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source term in Zygmund-space
Boussad Hamour
To cite this version:
Boussad Hamour. Some existence results for a quasilinear problem with source term in Zygmund-
space. Portugaliae Mathematica, European Mathematical Society Publishing House, 2019, 76 (3),
pp.259-286. �10.4171/PM/2035�. �hal-02905872�
with source term in Zygmund-space
Boussad Hamour
Abstract. In this paper we study the existence of solution to the problem ( u∈H01(Ω),
−div (A(x)Du) =H(x, u, Du) +f(x) +a0(x)u in D0(Ω),
where Ω is an open bounded set ofR2,A(x) a coercive matrix with coefficients inL∞(Ω), H(x, s, ξ) a Carath´eodory function satisfying, for someγ >0,
−c0A(x)ξξ≤H(x, s, ξ) sign(s)≤γ A(x)ξξ a.e. x∈Ω, ∀s∈R, ∀ξ∈R2. Heref belongs to L1(logL1)(Ω) anda0 ≥0 to Lq(Ω),q >1. Forf and a0 sufficiently small, we prove the existence of at least one solutionuof this problem which is such that eδ0|u|−1 belongs toH01(Ω) for someδ0≥γand satisfies ana priori estimate.
Keywords. Quasilinear problems; Existence; Zygmund-space.
1. Introduction
In this paper, we consider the quasilinear problem ( u∈H01(Ω),
−div (A(x)Du) =H(x, u, Du) +f(x) +a0(x)u in D0(Ω), (1.1) where Ω is a bounded open subset ofR2, Aa coercive matrix with bounded mea- surable coefficients, H(x, s, ξ) a Carath´eodory function having quadratic growth inξ, more precisely, for someγ >0 andc0≥0, one has
−c0A(x)ξ·ξ≤H(x, s, ξ) sign(s) ≤ γ A(x)ξ·ξ
a.e. x∈Ω, ∀s∈R, ∀ξ∈R2, (1.2) where
f ∈L1(logL1)(Ω), f6= 0, and 0≤ a0∈Lq(Ω), q >1 with a0>6= 0. (1.3)
For f and a0 are sufficiently small, in the sense that f and a0 satisfy the two smallness conditions (3.7) and (3.8), we prove that problem (1.1) has at least one solution, which is such that
eδ0|u|−1∈H01(Ω), (1.4)
with
eδ0|u|−1 δ0
H1
0(Ω)
≤Zδ0, (1.5)
where δ0 ≥ γ and Zδ0 are two constants (see (7.9), (7.10), and (7.11) for their definitions) depending only on the data of the problem.
The main originality of our result is the fact that we assume that a0 satis- fies (1.3), namely that a0 is a nonnegative function and f ∈ L1(logL1)(Ω), with f 6= 0. The originality of this paper is twofold. First, we consider the case of source termf in a class of Zygmund-space and no more in Lebesgue space [13]. Second, the fact that we assume that a0 satisfies (1.3), namely that a0 is a nonnegative function.
Let us begin with vast literature concerned with problems like (1.1) which has been studied in many papers in the case wherea0≤0.
Among these papers is a series of papers [8], [9], [10], [11] and [12] by L. Boc- cardo, F. Murat and J.-P. Puel (see also the paper [26] by J.-M. Rokotoson), which are concerned with the case where
a0(x)≤ −α0<0. (1.6)
In these papers, the authors prove that whena0satisfies (1.6) and whenf belongs to Lq(Ω),q > N2, there exists at least one solution of (1.1) belonging to L∞(Ω) and satisfying somea priori estimates.
The case where
a0= 0 (1.7)
was considered, among others, by A. Alvino, P.-L. Lions and G. Trombetti in [1], by C. Maderna, C. Pagani and S. Salsa in [23], by V. Ferone and M.-R. Posteraro in [18], and by N. Grenon-Isselkou and J. Mossino in [19]. In these papers (which consider also nonlinear monotone operators), the authors prove that ifa0≡0 and f belonging toLq(Ω),q > N2, withkfkLq(Ω)sufficiently small, there exists at least oneL∞(Ω)-solution of (1.1) which moreover belongs toL∞(Ω) and which satisfies somea priori estimates.
The case wherea0satisfies (1.7) butf belongs only toLN/2(Ω) forN ≥3 (and no more toLq(Ω) with q > N2) was considered by V. Ferone and F. Murat in [15]
(and in [16] in the nonlinear monotone case). These authors proved that when kfkLN/2(Ω)is sufficiently small, there exists at least one solution of (1.1) such that
eδ|u|−1∈H01(Ω) for someδ > γ, and that such a solution satisfies ana priori es- timate. Similar results were obtained in the case of nonlinear monotone operators wheref ∈L1(logL)N−1(Ω) and a0satisfies (1.7) by F. Chiacchio in [13].
Finally, in [17], V. Ferone and F. Murat considered (also in the case of nonlin- ear monotone operators) the case where a0 satisfies a0 ≤0 and f belongs to the Zygmund-space LN/2,∞(Ω); in this case two smallness conditions should be ful- filled.
To finish with the case wherea0satisfiesa0≤0 andf belongs toH−1(Ω), let us quote the papers [7] by L. Boccardo, F. Murat and J.-P. Puel and [8] by A. Ben- soussan, L. Boccardo and F. Murat, where these authors proved the existence of at least one solution of (1.1).
In contrast with the cases (1.6) and (1.7), the present paper is concerned with the case (1.3) wherea0≥0, anda06= 0. In this setting we are only aware of four recent papers. In [20], B. Hamour and F. Murat proved a similar result to the one of the present paper, for kfkLN/2(Ω) and ka0kq are sufficiently small, with N ≥3 andq > N2. In [22], L. Jeanjean and B. Sirakov proved too a result similar to the one of the present paper whenf ∈Lq(Ω),q > N2, they also proved the existence of at least two solutions of (1.1). In [3], D. Arcoya, C. De Coster, L. Jeanjean and K. Tanaka proved the existence of a continuum (u, λ) of solutions (withubelonging moreover to L∞(Ω)) when A(x) = Id, H(x, s, ξ) = µ(x)|ξ|2, with µ ∈ L∞(Ω), µ(x) ≥ µ > 0, f ∈ Lq(Ω), q > N2, f ≥ 0, f 6= 0 and a0(x) = λa?0(x) with a?0 ∈ Lq(Ω), a?0 ≥ 0 and a?0 6= 0; moreover, under some further conditions on f, and that there are at least two nonnegative solutions of (1.1) when λ > 0 is sufficiently small.
In [27], in a similar setting, assuming only thatµ(x)≥0 but that the supports ofµ and of a?0 have a nonempty intersection and that N ≤5, P. Souplet proved the existence of a continuum (u, λ) of solutions, and that there are at least two nonnegative solutions of (1.1) whenλ >0 is sufficiently small.
With respect to the results obtained in the four latest papers, we prove in this paper, as said above, the existence of (only) one solution of (1.1) in the casea0≥0 whena0andf satisfy the two smallness conditions (3.7) and (3.8), but our result is obtained in the general case of a nonlinearityH(x, s, ξ) which satisfies only (1.2), withf ∈L1(logL1)(Ω) and with a0∈Lq(Ω),q >1.
The paper is organized as follows:
The (short) Section 2 deals with preliminaries results about Zygmund-spaces.
The precise statement of our result is given in Section 3 (Theorem 3.1), as well as the precise assumptions under which we are able to prove it. These assumptions include in particular the two smallness conditions (3.7) and (3.8).
Our method for proving Theorem 3.1 is based on an equivalence result (see Theorem 4.2) stated in Section 4 once we have introduced the functionsKδ(x, s, ζ) and gδ(s) (see (4.5) and (4.6)) and made some technical remarks on them. This equivalence result is very close to that given in the paper [16] by V. Ferone and F. Murat.
Our equivalence result permuts us to reduce the proof of existence of a solution uof (1.1), which satisfies (1.4) and (1.5), to the existence (see Theorem 4.3) of a functionwrelated touby
w= 1
δ0(eδ0|u|−1) sign(u), which solve the problem (see (4.13))
w∈H01(Ω),
−div(A(x)Dw) +Kδ0(x, w, Dw) sign(w)
= (1 +δ0|w|)f(x) +a0(x)w+a0(x)gδ0(w) sign(w) in D0(Ω), and satisfies the estimate see (4.14)
kwkH1
0(Ω)≤Zδ0.
Our goal thus becomes to prove Theorem 4.3, namely to prove the existence of a solutionwwhich satisfies (4.13) and (4.14).
Problem (4.13) is very similar to Problem (1.1), since it involves a term
−Kδ0(x, w, Dw) sign(w) which has quadratic growth in Dw, as well as a zeroth order term is given byδ0|w|f(x) + a0(x)w+a0(x)gδ0(w) sign(w). But this prob- lem is also very different from (1.1), since the term−Kδ0(x, w, Dw) sign(w) with quadratic growth has the “good sign property”, sinceKδ0(x, s, ξ) satisfies
Kδ0(x, s, ξ)≥0,
while the zeroth order term is now no more a linear but a semilinear term with
|s|2 growth due to presence of the terma0(x)gδ0(w) sign(w).
We will prove Theorem 4.3 essentially by applying Schauder’s fixed point theo- rem. But there are some difficulties to do it directly, since the term with quadratic growthKδ0(x, w, Dw) sign(w) only belongs toL1(Ω) in general. We therefore be- gin by defining an approximate problem (see (5.1)) whereKδ(x, w, Dw) is replaced by its truncation at height k, namely Tk(Kδ(x, w, Dw)), and we prove (see The- orem 5.1) that if f and a0 satisfy the two smallness conditions (3.7) and (3.8),
this approximate problem has at least one solutionwk which satisfies the a priori estimate
kwkkH1
0(Ω)≤Zδ0. (1.8)
This result is obtained in Section 5 by applying Schauder’s fixed point theorem in a classical way.
We then pass to the limit asktends to infinity and we prove in Section 6 that (for a subsequence ofk)wk tend to some w? strongly inH01(Ω) and that thisw? is a solution of (4.13) which also satisfies (4.14) (see End of the proof of Theorem 4.3).
This completes the proof of Theorem 4.3, and therefore proves Theorem 3.1, as announced.
This proof follows along the lines of the proof used by V. Ferone and F. Murat in [15] in the case wherea0= 0. As mentionned above, this method can be applied to the nonlinear case where the linear operator −div(A(x)Du) is replaced by a Leray-Lions operator−div(a(x, u, Du)) working in W01,p(Ω) for some 1< p < N andp=N, and where the quasilinear termH(x, u, Du) hasp-growth in|Du|, as it was done in [16] by V. Ferone and F. Murat in this nonlinear setting whena0= 0.
This will be the goal of our next paper [21].
2. Preliminary result
In this section we recall some definitions and classical properties about rearrange- ments and introduce the Zygmund-spaces. Let Ω be an open bounded subset of RN equipped with the Lebesgue’s measure and let φ : Ω→ Rbe a measurable function, we define the distribution functionµφ by
µφ(t) = meas{x∈Ω : |φ(x)|> t}, ∀t≥0, and the decreasing rearrangement ofφby
φ∗(s) = sup{t >0 : µφ(t)> s}, ∀s∈(0,|Ω|) (|Ω|= meas (Ω)).
We recall only the following Hardy-Littlewood inequality (see [24]) Z
Ω
|g(x)h(x)|dx≤ Z |Ω|
0
g∗(s)h∗(s)ds, (2.1) for anygand hreal measurable functions defined in Ω.
The Zygmund-space Lp(logL)q, p > 0 and q ∈ R, consists of all measurable functionsφ : Ω→Rsuch that the following quantity is finite.
kφkLp(logL)q = Z |Ω|
0
log|Ω|
s q
φ∗(s) p
ds
!1/p
. (2.2)
Finally, we recall the following Theorem (see [2]).
Theorem 2.1. If uis a real function in W01,N(Ω), then u∗(s)≤ kDukN
N ωN1/N
log|Ω|
s N−1N
∀s∈(0,|Ω|), (2.3) wherewN is the the measure of unit sphere ofRN such that
wN = πN/2 Γ(1 +N/2). In particular, forN = 2one has
N ω1/NN = (2π)1/2.
3. The main result
In this paper we consider the following quasilinear problem ( u∈H01(Ω),
−div (A(x)Du) =H(x, u, Du) +a0(x)u+f(x) in D0(Ω), (3.1) where Ω is bounded open subset ofR2, whereAis a coercive matrix with bounded mesurable coefficients, i.e.,
( A∈(L∞(Ω))2×2,
∃α >0, A(x)ξξ≥α|ξ|2 a.e.x∈Ω, ∀ξ∈R2, (3.2) where H : Ω×R×R2 → R is a Carath´eodory function which satisfies the condition:
−c0A(x)ξξ≤H(x, s, ξ) sign(s)≤γ A(x)ξξ, a.e. x∈Ω, ∀s∈R, ∀ξ∈R2, (3.3) for a givenγ >0 andc0≥0, where the function sign(s) is defined by:
sign(s) =
+1 if s >0, 0 if s= 0,
−1 if s <0,
(3.4)
where the coefficienta0 satisfies
a0∈Lq(Ω) for some q >1, a0≥0, a06= 0, (3.5)
and finally where
f ∈L1(logL1)(Ω), f 6= 0. (3.6) Since Ω is bounded, we equip the spaceH01(Ω) with the gradient norm
kukH1
0(Ω)=kDukL2(Ω)2.
We finally assume thatf anda0are sufficiently small, and more precisely that α−C1(q)ka0kq−γkfkL1(logL)(Ω)>0, (3.7)
kfkH−1(Ω)≤ 2πα−C1(q)ka0kq−δkfkL1(logL)(Ω)
2
16π2δ1C2(q)ka0kq
, (3.8)
where the constant δ1 is defined by (7.5) and where some constants C1(q) > 0 and C2(q)>0, as well as
C1(q) = Z |Ω|
0
log|Ω|
s q0
ds 1/q0
, C2(q) = Z |Ω|
0
log|Ω|
s 3q
0 2 ds
1/q0
(3.9) whereq0 the H¨older’s conjugate of the exponentq, i.e.
1 q + 1
q0 = 1.
Our aim is to prove the following Theorem:
Theorem 3.1. Assume that(3.2), (3.3), (3.4), (3.5), (3.6)hold true. Assume moreover that the two smallness conditions(3.7)and(3.8)hold true.
Then there exist a constantδ0 withδ0≥γand a constantZδ0, such that there exists at least one solutionuof(3.1) which further satisfies
(eδ0|u|−1)∈H01(Ω), (3.10)
with
keδ0|u|DukL2(Ω)2 =
eδ0|u|−1 δ0
H10(Ω)
≤Zδ0. (3.11)
Remark 3.2. The definitions of the constantsδ0 andZδ0 which appear in Theorem 3.1 are given in Appendix 7 (see Lemma 7.2). These definitions are based on the properties of the family of functionsφδ which look like convex parabolas (see Fig. 2). The constant δ0 is the unique value of the parameter δ0 for which the function φδ0 admits to double zero andZδ0 is the value such thatφδ(Zδ0) = 0.
4. An equivalence result
The main results of this Section are Theorems 4.2 and 4.3.
Indeed, as said above, the proof of Theorem 3.1 is based on the equivalence result of Theorem 4.2 that we state and we prove in this section. Here we always assume that
δ >0 and let us first proceed with a formal computation. Ifuis a solution of the following problem:
( −div (A(x)Du) =H(x, u, Du) +f(x) +a0(x)u in Ω,
u= 0 on ∂Ω, (4.1)
and if we formally define the functionwδ by wδ= 1
δ
eδ|u|−1
sign(u), (4.2)
where the function sign is defined by (3.4), we have, at least formally,
eδ|u|= 1 +δ|wδ|, |u|=1
δ log(1 +δ|wδ|), sign(u) = sign(wδ), Dwδ=eδ|u|Du,
−div (A(x)Dwδ) =−δeδ|u|A(x)DuDusign(u)−eδ|u| div (A(x)Du) ,
(4.3)
and thereforewδis, at least formally, a solution of
−div (A(x)Dwδ) =−Kδ(x, wδ, Dwδ) sign(wδ) +(1 +δ|wδ|)f(x) + a0(x)wδ
+a0(x)gδ(wδ) sign(wδ) in Ω, wδ= 0 on ∂Ω,
(4.4)
where the functionsKδ: Ω×R×RN →Randgδ :R→Rare defined by the following formulas:
Kδ(x, t, ζ) =
= δ
1 +δ|t|A(x)ζζ−(1 +δ|t|)H
x,1
δlog(1 +δ|t|) sign(t), ζ 1 +δ|t|
sign(t), a.e. x∈Ω, ∀t∈R, ∀ζ∈RN,
(4.5)
and
gδ(t) =−|t|+1
δ(1 +δ|t|) log(1 +δ|t|), ∀t∈R. (4.6) Conversely, ifwδ is a solution of (4.4), and if we formally define the function by
u=1
δlog(1 +δ|wδ|) sign(wδ), (4.7) the some formal computation easily shows thatuis solution of 4.1.
Remark 4.1. Observe that the functions Kδ(x, w, Dw) and Kδ(x, w, Dw)sign(w) are correctly defined and are mesurable functions when w ∈ H1(Ω), and continuous with respect to the almost everywhere convergence ofwandDw(see Lemma 3.3 in [20])
Theorem 4.2. Assume that(3.2), (3.3), (3.4), (3.5), (3.6) hold true, and let δ >0be fixed. Let the functionsKδ andgδ be defined by(4.5)and(4.6).
Ifuis any solution of(3.1) which satisfies
(eδ|u|−1)∈H01(Ω), (4.8)
then the functionwδ defined by(4.2), namely by wδ=1
δ
eδ|u|−1
sign(u), satisfies
wδ∈H01(Ω),
−div (A(x)Dwδ) +Kδ(x, wδ, Dwδ) sign(wδ)
= (1 +δ|wδ|)f(x) +a0(x)wδ+a0(x)gδ(wδ) sign(wδ) in D0(Ω).
(4.9)
Conversely, ifwδ is any solution of(4.9), then the functionudefined by u=1
δlog(1 +δ|wδ|) sign(wδ), is a solution of(3.1)which satisfies(4.8).
Proof of Theorem4.2. Define the function ˆfby
f(x) =ˆ f(x) +a0(x)u(x), (4.10) In view of (4.2) and of the definition (4.6) ofgδ(s), one has
(1 +δ|wδ|)f(x) +a0(x)wδ+a0(x)gδ(wδ) sign(wδ)
= (1 +δ|wδ|) f(x) +a0(x)1
δ log(1 +δ|wδ|) sign(wδ)
= (1 +δ|wδ|) (f(x) +a0(x)u(x)) = (1 +δ|wδ|) ˆf(x).
(4.11)
Then Theorem 4.2 becomes an immediate application of Proposition 1.8 of [15], once one observes that
fˆ∈L1(Ω); (4.12)
Indeeda0u∈L1(Ω), sincea0 is assumed to belong toLq(Ω),q >1, whileu∈Lr(Ω) for everyr <∞, since (eδ|u|−1)∈H01(Ω), hence in particular toL1(Ω), which implies that eδ|u|belongs toL1(Ω). Theorem 4.2 is therefore proved.
From the equivalence Theorem 4.2 one immediately deduces that Theorem 3.1 is equiv- alent to the following Theorem.
Theorem 4.3. Assume that(3.2), (3.3), (3.4), (3.5), (3.6)hold true. Assume moreover that the two smallness conditions(3.7)and(3.8)hold true.
Then there exist a constantδ0 withδ0≥γ, and a constantZδ0, such that there exists at least one solutionwof
w∈H01(Ω),
−div(A(x)Dw) +Kδ0(x, w, Dw) sign(w)
= (1 +δ0|w|)f(x) +a0(x)w+a0(x)gδ0(w) sign(w) in D0(Ω),
(4.13)
which satisfies
kwkH1
0(Ω)≤Zδ0. (4.14)
The rest of this paper will therefore be devoted to the proof of Theorem 4.3. This will be done in two steps: first, we will prove the existence of a solution for an approximate problem; second, we will pass to the limit in this approximate problem.
Remark 4.4. We assume that (3.2), (3.3), (3.5), (3.6) hold true. Assume moreover that the two smallness conditions (3.7) and (3.8) hold true, we try to explain how these two conditions come from an “a priori estimate” that one can obtain on the solutions of (4.9).
Ifwδis any solution of (4.9), usingTk(wδ)∈H01(Ω)∩L∞(Ω) as test function, where Tk:R→Ris the usual truncation at heightkdefined by
Tk(s) =
−k if s≤ −k, s if −k≤s≤+k, +k if +k≤s,
(4.15) one has
Z
Ω
A(x)DwδDTk(wδ)dx+ Z
Ω
Kδ(x, wδ, Dwδ)|Tk(wδ)|dx
= Z
Ω
f(x)Tk(wδ)dx+ Z
Ω
δf(x)|wδ|Tk(wδ)dx +
Z
Ω
a0(x)wδTk(wδ)dx+ Z
Ω
a0(x)gδ(wδ)|Tk(wδ)|dx.
(4.16)
We assume thatδsatisfies
γ≤δ≤δ1, (4.17)
whereδ1 is defined by (7.5).
Sinceδ≥γby (4.17), we deduce thatKδ(x, s, ζ)≥0 (see Remark 3.1 in [20]). Using this fact and passing to the limit asktends to +∞, we obtain
Z
Ω
A(x)DwδDwδdx ≤ Z
Ω
f(x)wδdx+ Z
Ω
δf(x)|wδ|2dx
+ Z
Ω
a0(x)|wδ|2dx+ Z
Ω
a0(x)gδ(wδ)|wδ|dx.
(4.18)
In (4.18) we use the left-hand side the fact that the matrix is coercive, in all terms of the right-hand side the fact thatf∈L1(logL)(Ω), Holder’s inequality, Hardy-Littlewood’s
inequality, classical properties about rearrangements and the fact 0≤gδ(t)≤δ|t|2, ∀t∈R, t6= 0, ∀δ, 0< δ≤δ1.
This allows us to obtain an estimate onTk(wδ), in which we pass to the limit inkto get
αkDwδk22<kfkH−1(Ω)kDwδk2+δkfkL1(logL)(Ω)
2π kDwδk22 +C1(q)ka0kq
2π kDwδk22+δC2(q)ka0kq
(2π)3/2 |Dwδk32 ifwδ6= 0,
(4.19)
where these constantsC1(q) andC2(q) depend only onq(see (3.9)).
Dividing bykDwδk2 this implies that
αkDwδk2<kfkH−1(Ω)+δkfkL1(logL)(Ω)
2π kDwδk2
+C1(q)ka0kq
2π kDwδk2+δ C2(q)ka0kq
(2π)3/2 |Dwδk22 ifwδ6= 0,
(4.20)
Using (4.17) in the fourth term of the right-hand side of (4.20), we have
αkDwδk2<kfkH−1(Ω)+δkfkL1(logL)(Ω)
2π kDwδk2
+C1(q)ka0kq
2π kDwδk2+δ1C2(q)ka0kq
(2π)3/2 |Dwδk22 ifwδ6= 0,
(4.21)
In view of the definition (7.7) of the function Φδ (see also Fig. 2), we have proved that ifwδ is any solution of (4.9), one has
Φδ(kDwδk2)>0 if γ≤δ≤δ1. (4.22) But by the definition ofδ0, one has
Φδ(X)>0, ∀X≥0, ∀δ, 0< δ≤δ1,
and therefore inequality (4.22) does not imply anything on kDwδk2 when δ satisfies δ0< δ≤δ1. In contrast, whenδ < δ0, the strict inequality (4.22) implies that
either kDwδk2< Yδ− or kDwδk2> Yδ+ if δ < δ0, (4.23) whereYδ−< Yδ+are the two distinct zeros of the function Φδ(see Remarks 7.3 and Fig.
2), while whenδ=δ0, the strict inequality (4.22) implies that
either kDwδ0k2< Zδ0 or kDwδ0k2> Zδ0 if δ=δ0. (4.24) Inequalities (4.23) and (4.24) are not a priori estimates, since they do not imply any bound onkDwδk2. Nevertheless these inequalities exclude the closed interval [Yδ−, Yδ+] or the point Zδ0 forkDwδk2, and they give the hope to prove the existence of a fixed point in the setkDwδk2≤Yδ−, whenδ < δ0, or in the setkDwδ0k2≤Zδ0, whenδ=δ0. These inequalities also explain where the two smallness conditions (3.7) and (3.8) come from. Indeed (see Remark 7.3), these two smallness conditions imply that the value δ0 ofδ for which Φδ has a double zero satisfiesδ0≥γ, which is the case where, as said just above, some hope is allowed.
5. Existence of a solution for an approximate problem
In this Section we introduce an approximation of problem (4.13). Under the two small- ness conditions (3.7) and (3.8), we prove by applying Schauder’s fixed point theorem that this approximate problem has at least one solution which satisfies the estimate (4.14).
Letδ0 be defined by (7.9), (7.10) and (7.11). For anyk >0, we consider the following approximate problem.
wk∈H01(Ω),
−div (A(x)Dwk) +Tk(Kδ0(x, wk, Dwk)) signk(wk)
= (1 +δ0|wk|)f(x) +a0(x)wk+a0(x)gδ0(wk) sign(wk) in D0(Ω),
(5.1)
whereTkis the usual truncation at heightk defined by (4.15) and where signk:R→R is the approximation of the function sign which is defined by
signk(s) =
ks if |s| ≤ 1 k, sign(s) if |s| ≥ 1
k.
(5.2)
Theorem 5.1. Assume that (3.2), (3.3), (3.5) and (3.6) hold true. Assume moreover that the two smallness conditions(3.7) and(3.8) hold true. Letδ0 be defined in Lemma 7.2(see (7.9)and(7.10)), and letk >0be fixed.
Then there exists at least one solution of(5.1)such that kwkkH1
0(Ω)≤Zδ0, (5.3)
whereZδ0 is defined in Lemma 7.2(see(7.9),(7.10) and(7.11)).
The proof of Theorem 5.1 consists in applying Schauder’s fixed point theorem. First we prove the two following lemmas.
Lemma 5.2. Assume that(3.2), (3.3), (3.4), (3.5), (3.6)hold true. Letk >0be fixed.
Then, for anyw∈H01(Ω), there exists a unique solutionW of the following semilin- ear problem
W ∈H01(Ω),
−div(A(x)DW) +Tk(Kδ0(x, w, Dw)) signk(W)
= (1 +δ0|w|)f(x) +a0(x)w+a0(x)gδ0(w) sign(w) in D0(Ω).
(5.4)
MoreoverW satisfies
αkDWk2<kfkH−1(Ω)+δ0kfkL1(logL)(Ω)
2π kDwk2
+C1(q)ka0kq
2π kDwk2+δ0C2(q)ka0kq
(2π)3/2 |Dwk22 ifw6= 0,
(5.5)
whereC1(q)andC2(q)are the constants given by(3.9).
Proof. Problem (5.4) is of the form
W ∈H01(Ω),
−div (A(x)DW) + ˆb(x) signk(W) =Fb(x) in D0(Ω),
(5.6)
where ˆb(x) and ˆF(x) are given. Since ˆb(x) =Tk(Kδ0(x, w, Dw)) belongs toL∞(Ω) and is nonnegative (see Remark 3.1 in [20]) andδ0≥γ(see (7.9)), since the function signkis continuous and nondecreasing, and sinceFbbelong toH−1(Ω) (see e.g. the computation which allows one to obtain (5.8) below), this problem has a unique solution.
SinceW ∈H01(Ω), the use ofW as a test function in (5.4) is licit. Since the function Tk(Kδ(x, t, ζ)) is nonnegative, this gives
Z
Ω
A(x)DW DW dx
≤ Z
Ω
(1 +δ0|w|)f(x)W dx
+ Z
Ω
a0(x)w W dx+ Z
Ω
a0(x)gδ0(w) sign(w)W dx.
(5.7)
As in the computation made in Remark 4.4 to obtain the inequality (4.19), we use in (5.7) the coerciveness (3.3) of the matrixA, and classical properties about rearrangements. We obtain
αkDWk22≤ kfkH−1(Ω)kDwδk2kDWk2+δ0kfkL1(logL)(Ω)
2π kDwk2kDWk2
+C1(q)ka0kq
2π kDwk2kDWk2+C2(q)δ0ka0kq
(2π)3/2 kDwk22kDWk2,
(5.8)
which implies (5.5).
Lemma 5.3. Assume (3.2), (3.3), (3.4), (3.5), (3.6)hold true. Assume moreover that the two smallness conditions(3.7) and(3.8)hold true. Letk >0be fixed.
Letwnbe a sequence such that
wn* w in H01(Ω) weakly and a.e. in Ω. (5.9) DefineWn as the unique solution of(5.4) forw=wn, i.e.
Wn∈H01(Ω),
−div(A(x)DWn) +Tk(Kδ0(x, wn, Dwn)) signk(Wn)
= (1 +δ0|wn|)f(x) +a0(x)wn+a0(x)gδ0(wn) sign(wn) in D0(Ω).
(5.10)
Assume moreover that for a subsequence, still denoted byn, and for someW?∈H01(Ω), Wnsatisfies
Wn* W? in H01(Ω) weakly and a.e. in Ω. (5.11) Then for the same subsequence one has
Wn→W? in H01(Ω) strongly. (5.12)
Proof. SinceWn−W?∈H01(Ω), the use of (Wn−W?) as test function in (5.10) is licit.
This gives
Z
Ω
A(x)D(Wn−W?)D(Wn−W?)dx
=− Z
Ω
A(x)DW?D(Wn−W?)dx
− Z
Ω
Tk(Kδ0(x, wn, Dwn)) signk(Wn) (Wn−W?)dx
+ Z
Ω
(1 +δ0|wn|)f(x) (Wn−W?)dx + Z
Ω
a0(x)wn(Wn−W?)dx
+ Z
Ω
a0(x)gδ0(wn) sign(wn) (Wn−W?)dx.
(5.13)
We claim that every term of the right-hand of (5.13) tends to zero as n tends to infinity.
For the first term, we just use the fact thatWn−W?tends to zero inH01(Ω) weakly.
For the second term, we use the factTk(Kδ0(x, wn, Dwn)) signk(Wn) is bounded in L∞(Ω), sincekis fixed, whileWn−W?tends to zero inL1(Ω) strongly.
For the last three terms we observe that, since wn and Wn respectively converge almost everywhere towand toW∗(see (5.9) and (5.11)), we have
(1 +δ0|wn|)f(x) (Wn−W?)→0 a.e. in Ω, a0(x)wn(Wn−W?)→0 a.e. in Ω, a0(x)gδ0(wn) sign(wn) (Wn−W?)→0 a.e. in Ω.
(5.14)
We will now prove that each of the three sequences which appear in (5.14) are equi- integrable. Together with (5.14), this will imply that these sequences converge to zero inL1(Ω) strongly, and this will prove that the three last terms of the right-hand side of (5.13) tend to zero asntends to infinity.
In order to prove that the sequence (1 +δ0|wn|)f(x) (Wn−W?) is equiintegrable, we use the classical properties about rearrangements (2.1), (2.2) and (2.3).
For any measurable setE,E⊂Ω, we have
Z
E
|(1 +δ|wn|)f(x) (Wn−W?)|dx
= Z
E
f(x) (Wn−W?)dx+δ Z
E
|wn|f(x) (Wn−W?)|dx
≤ kfk1/2L1(Ω)kfk1/2L1(logL)(E)
(2π)1/2 kD(Wn−W?)k2+δkfkL1(logL)(E)
2π kDwnk2kD(Wn−W?k2
≤c1kfk1/2L1(logL)(E)+c2kfkL1(logL)(E).
wherec1,c2 denote the constants which are independents ofn.
Proving that the sequencea0(x)wn(Wn−W?) is equiintegrable is similar, since for any measurable setE,E⊂Ω, we have
Z
E
|a0(x)wn(Wn−W?)|dx
≤ Z
E
|a0(x)|qdx 1/q
C1(q)
2π kDwnk2kD(Wn−W?)k2
≤c Z
E
|a0(x)|qdx 1/q
.
Finally, in order to prove that the sequence a0(x)gδ0(wn) sign(wn) (Wn−W?) is equiintegrable, we use as in (5.8), Hardy-Littlewood inequality; for any measurable set E,E⊂Ω, we have
Z
E
|a0(x)gδ0(wn) sign(wn) (Wn−W?)|dx
≤ Z
E
|a0(x)|δ0|wn|2|Wn−W?|dx
≤ Z
E
|a0(x)|qdx 1/q
δ0
C2(q)
(2π)3/2kDwnk22kD(Wn−W?)k2≤c Z
E
|a0(x)|qdx 1/q
. We have proved that the right-hand side of (5.13) tends to zero. Since the matrixA is coercive (see (3.3)), this proves thatWn tends toW? inH01(Ω) strongly. Lemma 5.3 is proved.
Proof of Theorem5.1. Recall that in this Theoremk >0 is fixed.
Consider the ballBofH01(Ω) defined by
B={w∈H01(Ω) :kDwk2 ≤Zδ0}, (5.15) whereZδ0 is defined fromδ0 by (7.11).
Consider also the mappingS:H01(Ω)→H01(Ω) defined by
S(w) =W, (5.16)
where for everyw∈H01(Ω),W is the unique solution of (5.4) (see Lemma 5.2).
We will apply Schauder’s fixed point theorem in the Hilbert spaceH01(Ω) to the map- pingS and to the ballB.
First step : In this step we prove thatS mapsB into itself.
Indeed by Lemma 5.2,W =S(w) satisfies (5.5); therefore, whenkDwk2≤Zδ0, one has, in view of the definition (7.7) the function Φδ and of the property (7.10) ofZδ0,
αkDWk2≤ kfkH−1(Ω)+δ0kfkL1(logL)(Ω)
2π kDwk2
+ C1(q)ka0kq
2π kDwk2+δ1C2(q)ka0kq
(2π)3/2 kDwk22
≤ kfkH−1(Ω)+δ0kfkL1(logL)(Ω)
2π Zδ0
+ C1(q)ka0kq
2π Zδ0+δ1C2(q)ka0kq
(2π)3/2 Zδ20
= αZδ0+ Φδ0(Zδ0) =αZδ0,
(5.17)
i.e. kDWk2≤Zδ0, or in other termsW ∈B, which proves thatS(B)⊂B.
Second step : In this step we prove that S is continuous from H01(Ω) strongly into H01(Ω) strongly.
For this we consider a sequence such that
wn∈B, wn→w in H01(Ω) strongly, (5.18) and we defineWnasWn=S(wn), i.e. as the solution of (5.10).
The functionswn belong toB, and therefore the functionsWnbelong toB in view of the first step. We can therefore extract a subsequence, still denoted byn, such that for someW?∈H01(Ω),
Wn* W? inH01(Ω) weakly and a.e. in Ω. (5.19) We can moreover assume that for a further subsequence, still denoted byn,
wn* w a.e. in Ω and Dwn* Dw a.e. in Ω. (5.20) Since the assumptions of Lemma 5.3 are satisfied by the subsequenceswn andWn, the subsequenceWnconverges toW?inH01(Ω) strongly.
We now pass to the limit in equation (5.10) asn tends to infinity by using the fact that signk(s) and gδ0(s) sign(s) are Carath´eodory functions, and the result of almost everywhere convergence(see Lemma 3.3 in [20]) as far asTk(Kδ0(x, wn, Dwn)) is con- cerned (this point is the only point of the proof of Theorem 5.1 where the assumption
of strongH01(Ω) convergence in (5.18), or more exactly its consequence (5.20), is used).
This implies thatW?is a solution of (5.4). Since the solution of (5.4) is unique, one has W?=S(w).
In view of the fact that W? is uniquely determined, we conclude that it was not necessary to extract a subsequence in (5.19) and (5.20), and that the whole sequence Wn=S(wn) converges inH01(Ω) strongly toW?=S(w). This proves the continuity of the applicationS.
Third step : In this step we prove thatS(B) is precompact inH01(Ω).
For this we consider a sequencewn∈B and we defineWnasWn=S(wn); in other termsWnis the solution of (5.10). Sincewn andWn belong toB, they are bounded in H01(Ω), and we can extract a subsequence, still denoted byn, such that
wn* w in H01(Ω) weakly and a.e. in Ω, Wn* W? in H01(Ω) weakly and a.e. in Ω.
SincewnandWnsatisfies the assumptions of Lemma 5.3, we have Wn→W? in H01(Ω) strongly.
This proves thatS(B) is precompact inH01(Ω) (note that in contrast with the second step, we do not need here to prove thatW?=S(w)).
End of the proof of Theorem 5.1We have proved that the application S and the ballBsatisfy the assumptions of Schauder’s fixed point theorem. Therefore there exists at least onewk∈B such thatS(wk) =wk. This proves Theorem 5.1.
6. Proof of Theorem 4.3
Theorem 5.1 asserts that for for everyk >0 fixed there exists at least a solutionwk of (5.1) which satisfies (5.3). We can therefore extract a subsequence, still denoted byk, such that for somew?∈H01(Ω)
wk* w? in H01(Ω) weakly and a.e. in Ω, (6.1) wherew?satisfies
kw?kH1
0(Ω)≤Zδ0, (6.2)
i.e. (4.14).
In this Section we will first prove that for this subsequence
wk→w? in H01(Ω) strongly, (6.3)
and then that w? is a solution of (4.13) (which satisfies (4.14)). This will prove Theorem 4.3.
To prove (6.3), we use a technique which traces back to [7] (see also [15]).
For n >0, we define Gn : R→R as the remainder of the truncation at heightn, namely
Gn(s) =s−Tn(s), ∀s∈R (6.4)
whereTnis the truncation at heightndefined by (4.15), or in other terms Gn(s) =
s+n if s≤ −n, 0 if −n≤s≤n, s−n if s≥n.
(6.5) First we use the two following Lemmas.
Lemma 6.1. [15]Assume that(3.2), (3.3), (3.4), (3.5), (3.6)hold true. Assume moreover that the two smallness conditions(3.7)and(3.8)hold true. Letwkbe a solution of (5.1).
Assume finally that the subsequencewk satisfies(6.1).
Then for this subsequence we have lim sup
k→+∞
Z
Ω
|DGn(wk)|2dx→0 as n→+∞. (6.6) Lemma 6.2. [15]Assume that(3.2), (3.3), (3.4), (3.5), (3.6)hold true. Assume moreover that the two smallness conditions(3.7)and(3.8)hold true. Letwkbe a solution of (5.1).
Assume finally that the subsequencewk satisfies(6.1).
Then for this subsequence we have for everyn >0fixed
Tn(wk)→Tn(w?) in H01(Ω) strongly as k→+∞. (6.7) End of the proof of Theorem 4.3
First stepSince we have
wk−w?=Tn(wk) +Gn(wk)−Tn(w?)−Gn(w?), and since by Lemma 6.2 (see (6.7)) we have
kTn(wk)−Tn(w?)kH1
0(Ω)→0 as k→+∞ for every n >0 fixed, while by Lemma 6.1 (see (6.6)) we have
lim sup
n→+∞
lim sup
k→+∞
kGn(wk)kH1 0(Ω)= 0, and while we have
lim sup
n→+∞
kGn(w?)kH1 0(Ω)= 0, sincew?∈H01(Ω), we conclude that
wk→w? in H01(Ω) strongly as k→+∞, (6.8) which is nothing but (6.3).
Second stepLet us now pass to the limit in (5.1) asktends to infinity. This is easy for the first term of the left-hand side of (5.1) as well as for the three terms of the right-hand
side of (5.1), which is similar to the one that we used in the proof of Lemma 5.3.
It remains to pass to the limit in the second term of the left-hand side of (5.1), namely in
Tk Kδ0(x, wk, Dwk)
signk(wk).
We first observe thatKδ(x, s, ζ)>0 andδ0≥γ (see (7.9)), we have
|Tk Kδ0(x, wk, Dwk)
signk(wk)| ≤ |Kδ0(x, wk, Dwk)|
≤(c0+δ0)kAk∞|Dwk|2 a.e. in Ω, which implies that the functionsTk Kδ0(x, wk, Dwk)
signk(wk) are equiintegrable since Dwk converges strongly toDw?inL2(Ω)2.
Extracting if necessary a subsequence, still denoted byk, such that Dwk→Dw? a.e. in Ω,
we claim that
Tk Kδ0(x, wk, Dwk)
signk(wk)→Kδ0(x, w?, Dw?) sign(w?) a.e. in Ω. (6.9) On the first hand we use the following almost everywhere convergence, which asserts that
Kδ0(x, wk, Dwk)→Kδ0(x, w?, Dw?) a.e. in Ω, and the fact that for everys∈R
Tk(sk)→s if k→+∞ when sk→s, to deduce that
Tk Kδ0(x, wk, Dwk)
→Kδ0(x, w?, Dw?) a.e. in Ω. (6.10) On the other hand we use the fact that
signk(wk)→sign(w?) a.e. in {y∈Ω : w?(y)6= 0},
which together with (6.10) proves convergence (6.9) in the set{y∈Ω : w?(y)6= 0}.
Finally, as far as convergence in the set{y∈Ω :w?(y) = 0}is concerned, convergence (6.10), the fact that
Kδ0(x, w?, Dw?) = 0 a.e. in {y∈Ω : w?(y) = 0}, and the fact that|signk(s)| ≤1 for everys∈Rtogether prove that
Tk Kδ0(x, wk, Dwk)
signk(wk)→0 =Kδ0(x, w?, Dw?) sign(w?) a.e. in {y∈Ω : w?(y) = 0}.
This completes the proof of (6.9).
The equiintegrability and the almost everywhere convergence of the functions Tk Kδ0(x, wk, Dwk)
signk(wk) then imply that Tk Kδ0(x, wk, Dwk)
signk(wk)→Kδ0(x, w?, Dw?) sign(w?) in L1(Ω) strongly.
This proves thatw?satisfies (4.13). Sincew?also satisfies (4.14) (see (6.2)), Theorem 4.3 is proved.
7. Appendix
In this Appendix, we give an estimate of the function gδ defined by (4.6) (see Lemma 7.1), and the definitions of the constantsδ0 and Zδ0 which appear in Theorem 3.1 (see Lemma 7.2).
7.1. An estimate for the function gδ.
Lemma 7.1. Forδ >0, letgδ: R→Rbe the function defined by(4.6), i.e. by gδ(t) =−|t|+1
δ(1 +δ|t|) log(1 +δ|t|), ∀t∈R. (7.1) Then
0≤gδ(t)≤δ|t|2, ∀t∈R, ∀δ, 0< δ≤δ1. (7.2) Proof. Letg:R+→Rbe the function defined by
g(τ) =−τ+ (1 +τ) log(1 +τ), ∀τ≥0.
Sinceg(0) = 0 andg0(τ)≥0, one has
g(τ)≥0, ∀τ ≥0. (7.3)
Since log(1 +τ)< τ forτ >0, one has
g(τ)< τ2, ∀τ, τ >0. (7.4) Since
gδ(t) = 1
δg(δ|t|), ∀t∈R, one deduces from (7.4) thatgδ satisfies (7.2).
7.2. Definition ofδ0andZδ0. The goal of this Subsection is to define the constants δ0 andZδ0 which appear in Theorem 3.1. We will prove the following result.
Lemma 7.2. Assume that(3.2), (3.3), (3.4), (3.5) and (3.6)hold true. Assume moreover that the two smallness conditions(3.7)and(3.8)hold true.
Letδ1 be the number defined by
δ1= 2πα−C1(q)ka0kq
kfkL1(logL)(Ω)
. (7.5)
One has
δ1> γ. (7.6)
Forδ≥0, letΦδ : R+→R(see Fig. 2) be the function defined by Φδ(X) = δ1C2(q)ka0kqX2
−2πα−C1(q)kakq−δkfkL1(logL)(Ω)
2π X+kfkH−1(Ω),
(7.7)
whereC1(q)andC2(q)are the constants given in(3.9).
Then, for 0≤δ≤δ1, the function Φδ has a unique minimizerZδ onR+, which is given by
Zδ=2πα−C1(q)ka0kq−δkfkL1(logL)(Ω)
4πδ1C2(q)ka0kq
, f or 0≤δ≤δ1. (7.8) Moreover, there exists a unique numberδ0 such that
γ≤δ0< δ1, (7.9)
and
Φδ0(Zδ0) = 0. (7.10)
This number is the numberδ0 which appear in Theorem 3.1, andZδ0 is then defined fromδ0 through formula(7.8), namely by
Zδ0 =2πα−C1(q)ka0kq−δ0kfkL1(logL)(Ω)
4πδ1C2(q)ka0kq
. (7.11)
Remark 7.3. Let us explain the meaning of the results stated in Lemma 7.2.
As we will see in the proof of Lemma 7.2 (see also Fig. 2), the function Φδ is the restriction toR+ of a function which looks like a convex parabola. This function attains its minimum at a unique pointZδ, and forδwhich satisfiesδ < δ1withδ1 given by (7.5), one hasZδ>0.
The smallness condition (3.7) is equivalent to the fact thatδ1> γ, and the smallness condition (3.8) to the fact that the minimum Φγ(Zγ) of Φγ is nonpositive. Forδ=δ1, the minimum Φδ1(Zδ1) of Φδ1 is equal tokfkH−1(Ω), which is strictly positive. Therefore it can be proved that there exists someδ0 with γ ≤δ0 < δ1 (see (7.9)) such that the minimum Φδ0(Zδ0) of Φδ0 is equal to zero (see (7.10)), or in other terms such that the function Φδ0 has a double zero in Zδ0. Moreover, when γ < δ0, for every δ with γ ≤δ < δ0, the function Φδ has two distinct zeros Yδ− and Yδ+ with Yδ− < Yδ+ which satisfy 0< Yδ−< Zδ0< Yδ+
Fig. 1: The graph of the straight lineLδ Fig. 2: The graphs of the functions Φδ(X)
