NONLINEAR ANISOTROPIC ELLIPTIC AND PARABOLIC EQUATIONS IN ℝᴺ WITH ADVECTION AND LOWER ORDER TERMS AND LOCALLY INTEGRABLE DATA

Pure Mathematics

ISBN 82–553–1375–3 No. 13 ISSN 0806–2439 April 2003

NONLINEAR ANISOTROPIC ELLIPTIC AND PARABOLIC EQUATIONS IN R^N WITH ADVECTION AND LOWER

ORDER TERMS AND LOCALLY INTEGRABLE DATA

MOSTAFA BENDAHMANE AND KENNETH H. KARLSEN

Abstract. We prove existence and regularity results for distributional solutions inR^Nfor non- linear elliptic and parabolic equations with general anisotropic diffusivities as well as advection and lower-order terms that satisfy appropriate growth conditions. The data are assumed to be merely locally integrable.

1. Introduction

In this paper we prove existence and regularity of distributional solutions in an appropriate function space for nonlinear anisotropic elliptic equations. A prototype example is

(1.1) −

N

X

`=1

∂

∂xl

βl(x)

∂u

∂xl

pl−2

∂u

∂xl

!

−divg(u) +|u|^s−1u=f inR^N, N≥2, where each βl : R^N → R is a strictly positive and bounded function; g = (g1, . . . , gN) is a continuous vector field with components that grow like|u|^s−η fors >1 and some η ∈(1, s); and f is locally integrable. We also prove corresponding results for nonlinear anisotropic parabolic equations. For (1.1) we assume that the exponentsp₁, . . . , p_N andsare restricted as follows:

(1.2)















p < N, 1 p= 1

N

N

X

l=1

1 pl

, p_l>1 and p_l> p(N−1)

N(p−1), l= 1, . . . , N, s > p_l, l= 1, . . . , N.

We recall that for isotropic elliptic equations with pl = 2 for l = 1, . . . , N ands > 1, and no advection field, existence and uniqueness results for distributional solutions are proved in [7]. In the isotropic case withp_l=p >2−_N¹ forl= 1, . . . , N ands > p−1, still with no advection field, existence and regularity results for distributional solutions are proved in [5]. The corresponding results for isotropic parabolic equations are developed in [6].

Compared to [5, 6], the main feature of of the present paper is the combination of an anisotropic diffusion operator, nonlinear advection and lower-order terms, a locally integrable right-hand side f, and an unbounded domain. In the case of the Dirichlet problem on a bounded domain, existence and regularity results for distributional solutions withL¹-data have been obtained in [4, 11] for a class of anisotropic elliptic and parabolic equations. For an anisotropic parabolic reaction-diffusion- advection system with a zero-flux boundary condition, still on a bounded domain, similar results are established in [2].

Date: April 29, 2003.

Key words and phrases. Elliptic equation, parabolic equation, anisotropic diffusion,L¹data.

This work is supported by the BeMatA program of the Research Council of Norway and the European network HYKE, funded by the EC as contract HPRN-CT-2002-00282. This work was done while M. Bendahmane visited the Centre of Mathematics for Applications (CMA) at the University of Oslo, Norway.

1

Our main purpose is to prove the existence of at least one functionu∈L^s_loc(R^N) that possesses the regularity

(1.3) u∈

N

\

l=1

W_loc^1,ql(R^N), 1≤ ql< N(p−1) p(N−1)pl,

where pis defined in (1.2), and solves (1.1) in the distributional sense. The anisotropic Sobolev spaces appearing in (1.3) are defined in the next section. Observe that (1.2) impliesp >2−_N¹ and thus ^N_p(N^(p−1)₋₁₎p_l>1, which is in accordance with the “isotropic” theory [5]. On the other hand, the conditions > pl in (1.2) is stronger than in [5]. This is a consequence of the anisotropic Sobolev inequality [17] that we have at our disposal here.

As in [5, 6], the strategy of an existence proof consist of deriving “good” a priori estimates for suitable approximate solutions (uε)0<ε<1 (to which the standard variational framework applies) and passing to the limit asε→0. There are two difficulties associated with this strategy. In view of the assumption thatf is only locally integrable on R^N, the first difficulty is to obtain suitable local a priori estimates onu_εand the partial derivatives ^∂u_∂x^ε

l,l= 1, . . . , N, that are independent of ε. The second difficulty lies in passing to the limit in the nonlinear vector fieldA(x,∇uε) +g(uε) and the nonlinear term|uε|^s−1u_ε.

The remaining part of the paper is organized as follows: In Section 2 we recall some basic notations and a Sobolev inequality for anisotropic Sobolev spaces. In addition, we prove an

“interpolation” lemma that will be used later to obtain local a priori estimates. Our main “elliptic”

results are stated in Section 3, while the proofs are given in Section 4. In Section 5 we briefly discuss the Dirichlet problem on a bounded domain. Finally, we convert our “elliptic” results to

“parabolic” results in Section 6.

2. Anisotropic Sobolev spaces and a technical lemma

We start by recalling the notion of anisotropic Sobolev spaces. These spaces were introduced and studied by Nikolskii [14], Slobodeckii [16], and Troisi [17], and later by Trudinger [18] in the framework of Orlicz spaces.

Let Ω be a bounded domain in R^N with Lipchitz boundary ∂Ω. Let p1, . . . , pN be N real numbers with pl ≥ 1, l = 1, . . . , N. With a slight abuse of the notation, we introduce the anisotropic Sobolev space

W^1,pl(Ω) =

u∈L^pl(Ω) : ∂u

∂xl

∈L^pl(Ω)

, which is a Banach space under the norm

kuk_W1,pl(Ω)=kuk_Lpl(Ω)+

∂u

∂xl

_Lpl(Ω)

, forl= 1, . . . , N.

We recall the anisotropic Sobolev imbedding theorem due to Troisi [17] (see also [1]).

Theorem 2.1. Supposeu∈

N

\

l=1

W^1,pl(Ω), and set 1

p= 1 N

N

X

l=1

1

p_l, r=

(p^?:= _N−p^{N p} , if p^?< N , any number from[1,∞), if p^?≥N .

Then there exists a constantC, depending on N,p1, . . . , pN ifp < N and also onr andmeas(Ω) if p≥N, such that

(2.1) kuk_Lr(Ω)≤C

N

Y

l=1

"

∂u

∂xl

_Lpl(Ω)

+kuk_Lpl(Ω)

#_N¹ .

Theorem 2.1 is used to prove the “interpolation” lemma below, which is a technical result we will use later to obtain a priori estimates. A similar result is found in [4] withW^1,pl(Ω) replaced byW₀^1,pl(Ω) in the case of a Dirichlet boundary condition.

Lemma 2.2. Let (uε)_0<ε≤1 be a sequence in

N

\

l=1

W^1,pl(Ω) withp≤N. Suppose that there exists a constant c, independent ofε, such that

(2.2) kuεk_Lpl(Ω)≤c, l= 1, . . . , N, and

(2.3) sup

γ>0 N

X

l=1

Z

B_γ

∂u_ε

∂xl

pl

dx≤c, whereBγ={x∈Ω : γ≤ |uε| ≤γ+ 1}forγ >0, or

(2.4)

N

X

l=1

Z

Ω

∂uε

∂x_l

p_l

(1 +|uε|)^γ dx≤c.

Then for everyql such that

(2.5) 1≤ql<N(p−1)

p(N−1)pl,

there exists a constant C, depending on Ω,N,p1, . . . , pN,q1, . . . , qN, andc, but notε, such that (2.6)

∂uε

∂x_{l Lql(Ω)}

≤C, l= 1, . . . , N, and

(2.7) kuεk_Lq(Ω)≤C, 1

q = 1 N

N

X

l=1

1 ql

.

Proof. We adapt the proof in [3, 4] to our setting. Letq_l< p_landγ₀≥1. Then, using (2.3), Z

Ω

∂u_ε

∂xl

q_l

dx=

γ₀−1

X

γ=0

Z

B_γ

∂u_ε

∂xl

q_l

dx+

∞

X

γ=γ0

Z

B_γ

∂u_ε

∂xl

q_l

dx

≤Cγ₀+

∞

X

γ=γ₀

Z

Bγ

∂uε

∂x_l

q_l

dx

≤C_γ0+

∞

X

γ=γ0

Z

B_γ

∂u_ε

∂xl

p_l

dx

!^ql_pl

(meas(B_γ))^1−qlpl

≤Cγ₀+C1

∞

X

γ=γ₀

(meas(Bγ))^pl

−ql pl . (2.8)

Clearly, 1 γ^q?

Z

B_γ

|uε|^q? dx≥meas(B_γ). From this estimate and H¨older’s inequality, we deduce

Z

Ω

∂uε

∂x_l

q_l

dx≤C2+C3

∞

X

γ=γ₀

1 γ^pl

−ql pl q^?

Z

Bγ

|uε|^q? dx

!^pl

−ql pl

≤C₂+C₄

∞

X

γ=γ₀

1 γ^pl

−ql ql q^?

!_pl^ql _∞ X

γ=γ₀

Z

Bγ

|u_ε|^q? dx

!^pl

−ql pl

. (2.9)

The anisotropic Sobolev inequality (2.1) gives (2.10)

Z

Ω

|uε|^q? dx _q?¹

≤C₅

N

Y

l=1

"Z

Ω

∂u_ε

∂xl

ql

dx _ql¹

+ Z

Ω

|uε|^ql dx _ql¹#N¹

,

whereq^?:= N q

N−q (noteq∈(1, N)). Since ql< pl, it follows from (2.10) and (2.2) that (2.11)

Z

Ω

|uε|^q? dx _q?¹

≤C6 N

Y

l=1

Z

Ω

∂uε

∂x_l

q_l

dx _qlN¹

+C7. By (2.11), (2.9), and the fact that pl−ql

q_l q^?>1 thanks to (2.5), Z

Ω

|uε|^q? dx _q?¹

≤C9+C10 N

Y

l=1

Z

Ω

|uε|^q? dx ^pl

−ql qlplN

=C9+C10

Z

Ω

|uε|^q? dx

^PN_l=1^pl

−ql qlplN

=C9+C10

Z

Ω

|uε|^q? dx ¹_q⁻¹_p

. In other words,

kuεk_Lq?(Ω)≤C9+C10kuεk^a_Lq?(Ω), a:= p−q q p q^?.

One checks easily that the assumptionp < N impliesa <1, and we can therefore conclude that (2.7) holds. Moreover, (2.6) follows from (2.9) and (2.7).

Let ql =κpl, l = 1, . . . , N, for any κ ∈

0,^N_p(N^(p−1)₋₁₎pl

. Let λ= ^1−κ_κ q^?, so that γ_p^ql

l−ql =q^?. Recalling ^pl_q^−ql

l q^? >1, we see thatλ > 1. Using H¨older’s inequality and then estimate (2.4), we obtain

Z

Ω

∂uε

∂x_l

q_l

dx≤



 Z

Ω

∂u_ε

∂xl

p_l

(1 +|uε|)^γ dx





ql pl

Z

Ω

(1 +|uε|)^γplql−ql dx ^pl

−ql pl

≤C11

Z

Ω

(1 +|uε|)^γplql−ql dx ^pl

−ql pl ≤C12

Z

Ω

|uε|^q? dx ^pl

−ql pl

+C13. (2.12)

Inserting this into (2.11) and proceeding as above, we conclude that (2.6) and (2.7) hold under

condition (2.4) instead of (2.3).

3. Statements of results

Instead of (1.1) we will consider more general nonlinear anisotropic elliptic equations of the form

(3.1) −divA(x,∇u)−divg(x, u) +h(x, u) =f(x) inR^N.

The vector fieldA:R^N ×R^N →R^N has componentsa_l:R^N ×R^N →R,l= 1, . . . , N, and we assume that there exist two constantsC_AandC_A⁰ such that for allξ₁, ξ₂∈R^N and for a.e.x

A(x, ξ)·ξ≥C_A

N

X

l=1

|ξ|^pl, (3.2)

|al(x, ξ)| ≤C_A⁰ 1 +

N

X

`=1

|ξ`|<sup>p`−1</sup>

!

, l= 1, . . . , N, (3.3)

[A(x, ξ1)−A(x, ξ2)] [ξ1−ξ2]>0, ξ16=ξ2. (3.4)

The advection fieldg:R^N×R→R^N has continuous componentsg_l:R^N×R→R,l= 1, . . . , N, and satisfies the following conditions:

|g(x, σ)| ≤Cg|σ|^s−η, for a.e.x∈R^N and for allσ∈R. (3.5)

|divxg(x, σ)| ≤C_g⁰ |σ|^s−η, for a.e. x∈R^N and for allσ∈R, (3.6)

for some constantsCg,C_g⁰ and someη∈(1, s).

The nonlinear functionh:R^N ×R→Ris assumed to be measurable in x∈R^N for allσ∈R and continuous inσ∈Rfor a.e. x∈R^N. Furthermore,

h(x, σ)σ≥0, for allσ∈Rand a.e. x∈R^N, (3.7)

sup{|h(x, σ)| : |σ| ≤τ} ∈L¹_loc(R^N), ∀τ∈R. (3.8)

Finally, there should exists > pl,l= 1, . . . , N, such that

(3.9) h(x, σ)sign(σ)≥ |σ|^s, for allσ∈Rand a.e.x∈R^N. We look for distributional solutions to (3.1) in the following sense:

Definition 3.1. A distributional solution of (3.1) is a functionu:R^N →Rsuch u∈W_loc^1,1(R^N)∩L^s_loc(R^N), A(x,∇u)∈ L¹_loc(R^N)^N

, and

Z

R^N

(A(x,∇u) +g(x, u))· ∇ϕ dx+ Z

R^N

h(x, u)ϕ dx= Z

R^N

f ϕ dx, ∀ϕ∈C_c¹(R^N).

(3.10)

Note that (1.3) and the conditions ong,himply that all the terms in (3.10) are well-defined.

Our main results are collected in the following theorem:

Theorem 3.1. Assume (3.2)-(3.9) hold and that the corresponding exponents p₁, . . . , p_N ands are restricted as in (1.2). Letf ∈L¹_loc(R^N). Then (3.1)has at least one distributional solutionu.

If f ≥0, thenu≥0. Moreover, upossesses the regularity stated in (1.3). Finally, iff ∈L¹(R^N) andp > N, thenu∈L^∞_loc(R^N).

4. Proof of Theorem 3.1 For anyR > 0, let BR =

x∈R^N : |x|< R . In what follows, it is always understood that ε takes values in a sequence tending to zero. Let (fε)0<ε<1 ⊂ C_c^∞(Ω) be a sequence of smooth approximations off such that

(4.1)







|fε| ≤ 1

ε and |fε| ≤ |f|; fε→f in L¹_loc(R^N) as ε→0.

Then classical results, see, e.g., [13, 12, 10], provide us with the existence of a sequence of functions (uε)_0<ε≤1⊂

N

\

l=1

W₀^1,pl B1

ε

∩L^s(B1 ε), each of them satisfying the weak formulation

(4.2)

Z

B1 ε

(A(x,∇uε) +g(x, uε))· ∇ϕ dx+ Z

B1 ε

h(x, uε)ϕ= Z

B1 ε

fεϕ dx,

for allϕ∈

N

\

l=1

W₀^1,pl B1

ε

∩L^∞ B1

ε

, where

W₀^1,pl B1

ε

=

u∈W₀^1,1 B1

ε

: ∂u

∂x_l ∈L^pl B1

ε

.

The proof of Theorem 3.1 consists of three main steps. First, we proveε-uniform local a priori estimates for uε, which imply a.e. convergence of uε. Second, we prove strongL¹_loc convergence

of the nonlinear terms in (4.2). Finally, we complete the proof of Theorem 3.1 by passing to the limit in (4.1) asε→0.

In the remaining part of this paper, we use C, C₁, C₂, etc. to denote constants that are independent ofε.

4.1. A priori estimates.

Proposition 4.1. Assume (3.2)-(3.9)hold, and that the exponentsp1, . . . , pN andsare restricted as in (1.2). Set R := ¹_ε, and let ρ be any number such that 0 < 2ρ < R. Then, there exist a constant C, not depending onε, such that

(4.3) kuεk_Ls(B_ρ)≤C

and

(4.4) kh(x, uε)k_L1(Bρ)≤C.

Moreover, for every 1 ≤ ql < N(p−1)

p(N−1)pl there exists a constant C, depending on Bρ, N, p1, . . . , pN,q1, . . . , qN,kfk_L1(B_2ρ)but not ε, such that

(4.5)

∂uε

∂xl

_Lql(B

ρ)

≤C, l= 1, . . . , N, and

(4.6) ku_εk_Lq(Bρ)≤C, 1

q := 1 N

N

X

l=1

1 q_l. Proof. Following [5], we introduce forγ >1 the test function

(4.7) ϕ_γ(σ) =





 (γ−1)

Z σ 0

1

(1 +t)^γdt= 1− 1

(1 +σ)^γ−1, σ≥0,

−ϕγ(−σ), σ <0,

and a smooth cut-off functionθ=θ(x) that is supported in the ballB2ρ (recall 0<2ρ < R) such that 0 ≤θ ≤1,θ(x) = 1 for |x| ≤ρ, and |∇θ| ≤2/ρ. Observe that|ϕγ| ≤1 and, by assuming ρ≥2, there holds|∇θ| ≤1.

Letα >1. Insertingϕ=ϕγ(uε)θ^α into (4.2) gives Z

B_R

A(x,∇uε)· ∇uεϕ⁰_γ(u_ε)θ^αdx+ Z

B_R

g(x, u_ε)· ∇uεϕ⁰_γ(u_ε)θ^αdx +

Z

BR

h(x, uε)ϕγ(uε)θ^αdx+α Z

BR

A(x,∇uε)· ∇θ ϕγ(uε)θ^α−1dx +α

Z

B_R

g(x, u_ε)· ∇θϕ_γ(u_ε)θ^α−1dx= Z

B_R

f ϕ_γ(u_ε)θ^αdx.

(4.8)

Now we chooseγ andαso that (recall from (3.5) and (3.6) thatη∈(1, s)) (4.9) 1< γ < s

p_l−1, α >max

s, s η−1

and α > pls

s−γ(p_l−1), l= 1, . . . , N.

Let us introduce the vector fieldG= (G₁, ..., G_N) defined by G_l(x, σ) =

Z σ 0

g_l(x, t)ϕ⁰_γ(t)dt, l= 1, . . . , N.

Using the divergence theorem, G(0) = 0, (3.5) and (3.6), the condition (4.9) on α, |∇θ| ≤ 1, θ^α≤θ^α−1, and Young’s inequality, we estimate as follows:

Z

B_R

g(x, uε)· ∇uεϕ⁰_γ(uε)θ^αdx

= Z

B_R

divG(u_ε)θ^αdx− Z

B_R

Z uε

0

div_xg_l(x, t)ϕ⁰_γ(t)dt

θ^αdx

=

− Z

BR

αθ^α−1G(uε)· ∇θ dx− Z

BR

θ^α

Z uε

0

divxg(x, t)ϕ⁰_γ(t)dt

dx

≤C1

Z

B_R

|uε|^s−η+1θ^α−1dx+C2

Z

B_R

|uε|^s−η+1θ^αdx

≤C3

Z

BR

|uε|^s−η+1θ^α−1dx=C3

Z

BR

|uε|^s−η+1θ^{s−η+1s α}θ^{η−1s α−1}dx

≤1 8

Z

B_R

ϕ_γ(1)|uε|^sθ^αdx+C₄ Z

B_R

θ^α−η−1s dx

≤1 8

Z

BR

ϕγ(1)|uε|^sθ^αdx+C5meas (B2ρ). (4.10)

Similarly, we deduce the estimate

Z

B_R

g(x, u_ε)· ∇θϕ_γ(u_ε)θ^α−1dx

≤ Z

B_ρ

|g(x, uε)|θ^α−1dx≤ 1 8 Z

B_R

ϕγ(1)|uε|^sθ^αdx+C6meas (B2ρ). (4.11)

Using the structure conditions (3.2) and (3.3) in (4.8) along with (4.10) and (4.11), we get CA

Z

BR

N

X

l=1

∂uε

∂xl

p_l

ϕ⁰_γ(uε)θ^αdx+ Z

BR

h(x, uε)ϕγ(uε)θ^αdx

≤ Z

B_2ρ

|f|+C7

Z

B_R N

X

l=1

∂u_ε

∂xl

pl−1

θ^α−1dx +1

4 Z

B_R

ϕ_γ(1)|u_ε|^sθ^αdx+C₈meas (B_2ρ). (4.12)

An application of Young’s inequality gives

∂uε

∂xl

p_l−1

θ^α−1=

∂uε

∂xl

p_l−1

ϕ⁰_γ(uε)^pl_pl⁻¹ θ^αpl

−1

pl ϕ⁰_γ(uε)¹⁻_pl^pl θ^α

α−pl pl

≤ C_A 2C7

∂u_ε

∂xl

pl

ϕ⁰_γ(u_ε)θ^α+C₉ θ^α−pl ϕ⁰_γ(uε)^pl−1

= CA

2C7

∂uε

∂xl

p

ϕ⁰_γ(uε)θ^α+C10(1 +|uε|)^γ(pl−1)θ^α−pl

= C_A 2C7

∂u_ε

∂xl

p

ϕ⁰_γ(u_ε)θ^α+C₁₁|uε|^γ(pl−1)θ^α−pl+C₁₂θ^α−pl. (4.13)

We can estimate the last term in (4.13) by another application of Young’s inequality and (3.9):

C11|uε|^γ(pl−1)θ^α−pl=C11|uε|^γ(pl−1)θ^αγ(pl

−1) s θ^α

s−γ(pl−1) s −pl

≤ ϕγ(1)

4 |uε|^sθ^α+C13θ^{α−s−γ(plspl−1)}. (4.14)

From (4.7) and (3.9), it follows that

h(x, σ)ϕγ(σ)≥ |σ|^sϕγ(1), forσ≥1 and a.e.x∈R^N,

and hence

(4.15) |σ|^s≤h(x, σ)ϕ_γ(σ)

ϕγ(1) + 1, forσ∈Rand a.e.x∈R^N. Using (4.13), (4.14), and (4.15) in (4.12) we obtain

C_A 2

Z

B_R N

X

l=1

∂u_ε

∂xl

p_l

ϕ⁰_γ(u_ε)θ^αdx+1 2

Z

B_R

h(x, u_ε)ϕ_γ(u_ε)θ^αdx

≤ Z

B2ρ

|f|dx+C14meas (B2ρ). (4.16)

Using the definitions ofϕ_γ andθ, we obtain from (4.16) and (4.15) that (4.17)

Z

B_ρ

|uε|^sdx≤C15,

which proves (4.3) and, via (3.9), also (4.4). Moreover, it follows that (4.18)

N

X

l=1

Z

Bρ

∂u_ε

∂xl

p_l

(1 +|uε|)^γ dx≤C16.

Estimates (4.5) and (4.6) are direct consequences of (4.17), (4.18), and Lemma 2.2.

4.2. Strong convergence. Given anyρ >0, letεbe such that ¹_ε >2ρ. In view of Proposition 4.1,uεis uniformly (in ε) bounded inW^1,q0(Bρ), where

(4.19) q0:= min

1≤l≤Nql,

and q1, . . . , qN are restricted as in Proposition 4.1. Without loss of generality, we can therefore assume that

(4.20)

(uε→u strongly in L^q0(Bρ) and a.e. inBρ, h(x, uε)→h(x, u), g(x, uε)→g(x, u) a.e. inBρ.

By a standard diagonal process, we can in fact assume thatuε→uin L¹_loc(R^N) and a.e. inR^N, u_ε→uweakly inW_loc^1,q0(R^N), andh(x, u_ε)→h(x, u),g(x, u_ε)→g(x, u) a.e. inR^N.

For passing to the limit in (4.2), we prove first the convergence in L¹(B_ρ) of the sequences (h(x, u_ε))_0<ε≤1, (g(x, u_ε))_0<ε≤1 to respectivelyh(x, u),g(x, u).

Proposition 4.2. Assume (3.2)-(3.9)hold, and that the corresponding exponentsp1, . . . , pN and s are restricted as in (1.2). Then the sequences (h(x, uε))_0<ε≤1 and (g(x, uε))_0<ε≤1 converge to respectivelyh(x, u)andg(x, u)a.e. inR^N and strongly inL¹(Bρ)for any ρ >0.

Proof. In view of (4.20) and a theorem of Vitali (see, e.g., [8]), it is sufficient to establish the equi-integrability of (h(x, u_ε))_0<ε≤1onB_ρ. To this end, we follow [5, 6] and introduce forγ, β >1 the test functionϕγ,β defined by

(4.21) ϕγ,β(σ) =







ϕγ(σ−β), σ≥β 0, |σ|< β

−ϕγ,β(−σ), σ≤ −β,

whereϕγ is defined in (4.7). Letα >1. Insertingϕ=ϕγ,β(uε)θ^α into (4.2) and proceeding more or less as we did up to (4.16), we find

C_A 2

Z

B_R N

X

l=1

∂u_ε

∂xl

pl

ϕ⁰_γ,β(uε)θ^αdx+1 2

Z

B_R

h(x, uε)ϕγ,β(uε)θ^αdx

≤ Z

B_2ρ∩{|u_ε|≥β}

|f|dx+C₁meas (B_2ρ∩ {|u_ε| ≥β}). (4.22)

Sincef ∈L¹(B_2ρ) andu_εis bounded inL¹(B_2ρ) uniformly with respect to ε, (4.23)

Z

B_2ρ∩{|u_ε|≥β}

|f|dx+ meas(B_2ρ∩ {|u_ε| ≥β})→0, asβ→ ∞.

From (4.21), (3.7), (4.22), and (4.23), we conclude that Z

B_ρ∩{|uε|≥β+1}

|h(x, uε)| dx≤C Z

B_R

h(x, uε)ϕγ,β(uε)θ^αdx^β→∞−→ 0 (uniformly inε).

By (3.8), this implies the desired equi-integrability of (h(x, uε))0<ε≤1.

From (3.5) and the convergence proof just given, we deduce easily that g(x, uε) converges to g(x, u) a.e. inR^N and strongly inL¹(Bρ) for anyρ >0.

Proposition 4.3. Assume (3.2)-(3.9)hold, and that the corresponding exponentsp1, . . . , pN and s are restricted as in (1.2). Then the sequence (A(x,∇uε))_0<ε≤1 converges to A(x,∇u) a.e. in R^N and strongly in L¹(Bρ)for anyρ >0.

Proof. As in [5, 6], we prove first that the sequence (∇uε)_0<ε≤1 converges to∇uin measure on Bρ, which implies a.e. convergence after passing to a suitable subsequence. It suffices to show that (∇uε)_0<ε≤1 is a Cauchy sequence in measure onBρ, i.e., for anyµ >0,

meas ({x∈Bρ : |(∇uε⁰− ∇uε) (x)| ≥µ})→0, as ε, ε⁰→0.

For anyγ, δ >0, we have

{x∈Bρ : |(∇uε⁰− ∇uε) (x)| ≥µ} ⊂L1∪L2∪L3∪L4, whereL1={x∈Bρ : |∇uε(x)| ≥γ},L2={x∈Bρ : |∇uε⁰(x)| ≥γ},

L₃={x∈B_ρ : |(uε−u_ε⁰) (x)| ≥δ}, and

L4={x∈Bρ : |(∇uε− ∇uε⁰) (x)| ≥µ,|∇uε(x)| ≤γ,|∇uε⁰(x)| ≤γ,|(uε−uε⁰) (x)| ≤δ}. In view of Proposition 4.1, by choosing γ large we can make meas(L1) and meas(L2) arbitrarily small. Since (uε)_0<ε≤1is a Cauchy sequence inL¹(Bρ), then, forδ >0 fixed, meas(L3) tends to 0 asε, ε⁰ →0. It remains to control meas(L4). Since the set of (ξ1, ξ2) such that|ξ1| ≤γ,|ξ2| ≤γ, and|ξ1−ξ2| ≤µis a compact set andξ7→A(x, ξ) is continuous for a.e.x∈Bρ, the quantity

[A(x, ξ₁)−A(x, ξ₂)] [ξ₁−ξ₂]

reaches its minimum value on this compact set, and we will denote it byq(x). By (3.4), it is not hard to verify thatq(x)>0 a.e. inBρ. Consequently, for anyβ >0 there existsβ⁰>0 such that (4.24)

Z

L4

q(x)dx < β⁰ =⇒meas(L4)≤β.

Hence, it is sufficient to show that for any given β⁰ >0, one can produce a small enoughδ >0 such that

(4.25)

Z

L4

q(x)dx < β⁰.

For anyδ >0, defineTδ(z) = min (δ,max(z,−δ)). Note thatTδ is a Lipschitz continuous function satisfying 0≤ |Tδ(z)| ≤δ. By the definitions ofq(x) andL₄, we have

Z

L4

q(x)dx≤ Z

L4

[A(x,∇uε)−A(x,∇uε⁰)] [∇uε− ∇uε⁰]1_{{|uε−uε0|≤δ}}dx

= Z

L₄

[A(x,∇u_ε)−A(x,∇u_ε⁰)]∇T_δ(u_ε−u_ε⁰)dx.

(4.26)

Letθ be the cut-off function used in the proof of Proposition 4.1. Set p⁰ := max

1≤l≤Np_l, and letq₀ be the number defined in (4.19). Thanks to Proposition 4.1, we can find a q∈

p⁰−1, q0 such

that

∂uε

∂x_l

_Lq(B

2ρ)is bounded independently ofεfor alll= 1, . . . , N. SpecifyingTδ(uε−uε⁰)θas test function in the weak formulations foruεand uε⁰ and then subtracting the results, we find

Z

B_ρ

[A(x,∇uε)−A(x,∇uε⁰)]· ∇Tδ(u_ε−u_ε⁰)dx

≤2δ

"

C₁+C₂ Z

B_2ρ N

X

l=1

∂uε

∂x_l

p_l−1

dx+C₃ku_εk_Ls(B2ρ)+kfk_L1(B2ρ)

#

≤2δ

"

C1+C4

Z

B2ρ

∂uε

∂xl

q

dx+C3kuεk_Ls(B_2ρ)+kfk_L1(B_2ρ)

#

−→δ→00 (uniformly inεandε⁰).

(4.27)

For δ small enough, we have from (4.26) and (4.27) that (4.25) holds, and, by (4.24), also that meas(L4)≤β. Thus, we have the convergence of (∇uε)_0<ε≤1 to ∇uin measure. Thanks to this measure convergence and (4.5), we can finally conclude that along a subsequence

A(x,∇uε)→A(x,∇u) strongly in L¹(Bρ).

4.3. Completing the proof of Theorem 3.1. In view of the previous results, we can indeed sendε→0 in the weak formulation (4.2) withϕ∈C_c¹(R^N), thereby obtaining the existence of a distributional solution (in the sense of Definition 3.1) to (3.1), which possesses the regularity stated in (1.3). If f ≥0, then uε ≥0 a.e. in R^N for anyε >0. Hence the limit uis also nonnegative.

TheL^∞_loc-bound foruεis proved by replacingq^?in the proof Lemma 2.2 by any numberr∈[1,∞) and using Theorem 2.1.

5. The Dirichlet problem on a bounded domain

Let Ω be an open bounded domain in R^N (N ≥2). In this section we wish to point out that the existence result obtained in the previous section also applies to the Dirichlet problem on a bounded domain. In fact, on a bounded domain (under stronger assumptions) it is possible to prove that the constructed distributional solution has regularity corresponding to the limiting case of equality in the upper bound onq_lin (1.3). Our results generalize those obtained in [4] to general problems of the form

(5.1)

( −divA(x,∇u)−divg(x, u) +h(x, u) =f(x) in Ω, u= 0 on∂Ω.

whereA, g,hsatisfy the conditions stated in (3.2)-(3.9).

Theorem 5.1. Assume (3.2)-(3.9) hold, and that the exponents p1, . . . , pN and s are restricted as in (1.2). In addition, assume

(5.2) p_l> 1

1 +η−s, l= 1, . . . , N, η∈(s−1, s),

where η is given in (3.5) and (3.6). Let f ∈ L¹(Ω). Then the exists at least one function u ∈ W₀^1,1(Ω)∩L^s(Ω) such thatA(x,∇u)∈L¹(Ω), _∂x^∂u

l ∈L^ql(Ω)with 1≤ql< ^N(p−1)_p(N−1)pl,l= 1, . . . , N, and (5.1)holds in the distribution sense. Iff ∈L¹logL¹(Ω), i.e.,

Z

Ω

|f|log (1 +|f|)dx <∞, then there exists a distributional solutionuof (5.1)such that

(5.3) ∂u

∂x_l ∈L^ql(Ω), ql= N(p−1)

p(N−1)pl, l= 1, . . . , N.

Proof. Let (u_ε)_0<ε≤1be a sequence of approximate solutions satisfying the weak formulation (4.2) with B1

ε replaced by Ω. The first part of the theorem can be proved by adapting the proof of Theorem 3.1. Let us prove (5.3). Since, by (5.2), ^(s−η)p_p ^l

l−1 <1, we deduce from (3.5) Z

Ω

g(x, u_ε) ∇uε

(1 +|uε|)

dx≤C_g Z

Ω

∇uε

(1 +|u_ε|)^pl1

u^s−η_ε (1 +|u_ε|)^1−pl1

≤C_A 2

N

X

l=1

Z

Ω

∂uε

∂x_l

pl

(1 +|uε|)dx+C₁

N

X

l=1

Z

Ω

|uε|

(s−η)pl pl−1

(1 +|uε|) dx

≤C_A 2

N

X

l=1

Z

Ω

∂uε

∂x_l

pl

(1 +|uε|)dx+C₁

N

X

l=1

Z

Ω

|uε|

(s−η)pl pl−1 dx

≤C_A 2

N

X

l=1

Z

Ω

∂uε

∂x_l

pl

(1 +|uε|)dx+C₁

N

X

l=1

Z

Ω

(1 +|uε|)

(s−η)pl pl−1 dx

≤C_A 2

N

X

l=1

Z

Ω

∂uε

∂x_l

pl

(1 +|uε|)dx+C₂ Z

Ω

(1 +|uε|)dx (5.4)

Following [4], we shall modify the proofs of Proposition 4.1 and Lemma 2.2. Inserting the test functionϕ= log(1 +|u_ε|)sign(u_ε) into the weak formulation for u_ε and using (5.4), we find after some work the following a priori estimate:

CA

2

N

X

l=1

Z

Ω

∂u_ε

∂xl

p_l

(1 +|u_ε|)dx≤C3

Z

Ω

(1 +|uε|)dx+C4

Z

Ω

fεlog (1 +|uε|)dx

≤C4

Z

Ω

|fε|log (1 +|fε|)dx+ (C3+C4) Z

Ω

(1 +|uε|)dx

≤C5+C6

Z

Ω

(1 +|uε|)dx, (5.5)

where we have used the well-known inequalityxy≤xlog(1 +x) + exp(y) forx, y≥0.

To turn (5.5) into an L^ql(Ω) estimate on ^∂u_∂x^ε

l, we proceed as in (2.12). As in the proof of Proposition 4.1, one can prove thatuε is uniformly (inε) bounded in L^s(Ω) and thus L¹(Ω). By H¨older’s inequality and then (5.5),

Z

Ω

∂uε

∂xl

q_l

dx≤



 Z

Ω

∂u_ε

∂x_l

p_l

(1 +|uε|)dx





ql pl

Z

Ω

(1 +|uε|)^plql−ql dx ^pl_pl^−ql

≤C7

Z

Ω

(1 +|uε|)^plql−ql dx ^pl

−ql pl

. (5.6)

Inserting (5.6) into (2.11) and keeping in mind that ql

pl−ql

< q^?, we find Z

Ω

|u_ε|^q? dx _q?¹

≤C₈+C₉

N

Y

l=1

Z

Ω

(1 +|u_ε|)^plql−ql dx ^pl

−ql qlplN

≤C₁₀+C₁₁ Z

Ω

|uε|^q? dx ¹_q−¹_p

,

and then we obtain (5.3) as in the proof of Lemma 2.2.

6. Parabolic Case

We consider nonlinear anisotropic parabolic equations of the form (6.1)

(u_t−divA(t, x,∇u)−divg(t, x, u) +h(t, x, u) =f(t, x) (t, x)∈(0, T)×R^N, u(x,0) =u0(x), x∈R^N,

where T >0 is a fixed number. The vector field A : (0, T)×R^N ×R^N → R^N has components a_l: (0, T)×R^N ×R^N →R,l = 1, . . . , N, and we assume that there exist two constants C_A and C_A⁰ such that for allξ₁, ξ₂∈R^N and for a.e. (t, x)

A(t, x, ξ)·ξ≥CA N

X

l=1

|ξ|^pl, (6.2)

|al(t, x, ξ)| ≤C_A⁰ 1 +

N

X

`=1

|ξ`|<sup>p`−1</sup>

!

, l= 1, . . . , N, (6.3)

[A(t, x, ξ₁)−A(t, x, ξ₂)] [ξ₁−ξ₂]>0, ξ₁6=ξ₂. (6.4)

We assume that the advection field g : (0, T)×R^N ×R → R^N has continuous components gl: (0, T)×R^N×R→R,l= 1, . . . , N, and satisfies the following conditions:

|g(t, x, σ)| ≤C_g|σ|^s−η, for a.e. (t, x)∈(0, T)×R^N and for allσ∈R. (6.5)

|divxg(t, x, σ)| ≤C_g⁰|σ|^s−η, for a.e. (t, x)∈(0, T)×R^N and for allσ∈R, (6.6)

for some constantsCg,C_g⁰ and someη∈(1, s).

The functionh: (0, T)×R^N ×R→Ris assumed to be measurable in (t, x)∈(0, T)×R^N for allσ∈Rand continuous inσ∈Rfor a.e. (t, x)∈(0, T)×R^N. Furthermore,

h(t, x, σ)σ≥0, for allσ∈Rand a.e. (t, x)∈(0, T)×R^N, (6.7)

sup{|h(t, x, σ)| : |σ| ≤τ} ∈L¹(0, T;L¹_loc(R^N)), ∀τ ∈R. (6.8)

Finally, there should exists > p_l,l= 1, . . . , N, such that

(6.9) h(t, x, σ)sign(σ)≥ |σ|^s, for allσ∈Rand a.e. (t, x)∈(0, T)×R^N. The dataf,u0 are assumed to satisfy

(6.10) f ∈L¹(0, T;L¹_loc(R^N)), u₀∈L¹_loc(R^N).

We assume that the exponentsp1, . . . , pN andssatisfy the following conditions:

(6.11)















p < N+ N

N+ 1, 1 p = 1

N

N

X

l=1

1 p_l, 2− 1

N+ 1 < pl<p(N+ 1)

N , l= 1, . . . , N, s > p_l, l= 1, . . . , N.

We seek solutions to (6.1) in the following sense:

Definition 6.1. A distributional solution of (6.1) is a function u∈L¹

0, T;W_loc^1,1(R^N)

∩L^s 0, T, L^s_loc(R^N)

, A(t, x,∇u)∈ L¹ 0, T;L¹_loc(R^N)^N , that satisfies

− Z T

0

Z

R^N

uϕ_tdx dt+ Z T

0

Z

R^N

(A(t, x,∇u) +g(t, x, u))· ∇ϕ dx dt +

Z T 0

Z

R^N

h(t, x, u)ϕ dx dt= Z T

0

Z

R^N

f ϕ dx dt+ Z

R^N

u0(x)ϕ(0, x)dx, (6.12)

for allϕ∈C₀¹([0, T)×R^N).

Our main existence result for (6.1) is stated the following theorem:

Theorem 6.1. Assume (6.2)-(6.10) hold and that the corresponding exponentsp₁, . . . , p_N ands are restricted as in (6.11). Then (6.1)has at least one distributional solution u. Iff, u₀≥0, then u≥0. Moreover, upossesses the regularity

(6.13) u∈

N

\

l=1

L^ql

0, T, W_loc^1,ql(R^N)

, 1≤q_l<p_l p

p− N N+ 1

. Finally, iff, u0∈L¹(R^N)andp > N, thenu∈L^∞_loc((0, T)×R^N).

Proof. The proof is similar to the proof of Theorem 3.1, so we just sketch it. Let {fε}_0<ε≤1 and {u0,ε}_0<ε≤1be sequences functions satisfying

(6.14)











f_ε∈C_c^∞([0, T]×R^N) and u_0,ε∈C_c^∞(R^N);

|fε| ≤ 1

ε, |fε| ≤ |f|, fε→f inL¹(0, T;L¹_loc(R^N)) asε→0;

|u0,ε| ≤ 1

ε, |u0,ε| ≤ |u0|, u0,ε→u0 inL¹_loc(R^N) asε→0;

Set R = 1

ε. Then, classical results, see, e.g, [12, 9], provide the existence of a sequence of functions













 uε∈

N

\

l=1

L^pl(0, T;W₀^1,pl(BR))∩L^s((0, T)×BR)∩C([0, T];L²(BR)),

∂_tu_ε∈

N

X

l=1

L^p0l

0, T;

W₀^1,pl(B_R)0 , each of them satisfying the weak formulation

Z T 0

h∂_tu_ε, ϕidt+ Z T

0

Z

B_R

(A(t, x,∇u_ε) +g(t, x, u_ε))· ∇ϕ dx dt +

Z T 0

Z

Ω

h(t, x, u_ε)ϕ dx dt= Z T

0

Z

Ω

f_εϕ dx dt, (6.15)

for allϕ∈

N

\

l=1

L^pl

0, T;W₀^1,pl(BR)

∩L^∞((0, T)×BR). Moreover, the maximum principle holds, so thatu_0,ε≥0 andf_ε≥0 imply u_ε≥0.

We introduce the function

(6.16) ψγ(σ) =

Z σ 0

ϕγ(s)ds, whereϕγ is defined in (4.7).

As in the proof of Proposition (4.1), we takeϕ=ϕ_γ(u_ε)θ^α as a test function in (6.15) and find Z

B_R

ψ_γ(u_ε(x, T))θ^αdx+ Z T

0

Z

B_R

A(t, x,∇u_ε)· ∇u_εϕ⁰_γ(u_ε)θ^αdx dt +

Z T 0

Z

BR

g(t, x, u_ε)· ∇u_εϕ⁰_γ(u_ε)θ^αdx dt+ Z T

0

Z

BR

h(t, x, u_ε)ϕ_γ(u_ε)θ dx dt +α

Z T 0

Z

BR

A(t, x,∇uε)· ∇θϕγ(uε)θ^α−1dx dt +α

Z T 0

Z

BR

g(t, x, uε)· ∇θϕγ(uε)θ^α−1dx dt

= Z

BR

ψγ(u0,ε)θ^αdx+ Z T

0

Z

BR

fεϕγ(uε)θ^αdx dt.

We chooseγ andαaccording to (4.9).