z |Ω|

s

r⁻

p0r0 1 N0

Z r

0

f∗(σ)dσ

p0r0 1 p dri_r^q0

1ds s

≤CN

Z +∞

0

h s⁽

r0 1 +q

q −^p_N^0r₀¹⁰) ^p

p0r0 1

Z s

0

f_∗(σ)dσi^qp

0 p ds

s

≤C_N Z +∞

0

h s⁽

r0 1 +q

q −^p

0r0 0N1) ^p

p0r0 1

+1f_∗(s)i^qp

0 p ds

s

≤C_N Z +∞

0

h s⁽

r0 1 +q

q −^p

0r0 1 N0 ) ^p

p0r0 1

+1−_qp^p0

f_∗(s)i^qp

0 p ds,

where ^qp_p⁰ ≥ 1, ^p0_p^r1 = t, and CN a constant that vary from line to line. Since fε∈M^r(Ω) we conclude that

k(Aε(|uε|))^qk_L1(Ω)≤CN

Z +∞

0

(f_∗(s))^−rq(

1 r0 1

−_N^p0₀+^p_p⁰)+^qp_p⁰

ds

≤CNkf∗k^r_Lr([0,|Ω|]).

(3.12) where

r=−rq(1 r⁰₁ − p⁰

N⁰ +p⁰ p) +qp⁰

p , q= rtN(p−1) t(N−rp) +rN. Caser= 1. By Proposition 3.3, and H¨older inequality, we have

Aε(u_∗(s))≤CN

Z |Ω|

s

r^−p0/N0[D(r)]^p0/pZ r 0

f_∗(σ)dσp⁰/p

dr

≤C_NkDk_Lt[0,|Ω|]

Z |Ω|

s

r⁻

p0t(p−1) N0(tp−t−1)

^tp−t−1_t(p−1)

≤C_NkDk_Lt[0,|Ω|]s¹⁻

p0t(p−1) N0(tp−t−1)

which implies the result.

Remark 3.6. Sincep/N <1 + ¹_t, (see (2.3)), we have N tp

N t(p−1)−N+tp >1.

Proposition 3.7. Let uε be a solution of (3.3).

(a) If N t(p−1)−N+tp^{N tp} < r < _tp−N^tN , then

k∇Aε(|uε|)kL^p(Ω,ν)≤c₁. (3.13) (b) If

max 1, tN p

N t(p−1)p+pt−N

< r < tN p

N t(p−1) +pt−N, then

k∇Aε(|uε|)k_Lβ(Ω,ν^β/p)≤c2, (3.14) whereβ =_rN^{rN t(p−1)p}_{+N tp−ptr}.

(c) If

1≤r≤max 1, tN p

N t(p−1)p+pt−N , then

kν^1/p∇Aε(|uε|)k_Mβ(Ω)≤c3, (3.15) whereβ =_rN^{rN t(p−1)p}_{+N tp−ptr}.

Proof. Let uε is a solution of (3.3), by the definition of Aε we can use as test functionv= [Th(Aε(|uε|)−Tθ(Aε(|uε|)] sign(uε) and obtain

Z

θ<A_ε(|uε|)≤θ+h

ν(x)|∇Aε(|uε|)|^pdx≤ Z

A_ε(|uε|)>θ

|fε|dx, (3.16) Case 1: N t(p−1)−N+tp^{N tp} < r < _tp−N^tN . Passing to the limit in (3.16), we obtain

d dθ

Z

A_ε(|uε|)≤θ

ν(x)|∇Aε(|uε|)|^pdx≤ Z µ_ε(θ)

0

f_ε^∗(s)ds, (3.17) where we have denoted withµε(θ) the distribution functions ofAε(|uε|). Integrating (3.17) between 0 and +∞and using a H¨older inequality, we have

Z

Ω

ν(x)|∇Aε(|uε|)|^pdx≤ Z +∞

0

dθ Z µ_ε(θ)

0

f_ε^∗(s)ds

= Z |Ω|

0

Aε(u^∗_ε(s))f_ε^∗(s)ds

≤ kfk_Lr(Ω).kAε(|uε|)k_Lr0(Ω).

(3.18)

We observe that if r is such that _{N t(p−1)−N}^{N t} _+pt ≤r < _tp−N^tN , by (3.10) the right- hand side of the above inequality is controlled by a constant depending on the norm off_ε inL^r(Ω); so by (3.18) inequality (3.13) follows.

Case 2: max 1,N t(p−1)p+pt−N^{tN p}

< r < N t(p−1)+pt−N^{tN p} . Applying the H¨older inequality in (3.16) and reasoning as before, we obtain

Z

Ω

|∇Aε(|uε|)|^βν^β/p(x)dx

≤ Z +∞

0

Z µ_ε(θ)

0

f_ε^∗(s)ds^β/p

(−µ⁰_ε(θ))^1−βpdθ

≤Z +∞

0

(1 +θ)^q(−µ⁰_ε(θ))dθ1−^β_p

×Z +∞

0

(1 +θ)^q(1−pβ)Z µ_ε(θ) 0

f_ε^∗(s)ds dθβ/p

.

(3.19)

By the properties of rearrangements, we can write the first integral on the right- hand side of (3.19) as

Z +∞

0

(1 +θ)^q(−µ⁰_ε(θ))dθ= Z |Ω|

0

(1 +Aε(u^∗_ε))^qds, (3.20) and by (3.10) this quantity is bounded by a constant depending on the norm of fε in L^r(Ω). On the other hand, integrating by parts the second integral on the right-hand side of (3.19) we have

Z +∞

0

(1 +θ)^q(1−βp)Z µ_ε(θ) 0

f_ε^∗(s)ds dθ

≤c Z |Ω|

0

f_ε^∗(s)[(1 +Aε(u^∗_ε))^{(q(1−βp)+1)}]ds

≤ckfεk_Lr(Ω)

hZ |Ω|

0

[(1 +Aε(u^∗_ε))^q]dsi1−¹_r

.

(3.21)

Applying again (3.10), by (3.19) it follows the estimate (3.14).

Case 3: 1≤r≤max 1,N t(p−1)p+pt−N^{tN p}

. Integrating inequality (3.17) between 0 andk, we obtain

Z

Aε(|uε|)≤k

ν(x)|∇Aε(|uε|)|^pdx≤ Z k

0

dθ Z µ_ε(θ)

0

f_ε^∗(s)ds. (3.22) Ifr= 1, from (3.22) we obtain

Z

A_ε(|uε|)≤k

ν(x)|∇Aε(|uε|)|^pdx≤kkfεkL¹(Ω). by (3.11) and (2.3) we obtain the assertion.

If 1 ≤ r ≤ max(1,N t(p−1)p+pt−N^{tN p} ), then by (3.10) it follows that Aε(|uε|) ∈ M^q(Ω), withq=_tN+rN−ptr^{rN t(p−1)} ; so we obtain

Z

Aε(|uε|)≤k

ν(x)|∇Aε(|uε|)|^pdx≤ck^1−rq0

by Proposition 3.2, we conclude the result.

Replacing∇A(|u|) by∇uthe above estimates also hold; furthermore it follows that

Z

Ω

ν(x)|∇u|^γdx≤c,

withγ < _tN+N^{N t(p−1)}_−t, wherecis a constant depending on theL¹(Ω) norm off_ε. Using (3.5), theT_k(u_ε) are uniformly bounded inW₀^1,p(Ω, ν) for anyk >0. Hence, there exists a functionu∈W₀^1,γ(Ω, ν) such that

u_ε→u a.e. in Ω, (3.23)

and, for anyk >0,

Tk(uε)* Tk(u) weakly inW₀^1,p(Ω, ν). (3.24) Remark 3.8. Choosingk > l, we have

uε* u weakly inW₀^1,p(Ω, ν). (3.25) Indeed, let us suppose f ∈ L¹(Ω). Using T_2l(|uε|)−T_l(|uε|) as test function in (3.3), by (2.3) we obtain

b(l−ε) Z

Ω

(T_2l(|uε|)−T_l(|uε|))^p]dx≤lkfεkL¹(Ω).

Lettingε→0, from condition (2.1), we conclude that, for almost allxin Ω,|u| ≤l, which give the result by (3.24).

Next we prove a lemma needed for proving the existence result.

Lemma 3.9. Let uε be a weak solution to problem (3.3). Supposef ∈L¹(Ω), and letfε∈L^∞(Ω) be such thatfε→f inL¹(Ω). Then

∇uε→ ∇u a.e. in{|u|< l}.

Proof. We adapt the proof[presented in [11]. By Remark 3.8, we have uε →u in measure. We will prove thatuε→uin measure on{|u|< m}. Letλ >0 andη >0 for 0< k < l, andM >0, we set

E₁={|u|< l} ∩({|∇uε|> M} ∪ {|∇u|> M} ∪ {|uε|> k} ∪ {|u|> k}), E2={|u|< l} ∩ {|uε−u|> η},

E3={|uε−u| ≤η,|∇uε| ≤M,|∇u| ≤M,|uε| ≤k,|u| ≤k,|∇(uε−u)| ≥λ}

∩ {|u|< l}.

Observe that{|u|< l} ∩ {|∇u_ε| ≥λ} ⊂E₁∪E₂∪E₃.

Sinceu_εand∇u_ε are bounded inL¹(Ω), for anyσ >0 we can fix M andk < l such that|E1|< σ/3 independently of ε. By the monotonicity Assumption (2.5), there exists a real valued functionγ such that

meas({x∈Ω :γ(x) = 0}) = 0, (a(x, s, ξ)−a(x, s, ξ⁰))(ξ−ξ⁰)≥γ(x),

for anys∈(−l, l), ξ, ξ⁰ ∈R^N,|s| ≤k,|ξ|,|ξ⁰| ≤M, and|ξ−ξ⁰| ≥λ. Denoting by χ_η the characteristic function of [0, η], we obtain

Z

E₃

γ(x)dx≤ Z

E₃

[a_ε(x, u_ε,∇uε)−a_ε(x, u_ε,∇u)](∇uε−u)dx

≤ Z

{|uε|≤k,|u|≤k}

h

aε(x, uε,∇uε)−aε(x, uε,∇Tk(u))

×

∇u_ε−T_k(u))χ_η(|u_ε−T_k(u)|i dx

≤ Z

Ω

h

a_ε(x, u_ε,∇u_ε)−a_ε(x, u_ε,∇T_k(u))

×

∇uε−Tk(u))χη(|uε−Tk(u)|i dx

≤ Z

Ω

aε(x, uε,∇uε)(∇uε−Tk(u))χη(|uε−Tk(u)|)dx

− Z

Ω

aε(x, uε,∇Tk(u))·(∇uε−Tk(u))χη(|uε−Tk(u)|)dx :=J1−J2.

For the termJ1, using Tη(uε−Tk(u)), we have

|J1|= Z

Ω

fεTη(|uε−Tk(u)|)dx

≤ηkfk_L1(Ω).

Choosingη >0 such thatk+η < l, there existsε₀>0 such that for allε < ε₀, a_ε(x, u_ε,∇T_k(u)) =a(x, u_ε,∇T_k(u)) in{x∈Ω :|u_ε−T_k(u)| ≤η};

and since{x∈Ω :|u_ε−T_k(u)| ≤η} ⊂ {x∈Ω :|u_ε| ≤k+η} we obtain J2=

Z

Ω

a(x, uε,∇Tk(u))· ∇Tη(uε−Tk(u))dx

= Z

Ω

a(x, Tk+η(uε),∇Tk(u))·(∇Tk+η(uε−Tk(u)))χη(|uε−Tk(u)|)dx.

By (3.24), it follows that

Tk+η(uε)* Tk+η(u) weakly inW₀^1,p(Ω, ν), on the other hand

|a(x, T_k+η(u_ε),∇T_k(u))| ≤b(|T_k+η(u_ε|))ν(x)|∇T_k+η(u)|^p−1 using Vitali’s theorem we have

a(x, Tk+η(uε),∇Tk(u))→a(x, Tk+η(u),∇Tk(u)) strongly inL^p0(Ω, ν^−1/(p−1)).

Lettingεand ηtend to 0 respectively inJ2, we obtain

ε→0lim Z

Ω

a(x, uε,∇Tk(u))· ∇Tη(uε−Tk(u))dx

= Z

Ω

a(x, Tk+η(u),∇Tk(u))·(∇Tk+η(u−Tk(u)))χη(|uε−Tk(u)|)dx, and

η→0lim Z

Ω

a(x, Tk+η(u),∇Tk(u))·(∇Tk+η(u−Tk(u)))χη(|uε−Tk(u)|)dx= 0.

For η small enough ηkfk_L1(Ω) < δ/2, by Kolmogorov theorem, we have |E₃| < σ independently ofε. Fix η, by the fact thatuε→uin measure, we chooseε1 such that |E2| < η for ε ≤ε1. This implies that ∇uε → ∇uin measure in {|u| < l}, consequently

∇uε→ ∇u a.e. in{|u|< l}.

We observe that sinceuε→ua.e. in Ω (see (3.23)), we have

{x∈Ω :|u(x)|=l}=n

x∈Ω : lim

ε→0

Z |uε(x)|

0

b_ε(t)≥ Z l

0

b(t)dto

. (3.26) Theorem 3.10. Let f be a function in L^r(Ω), withr > tN/(tp−N). Assume that (2.1)–(2.5)hold. Then there exists a weak solutionu∈W₀^1,p(Ω, ν)of problem(2.2) such that kuk_L^∞_(Ω)< l.

Proof. For fε = f with ε > 0. By classical results see for example [2, 1]) there exists a solution u_ε ∈ W₀^1,p(Ω, ν) of the approximated problem (2.2). Estimate (3.4) implies

Aε(kuεkL^∞)≤C(f) =CN

Z |Ω|

0

r^−p0/N0[D(r)]^p0/pZ r 0

f_ε^∗(σ)dσp⁰/p

dr. (3.27) SinceAis bijective in [0, l), we can takeB=A⁻¹(C(f)) and then we chooseε0>0 such thatb(s)≤b(l−ε) for anys∈[0, B]. By definition ofb_εandA_εwe have, for anyε < ε₀,

A_ε(s) =A(s), s∈[0, B].

Moreover, beingA_ε increasing, it follows that, for anyε < ε₀, A_ε(s)≤C(f)⇔s∈[0, B], so by (3.27) we obtain

kuεkL^∞ ≤B < l.

By (2.3) and Lemma 3.9, we have

aε(x, uε₁(x),∇uε₁(x))→a(x, u,∇u) strongly inL^p0(Ω, ν^−1/(p−1)), f_ε→f strongly inL^∞(Ω).

Passing to the limit in the weak formulation of problem (3.3), we conclude thatu is a weak solution of (2.2), which satisfieskuk_L∞(Ω)< l.

Theorem 3.11. Letf ∈L^r(Ω), with _{N t(p−1)−N}^{N tp} _+tp< r < _tp−N^tN . Under hypothesis (2.1)-(2.5), there exists a weak solutionu∈W₀^1,p(Ω, ν)of problem (2.2), such that meas({x∈Ω :|u(x)|=l}) = 0.

Proof. Letuε∈W₀^1,p(Ω, ν) be a weak solution to the approximated problem (3.3).

By Remark (3.8), we haveuε→ua.e. in Ω, sinceA(l⁻) = +∞, (3.26) implies that Aε(|uε|)→A(|u|) a.e. in Ω. (3.28) By (3.13) and (3.28), we obtain

Aε(|uε|)→A(|u|) weakly inW₀^1,p(Ω, ν), (3.29) SinceA(|u|) is bounded inL¹(Ω) and meas({x∈Ω : |u(x)|=l}) = 0, by (2.3) we have

aε(x, uε,∇uε)→a(x, u,∇u) a.e. Ω.

On the other hand by (2.3) and (3.13)

|aε(x, uε,∇uε)| is bounded in L^p0(Ω, ν^−1/(p−1));

passing to the limit in the weak formulation (3.3), we obtain Z

Ω

a(x, u,∇u)· ∇ϕ dx= Z

Ω

f ϕ dx, for allϕ∈W₀^1,p(Ω, ν).

Theorem 3.12. Let f ∈ L^r(Ω), with 1 ≤ r < N t(p−1)−N+tp^{N tp} . Under hypothesis (2.1)−(2.5), there exists a solution u∈W₀^1,p(Ω, ν)of problem (2.2), in the sense of Definition (2.2)such that meas({x∈Ω :|u(x)|=l}) = 0.

Proof. Let u_ε be a weak solution of the approximate problem (3.3), by passing to the limit we can show that |u| < l a.e. in Ω. Take T_k(u_ε −ϕ), with ϕ ∈ W₀^1,p(Ω, ν)∩L^∞(Ω) as test function in (3.3) we obtain

Z

|uε−ϕ|≤k

a(x, T_l−ε(u_ε),∇uε)· ∇uεdx

− Z

|uε−ϕ|≤k

a(x, T_l−ε(uε),∇uε)· ∇ϕ dx

= Z

Ω

f_εT_k(u_ε−ϕ)dx.

(3.30)

Since {|uε−ϕ|} ⊆ {|uε| ≤k+kϕk_L∞(Ω) = M}, for 1 < k < l and kϕk_L∞(Ω) <

l−k, we obtainM < l and consequently|a(x, TM(uε),∇TM(uε))| is bounded in L^p0(Ω, ν^−1/(p−1)), and

ε→0lim Z

|uε−ϕ|≤k

a(x, T_l−ε(u_ε),∇u_ε)· ∇ϕ dx= Z

|u−ϕ|≤k

a(x, u,∇u)· ∇ϕ dx. (3.31)

Moreover since fε strongly convergent to f in L¹(Ω), and Tk(uε −ϕ) weakly*

convergent toTk(u−ϕ) inL^∞(Ω), we have

ε→0lim Z

Ω

fεTk(uε−ϕ)dx= Z

Ω

f Tk(u−ϕ)dx. (3.32) On the other handa(x, T_l−ε(u_ε),∇u_ε)· ∇u_εbeing non-negative, and almost every- where convergent toa(x, u,∇u)· ∇u, by Fatou’s lemma we conclude that

lim inf

ε→0

Z

|uε−ϕ|≤k

a(x, Tl−ε(uε),∇uε)·∇uεdx≤ Z

|u−ϕ|≤k

a(x, u,∇u)·∇u dx. (3.33) Combining (3.31), (3.32) and (3.33) we obtain

Z

Ω

a(x, u,∇u)· ∇Tk(u−ϕ)dx≤ Z

Ω

f Tk(u−ϕ)dx, for allϕ∈W₀^1,p(Ω, ν).

References

[1] L. Aharouch, Y. Akdim, E. Azroul;Quasilinear degenerate elliptic unilateral problems, Ab- stract and Applied Analysis, 1 (2005) 11–31.

[2] Y. Akdim, E. Azroul, A. Benkirane;Existence of solutions for quasilinear degenerate elliptic equations, Electronic J. Diff. Equ., 2001 (2001) No. 71, 1–19

[3] A. Alvino, L. Boccardo, V. Ferone, L. Orsina, G. Trombetti;Existence results for nonlinear elliptic equations with degenerate coercivity, Ann. Mat. Pura Appl., (4) 182 (1) (2003) 53–79.

[4] A. Alvino, V. Ferone, G. Trombetti;A priori estimates for a class of non-uniformly elliptic equations, Atti Semin. Mat. Fis. Univ. Modena, 46 (Suppl.) (1998) 381–391.

[5] A. Alvino and Trombetti;Sulle migliori costanti di maggiorazione per una classe di equazioni ellittiche degeneri.Ricerche Mat., 1978, 27, 413–428.

[6] C. Bandle;Isoperimetric inequalities and applications, in: Monographs and Studies in Math., Pitman, London; 1980.

[7] Ph. Benilan, L. Boccardo, T. Gallouet, R. Gariepy, M. Pierre, J. L. Vazquez;AnL¹-theory of existence and uniqueness of solutions of nonlinear elliptic equations, Ann. Scuola Norm.

Sup. Pisa Cl. Sci.; 4 (1995), 241-273.

[8] A. Benkirane, J. Bennouna;Existence of solutions for nonlinear elliptic degenerate equations, Nonlinear Analysis, 54 (2003) 9–37.

[9] D. Blanchard, O. Guib´e;Infinite valued solutions of non-uniformly elliptic problems, Anal.

Appl., 2(3) (2004) 227–246.

[10] L. Boccardo, A. Dall’Aglio, L. Orsina;Existence and regularity results for some degenerate elliptic equations, Atti. Semin. Mat. Fis. Univ. Modena, 46(suppl.) (1998) 51–81.

[11] F. Della Pietra, G. di Blasio;Existence results for nonlinear elliptic problems with unbounded coefficients, Nonlinear Analysis, 71 (2009) 72–87.

[12] R. Di Nardo, F. Feo;Existence and uniqueness for nonlinear anisotropic elliptic equations, Arch. Math., 102 (2014), 141–153.

[13] G. H. Hardy, J. L. Littlewood, G. Polya;Inequalities, Cambridge Univ. Press; 1964.

[14] J.-L. Lions;Quelques m´ethodes de r´eesolution des probl`emes aux limites non lin´eaires, Dunod et Gauthier-Villars, Paris, 1984.

[15] J. Mossino;In´egalities isop´erim`etriques et applications en physique, Collection travaux en cours, Herman, Paris; 1984.

[16] M. K. W Murthy, G. Stampacchia;Boundary value problems for some degenerate elliptic operators, Ann. Mat. Pura Appl.; 80 (1968), 1–122.

[17] A. Porretta, S. Segura de Leon; Nonlinear elliptic equations having a gradient term with natural growth, J. Math. Pures Appl.; 85 (2006) 465–492.

[18] G. Talenti; Linear Elliptic P.D.E.’s: Level sets, rearrangements and a priori estimates of solutions, Boll. Unione Mat. Ital., B (6) 4 (1985) 917–949.

Aharrouch Benali

Sidi Mohamed Ben Abdellah University, Faculty of Sciences Dhar El Mahraz, Labora- tory LAMA, Department of Mathematics, P.O. Box 1796 Atlas Fez, Morocco

Email address:[email protected]

Bennouna Jaouad

Sidi Mohamed Ben Abdellah University, Faculty of Sciences Dhar El Mahraz, Labora- tory LAMA, Department of Mathematics, P.O. Box 1796 Atlas Fez, Morocco

Email address:[email protected]