$W^{1,1}_0(\Omega)$ in some borderline cases of elliptic equations with degenerate coercivity

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W

_0(Ω) in some borderline cases of elliptic equations with degenerate coercivity

Lucio Boccardo, Gisella Croce

To cite this version:

Lucio Boccardo, Gisella Croce. W^1,1_0(Ω) in some borderline cases of elliptic equations with degen- erate coercivity. Analysis and Topology of Nonlinear Differential Equations, 85, 2014. �hal-00821278�

ELLIPTIC EQUATIONS WITH DEGENERATE COERCIVITY

LUCIO BOCCARDO, GISELLA CROCE

Abstract. We study a degenerate elliptic equation, proving ex- istence results of distributional solutions in some borderline cases.

Bernardo, come vide li occhi miei ...

(Dante, Paradiso XXXI) Acknowledgements.

This paper contains developments of the results presented by the first author at IX WNDE (Jo˜ao Pessoa, 18.9.2012), on the occasion of the sixtieth birthday of Bernhard Ruf”, and it is dedicated to him.

1. Introduction

The Sobolev spaceW₀^1,2(Ω) is the natural functional framework (see [10], [12]) to find weak solutions of nonlinear elliptic problems of the following type

(1)











−div

a(x)∇u (1 +|u|)^θ

=f, in Ω;

u= 0, on∂Ω,

where the function f belongs to the dual space of W₀^1,2(Ω), Ω is a bounded, open subset ofIR^N, withN >2,θ is a real number such that

(2) 0≤θ≤1,

and a : Ω→ IR is a measurable function satisfying the following con- ditions:

(3) α≤a(x)≤β ,

1991Mathematics Subject Classification. 35J61, 35J70, 35J75.

Key words and phrases. Elliptic equations; W^1,1 solutions; Degenerate equations.

1

2 LUCIO BOCCARDO, GISELLA CROCE

for almost every x ∈ Ω, where α and β are positive constants. The main difficulty to use the general results of [10], [12] is the fact that

A(v) =−div

a(x)∇v (1 +|v|)^θ

,

is not coercive. Papers [7], [4] and [3] deal with the existence and summability of solutions to problem (1) iff ∈L^m(Ω), for somem≥1.

Despite the lack of coercivity of the differential operator A(v) ap- pearing in problem (1), in the papers [7], [4] and [1], the authors prove the following existence results of solutions of problem (1), under as- sumption (3):

A) a weak solution u∈W₀^1,2(Ω)∩L^∞(Ω), if m > ^N₂ and (2) holds true;

B) a weak solution u ∈ W₀^1,2(Ω) ∩ L^{m∗∗(1−θ)}(Ω), where m^∗∗ = (m^∗)^∗ = _N^mN_−2m, if

0< θ <1, 2N

N + 2−θ(N −2) ≤m < N 2 ;

C) a distributional solutionu inW₀^1,q(Ω), q = N m(1−θ)

N −m(1 +θ) <2, if

1

N −1 ≤θ < 1, N

N+ 1−θ(N −1) < m < 2N

N + 2−θ(N −2). D) an entropy solution u ∈ M^{m∗∗(1−θ)}, with |∇u| ∈ M^q(Ω), for

1≤m ≤maxn

1,_N+1−θ(N^N ₋₁₎o .

The borderline case θ = 1 was studied in [3], proving the existence of a solution u∈ W₀^1,2(Ω)∩L^p(Ω) for every p < ∞. The case where the source is _|x|^A2 was analyzed too.

About the different notions of solutions mentioned above, we recall that the notion of entropy solution was introduced in [2]. Let

(4) T_k(s) =

s if |s| ≤k, k_|s|^s if |s|> k.

Then u is an entropy solution to problem (1) if T_k(u) ∈ W₀^1,2(Ω) for every k > 0 and

Z

Ω

a(x)∇u

(1 +|u|)^θ·∇T_k(u−ϕ)≤ Z

Ω

f T_k(u−ϕ), ∀ϕ∈W₀^1,2(Ω)∩L^∞(Ω).

Moreover, we say that u is a distributional solution of (1) if (5)

Z

Ω

a(x)∇u

(1 +|u|)^θ · ∇ϕ= Z

Ω

f ϕ , ∀ϕ∈C₀^∞(Ω).

The figure below can help to summarize the previous results, where the name of a given region corresponds to the results that we have just cited.

m N/2

A

B

C

D 2N/(N+2)

1/(N-1) !

Our results are the following.

Theorem 1.1. Let f be a function in L^m(Ω). Assume (3) and

(6) m= N

N + 1−θ(N −1), 1

N −1 < θ <1

Then there exists a W₀^1,1(Ω) distributional solution to problem (1).

Observe that this case corresponds to the curve between the regions C and D of the figure. Note that m >1 if and only if θ > _N−1¹ .

In the following result too, we will prove the existence of a W₀^1,1(Ω) solution.

Theorem 1.2. Let f be a function in L^m(Ω). Assume (3), flog(1 +

|f|)∈L¹(Ω) and θ = _N−1¹ . Then there exists a W₀^1,1(Ω) distributional solution of (1).

We end our introduction just mentioning a uniqueness result of so- lutions to problem (1) can be found in [13].

Moreover, in [4, 5, 11, 6] it was showed that the presence of a lower order term has a regularizing effect on the existence and regularity of the solutions.

4 LUCIO BOCCARDO, GISELLA CROCE

To prove our results, we will work by approximation, using the fol- lowing sequance of problems:

(7)











−div

a(x)∇u_n (1 +|u_n|)^θ

=T_n(f), in Ω;

u_n= 0, on ∂Ω.

The existence of weak solutions u_n∈W₀^1,2(Ω)∩L^∞(Ω) to problem (7) is due to [7].

2. W₀^1,1(Ω) solutions

In the sequelC will denote a constant depending onα, N,meas(Ω), θ and theL^m(Ω) norm of the source f.

We are going to prove Theorem 1.1, that is, the existence of a solution to problem (1) in the case where m = _N+1−θ(N^N ₋₁₎ and _N−1¹ < θ < 1.

Note thatm >1 if and only if θ > _N−1¹ .

Proof. (of Theorem 1.1) We consider T_k(u_n) as a test function in (7):

then (8)

Z

Ω

|∇T_k(u_n)|² ≤

(1 +k)^θkfk

L¹(Ω)

α by assumption (3) on a.

Choosing [(1 +|u_n|)^p −1]sign(u_n), for p = θ− _N¹₋₁, as a test func- tion in (7) we have, by H¨older’s inequality on the right hand side and assumption (3) on the left one

(9) α p

Z

Ω

|∇u_n| (1 +|u_n|)^2(N−1)N

!2

≤ Z

Ω

|f|[(1+|u_n|)^p−1]≤ kfk

L^m(Ω)

Z

Ω

[(1+|u_n|)^p−1]^m0 _m¹0

.

The Sobolev embedding used on the left hand side implies Z

Ω

(1 +|u_n|)^{2(N−1)N−2} −1

_N−2^2N ₂²∗

≤C Z

Ω

[(1 +|u_n|)^p−1]^m0 _m¹0

.

We observe that _2(N−1)^N−2 _N−2^2N =pm⁰; moreover ₂²∗ > _m¹0, since m < ^N₂. Therefore the above inequality implies that

(10)

Z

Ω

|u_n|^N−1N ≤C.

One deduces that (11)

Z

Ω

|∇u_n|²

(1 +|un|)^N−1N ≤C

from (10) and (9). Let v_n = ^2(N−1)_N−2 (1 +|u_n|)^{2(N−1)N−2} sign(u_n). Estimate (11) is equivalent to say that {v_n} is a bounded sequence in W₀^1,2(Ω);

therefore, up to a subsequence, there exists v ∈ W₀^1,2(Ω) such that v_n * v weakly in W₀^1,2(Ω) and a.e. in Ω. If we define the function u= h_2(N−1)

N−2 |v|i^2(N−1)_N−2

−1

sign(v), the weak convergence of∇v_n*∇v means that

(12) ∇u_n

(1 +|u_n|)^2(N−1)N * ∇u

(1 +|u|)^2(N−1)N weakly inL²(Ω).

Moreover, the Sobolev embedding forv_nimplies thatu_n →uinL^s(Ω), for every 1≤s < _N−1^N .

Hold¨er’s inequality with exponent 2 applied to Z

Ω

|∇un|= Z

Ω

|∇u_n|

(1 +|u_n|)^2N−2N (1 +|un|)^2N−2N gives

(13)

Z

Ω

|∇u_n| ≤C,

due to (10) and (11). We are now going to estimate Z

{k≤|un|}

|∇un|. By

using [(1 +|u_n|)^p−(1 +k)^p]⁺sign(u_n) as a test function in (7), we have Z

{k≤|un|}

|∇u_n|²

(1 +|u_n|)^N−1N ≤C Z

{k≤|un|}

|f|^m _m¹ Z

Ω

(1 +|u_n|)^N−1N _m¹0

,

which implies, by (10), (14)

Z

{k≤|un|}

|∇un|²

(1 +|un|)^N−1N ≤ C Z

{k≤|un|}

|f|^m _m¹

.

H¨older’s inequality, estimates (10) and (14) on Z

{k≤|un|}

|∇u_n|= Z

{k≤|un|}

|∇u_n|

(1 +|u_n|)^2N−2N (1+|u_n|)^2N−2N ≤ C Z

{k≤|un|}

|f|^m _2m¹

,

6 LUCIO BOCCARDO, GISELLA CROCE

give (15) Z

{k≤|u_n|}

|∇u_n|= Z

{k≤|u_n|}

|∇u_n|

(1 +|u_n|)^2N−2N (1+|u_n|)^2N−2N ≤ C Z

{k≤|u_n|}

|f|^m _2m¹

.

Thus, for every measurable subset E, due to (8) and (15), we have

(16) Z

E

∂u_n

∂x_i

≤ Z

E

|∇u_n| ≤ Z

E

|∇T_k(u_n)|+ Z

{k≤|u_n|}

|∇u_n|

≤meas(E)¹²

"(1 +k)^θkfk

L¹(Ω)

α

#¹2

+C Z

{k≤|un|}

|f|^m _2m¹

.

Now we are going to prove that u_n weakly converges to u in W₀^1,1(Ω) following [5]. Estimates (16) and (10) imply that the sequence {^∂u_∂xⁿ

i} is equiintegrable. By Dunford-Pettis theorem, and up to subsequences, there existsY_i inL¹(Ω) such that ^∂u_∂xⁿ

i weakly converges toY_i inL¹(Ω).

Since ^∂u_∂xⁿ

i is the distributional partial derivative of u_n, we have, for every n inIN,

Z

Ω

∂u_n

∂xi

ϕ=− Z

Ω

u_n ∂ϕ

∂xi

, ∀ϕ∈C₀^∞(Ω).

We now pass to the limit in the above identities, using that∂iunweakly converges toYi inL¹(Ω), and thatun strongly converges touinL¹(Ω):

we obtain

Z

Ω

Y_iϕ=− Z

Ω

u ∂ϕ

∂xi

, ∀ϕ∈C₀^∞(Ω). This implies that Y_i = _∂x^∂u

i, and this result is true for every i. Since Y_i belongs toL¹(Ω) for every i, ubelongs to W₀^1,1(Ω).

We are now going to pass to the limit in problems (7). For the limit of the left hand side, it is sufficient to observe that ^∇un

(1+|un|)

N 2(N−1)

*

∇u (1+|u|)

N 2(N−1)

weakly inL²(Ω) due to (12) and that|a(x)∇ϕ|is bounded.

We prove Theorem 1.2, that is, the existence of a W₀^1,1(Ω) solution in the case where θ= _N−1¹ and flog(1 +|f|)∈L¹(Ω).

Proof. (of Theorem 1.2) Let k ≥ 0 and take [log(1 +|u_n|)−log(1 + k)]⁺sign(u_n), as a test function in problems (7). By assumption (3) on

a one has α

Z

{k≤|un|}

|∇u_n|² (1 +|un|)^θ+1 ≤

Z

{k≤|un|}

|f|log(1 +|u_n|). We now use the following inequality on the left hand side:

(17) alog(1 +b)≤ a ρlog

1 + a

ρ

+ (1 +b)^ρ

where a, b are positive real numbers and 0< ρ < ^N_N⁻²₋₁. This gives, for any k ≥0

(18) α

Z

{k≤|u_n|}

|∇u_n|² (1 +|u_n|)^θ+1 ≤

Z

{k≤|u_n|}

|f| ρ log

1 +|f|

ρ

+ Z

{k≤|u_n|}

(1 +|un|)^ρ. In particular, for k≥1 we have

(19) α 2^θ+1

Z

{k≤|un|}

|∇u_n|²

|u_n|^θ+1 ≤ Z

{k≤|un|}

|f|

ρ log

1+|f|

ρ

+2^ρ Z

{k≤|un|}

|u_n|^ρ. Writing the above inequality fork= 1 and using the Sobolev inequality on the left hand side, one has

Z

Ω

(|un|^1−θ2 −1)²₊^∗ ₂²∗

≤C Z

{1≤|u_n|}

|f|

ρ log

1 + |f| ρ

+C

Z

{1≤|u_n|}

|un|^ρ,

which implies that Z

Ω

|u_n|^(1−θ)2

∗ 2

₂¹∗

≤C+C v u u t

Z

{1≤|un|}

|f|

ρ log

1 + |f| ρ

+C

v u u t

Z

Ω

|u_n|^ρ .

By using H¨older’s inequality with exponent ^(1−θ)2_2ρ ^∗ on the last term of the right hand side, we get

Z

Ω

|u_n|^(1−θ)2

∗ 2

₂¹∗

≤C+C v u u t

Z

{1≤|un|}

|f| ρ log

1 + |f| ρ

+C

Z

Ω

|u_n|^(1−θ)2

∗ 2

_(1−θ)2^ρ ∗

.

By the choice of ρ, this inequality implies that (20)

Z

Ω

|u_n|^(1−θ)2

∗

2 ≤ C .

Inequalities (20) and (18) written fork = 0 imply that{vn}={_1−θ² (1+

|u_n|)^1−θ2 sign(u_n)} is a bounded sequence in W₀^1,2(Ω), as in the proof

8 LUCIO BOCCARDO, GISELLA CROCE

of Theorem 1.1. Therefore, up to a subsequence there exists v ∈ W₀^1,2(Ω) such that v_n * v weakly in W₀^1,2(Ω) and a.e. in Ω. Let u = n

_1−θ

2 v_1−θ²

−1o

sign(v); the weak convergence of ∇v_n * ∇v means that

(21) ∇u_n

(1 +|u_n|)^θ+12 * ∇u

(1 +|u|)^θ+12 weakly in L²(Ω).

Moreover, the Sobolev embedding forv_nimplies thatu_n →uinL^s(Ω), s <

N N−1.

By (8) one has Z

Ω

|∇un|= Z

Ω

|∇T1(un)|+ Z

{1≤|u_n|}

|∇un| ≤C+ Z

{1≤|u_n|}

|∇u_n|

|u_n|^θ+12 |un|^θ+12 . H¨older’s inequality on the right hand side, and estimates (19) written with k = 1 and (20) imply that the sequence {u_n} is bounded in W₀^1,1(Ω).

Moreover, due to (19) Z

{k≤|un|}

|∇u_n|= Z

{k≤|un|}

|∇u_n|

|u_n|^θ+12 |u_n|^θ+12 ≤

≤C v u u t

Z

{k≤|un|}

|f| ρ log

1 + |f|

ρ

+ Z

{k≤|un|}

|u_n|^ρ.

For every measurable subsetE, the previous inequality and (8) imply Z

E

∂u_n

∂xi

≤ Z

E

|∇u_n| ≤ Z

E

|∇T_k(u_n)|+ Z

{k≤|un|}

|∇u_n|

≤Cmeas(E)¹²(1 +k)^θ2 +C v u u t

Z

{k≤|un|}

|f| ρ log

1 + |f|

ρ

+ Z

{k≤|un|}

|u_n|^ρ.

Since ρ < ^(1−θ)2₂ ^∗, by using H¨older’s inequality on the last term and estimate (20), one has

Z

E

∂u_n

∂x_i

≤Cmeas(E)¹²(1 +k)^θ2+

+C v u u t

Z

{k≤|un|}

|f|

ρ log

1 + |f| ρ

+ meas({|un| ≥k})^{1−(1−θ)22ρ ∗} .

One can argue as in the proof of Theorem 1.1 to deduce that un → u weakly in W₀^1,1(Ω).

To pass to the limit in problems (7), as in the proof of Theorem 1.1, it is sufficient to observe that ^∇un

(1+|u_n|)

N 2(N−1)

* ^∇u

(1+|u|)

N 2(N−1)

weakly in

L²(Ω), due to (21).

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L.B. – Dipartimento di Matematica, “Sapienza” Universit`a di Roma, P.le A. Moro 2, 00185 Roma (ITALY)

E-mail address: boccardo@mat.uniroma1.it

G.C. – Laboratoire de Math´ematiques Appliqu´ees du Havre, Univer- sit´e du Havre, 25, rue Philippe Lebon, 76063 Le Havre (FRANCE)

E-mail address: gisella.croce@univ-lehavre.fr