Differential and Integral Equations
Volume xx, Number xxx, , Pages xx–xx
QUASILINEAR ELLIPTIC EQUATIONS
WITH NATURAL GROWTH
Boumediene Abdellaoui∗
Département de Mathématiques, Université Aboubekr Belkaı̈d, Tlemcen
Tlemcen 13000, Algeria
Lucio Boccardo
Dipartimento di Matematica, Università di Roma I
Piazza A. Moro 2, 00185 Roma, Italy
Ireneo Peral∗ and Ana Primo∗
Departamento de Matemáticas, U. Autónoma de Madrid
28049 Madrid, Spain
(Submitted by: James Serrin)
Abstract. In this paper we deal with the problem
−div (a(x, u)∇u) + g(x, u, ∇u) = λh(x)u + f in Ω,
u = 0 on ∂Ω.
The main goal of the work is to get hypotheses on a, g and h such that
the previous problem has a solution for all λ > 0 and f ∈ L1 (Ω). In
particular, we focus our attention in the model equation with a(x, u) =
1
(1 + |u|m ), g(x, u, ∇u) = m
|u|m−2 u|∇u|2 and h(x) =
.
2
|x|2
1. Introduction
This work is devoted to some results concerning to the following quasilinear elliptic equations with natural growth:
−div (a(x, u)∇u) + g(x, u, ∇u) = λh(x)u + f in Ω,
(1.1)
u = 0 on ∂Ω.
Assume that
0 ∈ Ω, bounded open set in RN with N ≥ 3,
(1.2)
λ ≥ 0, f ∈ L1 (Ω),
(1.3)
Accepted for publication: June 2007.
AMS Subject Classifications: 35D05, 35D10, 35J20, 35J25, 35J70, 46E30, 46E35.
∗
Partially supported by projects MTM2004-02223, MEC, Spain and CCG06UAM/ESP-0340, CAM-UAM..
1
2
Boumediene Abdellaoui, Lucio Boccardo, Ireneo Peral, and Ana Primo
and a(x, s) : Ω×R → R, g(x, s, ξ) : Ω×R×RN → R are measurable functions
with respect to x and continuous with respect to (s, ξ). We assume the existence of α, σ, μ ∈ R+ and β, θ continuous, increasing (possibly unbounded)
functions of a real variable such that
0 < α ≤ a(x, s) ≤ β(s),
(1.4)
|g(x, s, ξ)| ≤ θ(s)|ξ| ,
(1.5)
2
g(x, s, ξ)s ≥ 0,
∀ s ∈ R,
(1.6)
and for |s| ≥ σ we have
(1.7)
|g(x, s, ξ)| ≥ μ|ξ|2 .
We focus our attention on the model equation with a(x, u) = (1 + |u|m ),
m−2 u|∇u|2 and h(x) = 1 . We have the associated
g(x, u, ∇u) = m
2 |u|
|x|2
functional of energy
u2
λ
1
m
2
(1 + |u| )|∇u| dx −
dx −
f (x)u(x) dx, m > 1,
J(u) =
2 Ω
2 Ω |x|2
Ω
whose Euler-Lagrange equation is
m−2 u|∇u|2 = λ u + f in Ω,
−div ((1 + |u|m )∇u) + m
2 |u|
|x|2
u = 0 on ∂Ω.
(1.8)
For λ ≡ 0, equations of the form (1.1) have been widely studied in the
literature. We refer to [3], [7], [8] and the references therein. In the case
where m > 0 and λ = 0, in [6], the existence of finite-energy solutions
with only the weak assumption f ∈ L1 (Ω) is proved. This is thanks to the
presence of the lower-order term with quadratic dependence with respect to
the gradient and the condition (1.6). Notice that this result is somewhat
surprising because it is not true in the linear case (m = 0).
If λ > 0, m = 0 and f ∈ Lp (Ω), in [10] it is proved that the classical result
∗∗
u ∈ Lm (Ω) holds only if λ < [N (p − 1)(N − 2p)]/[p2 ].
In the case where a(x, u) ≡ 1 and g(x, u, ∇u) = |∇u|q , q > 1, and under
the hypothesis (2.1) on h, it is proved in [1] (see also [2]) that problem (1.1)
has a positive solution for all λ > 0 and for all f ∈ L1 (Ω) such that f ≥ 0.
This means that the term |∇u|q produces a strong regularizing effect and
then breaks down any resonance effect of the linear zero-order term.
The main goal of this work is to extend the previous results to a general
operator under the conditions (1.4)–(1.7). Since we will not assume any sign
condition on f , we will then prove the existence of a nontrivial solution for
all λ > 0 and f ∈ L1 (Ω). We point out that the gradient-dependent term g
Quasilinear elliptic equations with natural growth
3
improves the regularity of the solutions, whereas the zero-order term makes
smaller the summability of the solutions.
The paper is organized as follows. We start by analyzing the meaning of
the hypothesis (2.1) on h and giving some preliminary results that we will
use later. Section 3 is devoted to proving results of existence. We begin by
proving an approximating existence result by taking a regularized version
of the main problem. The result follows using closely the arguments of [15]
and [9]. The second part of the section is devoted to proving the main
existence result by using a priori estimates and then passing to the limit
in the approximating problem. In the last section we study the problem of
regularity of the solution depending on the regularity of f .
2. Preliminary results
We say that h is an admissible weight in the sense that the following
hypothesis holds:
1
2 dx 2
|∇φ|
Ω
h 0, h ∈ L1 (Ω) and C(h) =
inf
> 0. (2.1)
1,2
φ∈W0 (Ω)\{0}
Ω h|φ| dx
Remark 2.1. It is obvious that if h satisfies (2.1), then h ∈ W −1,2 (Ω) ∩
L1 (Ω). Moreover, (2.1) implies that
h |u| dx < ∞ for all u ∈ W01,2 (Ω).
a)
Ω
b) Defining H : W01,2 (Ω) → R by
H, u ≡
hu dx,
(2.2)
Ω
we have that H is a linear continuous form on W01,2 (Ω). Thus we
get the existence of F
= (f1 , f2 , . . . , fN ) ∈ (L2 (Ω))N such that h =
−div(F
), and then
hu dx =
F
, ∇u dx.
H, u ≡
Ω
Ω
As a consequence, the duality product is equivalent to the first integral and H = F
(L2 (Ω))N .
Proposition 2.2. Assume that h satisfies (2.1) and consider H defined by
(2.2).
4
Boumediene Abdellaoui, Lucio Boccardo, Ireneo Peral, and Ana Primo
Define hn (x) = min{h(x), n} and the corresponding linear continuous
form
R
Hn : W01,2 (Ω) →
u
→ Hn (u) =
hn u dx.
Ω
The following statements hold.
i) Hn → H strongly in W −1,2 (Ω).
ii) Assume that un u weakly in W01,2 (Ω), un ≥ 0 and un → u almost
everywhere; then hn un → hu strongly in L1 (Ω).
Proof. i) As in Remark 2.1 we also have that
Hn , u =
hn u dx;
Ω
that is, the duality is realized by the integral, and moreover
1
2 dx 2
|∇φ|
Ω
inf
> 0.
C(hn ) =
φ∈W01,2 (Ω)\{0}
Ω hn |φ| dx
Notice that
Hn W −1,2 =
u
≤
u
sup
≤1
1,2
W0 (Ω)
|Hn , u| ≤
sup
≤1
1,2
W0 (Ω)
Ω
u
sup
≤1
1,2
W0 (Ω)
|
hn u|
Ω
h|u| ≤ HW −1,2 .
Then {Hn }n∈N , up to a subsequence, converges weakly in W −1,2 (Ω). As for
all u ∈ W01,2 (Ω), the Lebesgue theorem proves that
hn |u| → h|u|
strongly in L1 (Ω), then {Hn }n∈N H.
Therefore,
HW −1,2 ≤ lim inf Hn W −1,2 ≤ lim sup Hn W −1,2 ≤ HW −1,2 ,
n→∞
n→∞
and hence,
Hn → H, strongly in W −1,2 (Ω).
ii) According to the strong convergence proved in i), we have that if
un u weakly in W01,2 (Ω), then
hn un dx →
hu dx = H, u as n → ∞.
Hn , un =
Ω
Ω
Quasilinear elliptic equations with natural growth
5
If we assume that, moreover, un ≥ 0 and un → u almost everywhere, then
by using a well-known result in [5] we obtain that hn un → hu strongly in
L1 (Ω). (See also [14], Theorem 1.9, page 21.)
3. Existence of solution
In this subsection we prove the existence of a solution to problem (1.1).
The existence is obtained for admissible weights h; i.e., h satisfies (2.1), for
all λ > 0 and f ∈ L1 (Ω).
The existence of solution will be proved by approximation. Our techniques
will follow those of [4] and [1]. Define
gn (x, s, ξ) =
g(x, s, ξ)
,
1 + n1 |g(x, s, ξ)|
where
Tk (s) =
and an (x, s) = a(x, Tn (s)),
s,
sk
|s| ,
if |s| ≤ k,
if |s| > k.
First, we consider the problem (1.1) with f, h ∈ Ls (Ω), s >
N
2.
Theorem 3.1. Assume (1.2), (1.4), (1.5), (1.6), (1.7), λ ≥ 0 and f, h ∈
Ls (Ω), with s > N2 . Then there exists a solution u ∈ W01,2 (Ω) ∩ L∞ (Ω) to
problem (1.1) in the distributional sense,
⎧
⎪
u
∈ W01,2 (Ω) ∩ L∞ (Ω), g(x, u, ∇u) ∈ L1 (Ω),
⎪
⎪
⎪
⎨
a(x, u)∇u∇ϕ dx +
g(x, u, ∇u) ϕ dx
(3.1)
Ω
Ω ⎪
⎪
⎪
1,2
⎪
f ϕ dx, ∀ϕ ∈ W0 (Ω).
= λ Ω h(x)uϕ dx +
⎩
Ω
Proof. Thanks to the Leray-Lions existence theorem ([12]), for every fixed
k > 0 we find a sequence of solutions {um,k } to the problems
−div (am (x, um,k )∇um,k ) + gm (x, um,k , ∇um,k )
(3.2)
um ∈ W01,2 (Ω) ∩ L∞ (Ω),
= λh(x)Tk (um ) + f,
for all m ∈ N. Using the fact that f, h ∈ Ls (Ω) for s > N2 and using the
Stampacchia regularity theorem [15] and the assumption (1.6), we get the
existence of a positive constant C such that
um,k ∞ ≤ C(k, g, f, h, Ω).
(3.3)
6
Boumediene Abdellaoui, Lucio Boccardo, Ireneo Peral, and Ana Primo
Taking now um,k as test function in (3.2), we obtain
|∇um,k |2 dx +
gm (x, um,k , ∇um,k )um,k dx
α
Ω
Ω
h(x)um,k dx +
f um,k dx
≤ λk
Ω
Ω
2
|∇um,k | dx + Cε λk +
f um,k dx.
≤ λkε
Ω
Ω
Hence we get the existence of a positive constant C(k, g, f, Ω) such that
α
|∇um,k |2 dx ≤ C(k, g, f, Ω);
Ω
therefore, um,k uk weakly in W01,2 (Ω). It is clear that
uk ∞ ≤ C(k, g, f, h, Ω),
where C(k, g, f, h, Ω) is defined in (3.3). To get the strong convergence of
{um,k } in W01,2 (Ω) we follow closely the arguments used in [9].
2
Let φ(s) = seT s , where T is a positive constant that we will choose later.
Using φ(um,k − uk ) as a test function in (3.2) we get
am (x, um,k )∇um,k φ (um,k − uk )∇(um,k − uk ) dx
Ω
gm (x, um,k , ∇um,k )φ(um,k − uk ) dx
(3.4)
+
Ω
= (λh(x)Tk (um,k ) + f )φ(um,k − uk ) dx.
Ω
Direct computation shows that
am (x, um,k )∇um,k φ (um,k − uk )∇(um,k − uk ) dx
Ω
φ (um,k − uk )|∇(um,k − uk )|2 dx + o(1).
≥α
Ω
Using the fact that um,k and uk are bounded and that um,k uk weakly in
W01,2 (Ω) we obtain
gm (x, um,k , ∇um,k )φ(um,k − uk ) dx
Ω
|∇um,k |2 θ(um,k )|φ(um,k − uk )|dx
(3.5)
≤
Ω
Quasilinear elliptic equations with natural growth
7
≤
|∇(um,k − uk )|2 θ(C)|φ(um,k − uk )|dx,
Ω
where we have used the fact that um,k ∞ ≤ C. Since
(λh(x)Tk (um,k ) + f )φ(um,k − uk ) dx = o(1) as
m → ∞,
Ω
we conclude that
{αφ (um,k − uk ) − θ(C)|φ(um,k − uk )|}|∇(um,k − uk )|2 dx ≤ o(1).
Ω
Choosing T large enough such that
αT s2 + α − θ(C)s ≥
we obtain that
1
2
α
,
2
|∇(um,k − uk )|2 dx ≤ o(1).
Ω
Hence, the strong convergence holds. Then using the fact that {un } is
bounded in L∞ (Ω) and the above strong convergence we can pass to the
limit in problem (3.2) to conclude that uk is a solution to
−div (a(x, uk )∇uk ) + g(x, uk , ∇uk ) = λh(x)Tk (uk ) + f,
(3.6)
uk ∈ W01,2 (Ω) ∩ L∞ (Ω),
with f, h ∈ Ls (Ω), s > N2 .
Taking uk as test function in (3.6) it follows that
|∇uk |2 dx +
|g(x, uk , ∇uk )uk | ≤ λ
h(x)u2k dx +
f uk dx.
α
Ω
Ω
Ω
Ω
Applying Sobolev’s and Young’s inequalities we obtain that for all ε, ε̄ > 0
there exist Cε , Cε̄ , β > 0 such that
β
|∇uk |2 dx ≤ λ Cε̄ h N + Cε f N = C(ε, ε̄, f, h, λ) uniformly in k
Ω
L
2
L
2
and, since f, h ∈ Ls (Ω) with s > N2 , we can prove that the sequence {uk } is
uniformly bounded in L∞ (Ω). Moreover, uk u in W01,2 (Ω). As a consequence, we also have that u ∈ W01,2 (Ω) ∩ L∞ (Ω).
To get the existence result we have to prove that uk → u in W01,2 (Ω), but
this follows immediately using previous computations.
8
Boumediene Abdellaoui, Lucio Boccardo, Ireneo Peral, and Ana Primo
Assume now that hypotheses (1.2), (1.3), (1.4), (1.5), (1.6) and (1.7) hold,
and let h(x) be an admissible weight in the sense of hypothesis (2.1). Thanks
to Theorem 3.1 we can consider the approximated Dirichlet problems,
un ∈ W01,2 (Ω) ∩ L∞ (Ω) :
(3.7)
− div (a(x, un )∇un ) + g(x, un , ∇un ) = λhn (x)Tk (un ) + fn ,
where {fn } is a sequence of smooth functions converging uniformly to f in
L1 (Ω) (for example we can take fn = Tn (f )) and hn (x) = min{h(x), n}.
The use in (3.7) of the test function Tk (un ) yields for any k > 0 (see [6]),
2
a(x, un )|∇Tk (un )| dx +
g(x, un , ∇un )Tk (un )
Ω
Ω
hn (x)un dx + k
fn dx.
≤ λk
Ω
Ω
Since
g(x, un , ∇un )Tk (un ) dx
g(x, un , ∇un )un dx + k
=
Ω
{|un (x)|≤k}
{|un (x)|≥k}
g(x, un , ∇un )
un
dx,
|un |
from (1.6) and (1.7), by setting k = σ, we have that for all ∀ε > 0
2
|∇Tσ (un )| + μσ
|∇un |2
α
Ω
{x∈Ω:σ≤|un (x)|}
≤ σλεC(h)
|∇un |2 dx + C (ε, k, f ).
Ω
Choosing ε small enough, we conclude that {un } is bounded in W01,2 (Ω),
and then there exists u ∈ W01,2 (Ω) such that un u weakly in W01,2 (Ω) and
un → u strongly in Lp (Ω), ∀p < 2∗ = N2N
−2 .
+
−
−
Consider un = max{un , 0} and un = max{−un , 0}, then un = u+
n − un .
We prove the following lemma.
Lemma 3.2. It follows that
lim
|g(x, un , ∇un )| dx
k→∞ {x∈Ω: |un (x)|≥k+1}
= lim
|∇un |2 dx = 0.
k→∞ {x∈Ω: |un (x)|≥k+1}
(3.8)
Quasilinear elliptic equations with natural growth
9
Proof. Using T1 (Gk (un )) as a test function in (3.7) and by the condition
(1.6) we obtain that for all k > 0,
|g(x, un , ∇un )| dx
(3.9)
{x∈Ω: k+1≤|un (x)|}
≤λ
hn (x)|un | dx +
|f | dx
{x∈Ω: k≤|un (x)|}
{x∈Ω: k≤|un (x)|}
so that the choice k = σ − 1 gives the following estimate:
|∇un |2 dx
μ
{x∈Ω:k+1≤|un (x)|}
≤λ
hn (x)|un | dx +
{x∈Ω: k≤|un (x)|}
{x∈Ω: k≤|un (x)|}
(3.10)
|f | dx.
The measure of the set {x ∈ Ω : |un (x)| ≥ k} tends to zero as k tends to
infinity uniformly in n, since un → u strongly in Lp (Ω), ∀p < 2∗ .
Proposition 3.3. Fixing k > 0, it follows that Tk (un ) → Tk (u) strongly in
W01,2 (Ω).
Proof. We denote wn = Tk (un ) and w = Tk (u). It is immediate that
wn − wW 1,2 (Ω)
(3.11)
0
≤ (wn+ − w+ )+ W 1,2 (Ω) + (wn− − w− )+ W 1,2 (Ω) + (wn − w)− W 1,2 (Ω) .
0
0
0
We estimate each term of the second member. On the one hand, for every
l > 0,
(wn+ − w+ )+ W 1,2 (Ω) = Tl (wn+ − w+ )+ W 1,2 (Ω) + Gl (wn+ − w+ )+ W 1,2 (Ω) .
0
0
0
(3.12)
The first term on the right hand can be estimated in the following way.
We take Tl (wn+ − w+ )+ as test function in (3.7):
+
+ + 2
α
|∇Tl (wn − w ) | dx +
gn (x, un , ∇un )Tl (wn+ − w+ )+ dx
Ω Ω
+
+ +
hn (x)un Tl (wn − w ) dx +
fn Tl (wn+ − w+ )+ dx.
≤λ
Ω
Ω
Thanks to condition (1.6), it immediately follows that for every l > 0,
Tl (wn+ − w+ )+ W 1,2 (Ω) goes to zero as l and n tend to infinity.
0
10
Boumediene Abdellaoui, Lucio Boccardo, Ireneo Peral, and Ana Primo
The second term on the right hand can be estimated in the following way.
For every l > 0,
|∇(wn+ − w+ )|2 dx
Gl (wn+ − w+ )+ W 1,2 (Ω) =
+
0
+
{x∈Ω:l≤wn (x)−w (x)}
+ 2
≤2
|∇wn | dx + 2
|∇w+ |2 dx.
+
{x∈Ω:l≤wn
(x)}
+
{x∈Ω:l≤wn
(x)}
From (3.10),
Gl (wn+
2λ
≤
μ
− w ) W 1,2 (Ω)
hn (x)wn+ dx
+
0
{x∈Ω: l−1≤wn (x)}
2
+
|f | dx + 2
|∇w+ |2 dx.
+
μ {x∈Ω: l−1≤wn+ (x)}
{x∈Ω:l≤wn (x)}
+ +
Since the measure of the set {x ∈ Ω : wn+ (x) ≥ l} tends to zero as l tends to
infinity uniformly in n, then passing to the limit as l and n tend to infinity
1,2
+
+
+ +
and since u+
n u weakly in W0 (Ω), it follows that Gl (wn −w ) W01,2 (Ω)
goes to zero.
Therefore, it follows that
lim (wn+ − w+ )+ W 1,2 (Ω) = 0.
n→∞
0
(3.13)
Following the same arguments, we have that
lim (wn− − w− )+ W 1,2 (Ω) = 0.
n→∞
0
(3.14)
We continue by estimating the third term on the right hand in (3.11).
2
Consider the function φ(s) = seT s , used above, with T > 0 to be chosen
later. Using φ((wn − w)− ) = φ((w − wn )+ ) as test function in (3.7) it follows
that
a(x, un )∇un φ ((w − wn )+ )∇((w − wn )+ ) dx
(3.15)
Ω
g(x, un , ∇un )φ((w − wn )+ ) dx ≤
(λh(x)un + fn )φ((w − wn )+ ) dx.
+
Ω
Ω
We analyze term by term in the previous equation (3.15). On the left hand,
it follows that
a(x, un )∇un φ ((w − wn )+ )∇((w − wn )+ ) dx
Ω
a(x, un )∇un φ ((w − wn )+ )∇((w − wn )) dx
=
{x∈Ω:wn ≤w≤k}
Quasilinear elliptic equations with natural growth
=
+
{x∈Ω:wn ≤w≤k}
{x∈Ω:wn ≤w≤k}
11
a(x, un )|∇((w − wn ))|2 φ ((w − wn )+ ) dx
a(x, un )∇wφ ((w − wn )+ )∇((w − wn )) dx.
Since un u in W01,2 (Ω), then arguing by duality we conclude that
a(x, un )∇un φ ((w − wn )+ )∇((w − wn )+ ) dx
(3.16)
Ω
a(x, un )|∇((wn − w)− )|2 φ ((w − wn )+ ) dx + o(1).
=
Ω
Now we estimate the second term on the left hand. Taking φ((w − wn )+ )
as a test function in (3.7),
g(x, un , ∇un )φ((w − wn )+ ) dx
Ω
g(x, un , ∇un )φ((w − wn )+ ) dx
=
{x∈Ω:wn ≤w≤k}
≤
θ(un )|∇wn |2 φ((w − wn )+ ) dx
{x∈Ω:wn ≤w≤k}
=
θ(un )|∇(wn − w)|2 φ((w − wn )+ )dx
{x∈Ω:wn ≤w≤k}
−
θ(un )|∇w|2 φ((w − wn )+ )dx
{x∈Ω:wn ≤w≤k}
+2
θ(un )∇wn ∇w φ((w − wn )+ )dx.
{x∈Ω:wn ≤w≤k}
For every k > 0, we consider A = {x ∈ Ω : wn (x) ≤ w(x) ≤ k} and M =
maxs∈A θ(s). Recall that θ is continuous increasing function given in (1.5)
and that wn = Tk (un ) and w = Tk (u). Since φ((w − wn )+ ) converges to zero
in Lp (Ω), ∀p > 1, and thanks to an argument by duality, we have that
g(x, un , ∇un )φ((w − wn )+ ) dx
(3.17)
Ω
|∇((wn − w)− )2 |φ((w − wn )+ )|dx.
≤M
{x∈Ω:wn ≤w≤k}
12
Boumediene Abdellaoui, Lucio Boccardo, Ireneo Peral, and Ana Primo
From (3.16) and (3.17), we conclude that
|∇((wn − w)− )2 |(αφ − M |φ|)((w − wn )+ ) dx = o(1).
Ω
Choosing T in such a way that φ satisfies (αφ − M |φ|) ≥ 12 , it follows that
for every k > 0,
lim (wn+ − w+ )− W 1,2 (Ω) = 0.
n→∞
0
(3.18)
From (3.13), (3.14), and (3.18) we conclude that for every fixed k > 0 we
have that
Tk (un ) → Tk (u) strongly in W01,2 (Ω).
Proposition 3.4. The sequence g(x, un , ∇un ) converges to g(x, u, ∇u) in
L1 (Ω).
Proof. Since g(x, un , ∇un ) → g(x, u, ∇u) almost everywhere, we prove that
g(x, un , ∇un ) is uniformly equi-integrable. For any σ ∈ R+ ,
|g(x, un , ∇un )| dx
E
2
θ(σ)|∇Tσ (un )| dx +
|g(x, un , ∇un )| dx.
≤
{x∈E:|un (x)|≥σ}
E
Hence, choosing E with |E| small enough and applying Lemma 3.2 we obtain
the conditions to apply Vitali’s theorem, and we conclude.
Remark 3.5. In the same way, we can prove that |∇un |2 converges to |∇u|2
strongly in L1 (Ω). Since |∇un |2 converges to |∇u|2 almost everywhere, we
prove that |∇un |2 is uniformly equi-integrable. For any σ ∈ R+ ,
2
2
|∇un | dx =
|∇Tσ (un )| dx +
|∇un |2 dx.
E
E
{x∈E:|un (x)|≥σ}
Hence, choosing E with |E| small enough and applying Lemma 3.2 we get
the main conditions to apply Vitali’s Theorem, and then the result follows.
Thus, we can now pass to the limit in (3.7) to obtain that u is a solution
to (1.1) in a suitable sense. Assume ϕ ∈ W01,2 (Ω) ∩ L∞ (Ω); then using the
above regularity results we can use Tk [u − ϕ] as test function in (1.1). More
precisely, we have proved the next existence result.
Quasilinear elliptic equations with natural growth
13
Theorem 3.6. Assume (1.2), (1.3), (1.4), (1.5), (1.6) and (1.7), and let
h(x) be an admissible weight. There exists a solution u to problem (1.1) in
the following sense:
⎧
⎪
u
∈ W01,2 (Ω), g(x, u, ∇u) ∈
L1 (Ω),
⎪
⎪
⎪
⎨
a(x, u)∇u∇Tk [u − ϕ] +
g(x, u, ∇u) Tk [u − ϕ]
(3.19)
Ω Ω
⎪
⎪
⎪
1,2
∞
⎪
f Tk [u − ϕ], ∀ϕ ∈ W0 (Ω) ∩ L (Ω), ∀k > 0.
⎩ =
Ω
Remark 3.7. Notice that in this case we do not know in general if
a(x, u)|∇u|2 ∈ L1 (Ω) nor that ug(x, u, ∇u) ∈ L1 (Ω).
It is clear that in general, if u is a solution to problem (1.1) in the sense
of Theorem 3.6, then we don’t know if u is a distributional solution. To
overcome this difficulty we need to add extra conditions on a. More precisely,
we have the next result.
Theorem 3.8. Let α < 1 and assume that a(x, s) satisfies the following
hypothesis:
⎧ 2α
⎨ a (x, s)|ξ|2 ≤ C|g(x, s, ξ)|, ∀|s| ≥ σ,
(H) ≡
(3.20)
⎩ 2(1−α)
(x, s) ≤ C|s|β , β < 2∗ , ∀|s| ≥ σ.
a
Then if u ∈ W01,2 (Ω) is a solution to problem (1.1) in the sense of (3.19),
we have that u is a solution in the distributional sense.
Proof. It is sufficient to prove that a(x, un )|∇un | converges to a(x, u)|∇u|
in L1 (Ω). Let E ⊂ Ω be a measurable set,
a(x, un )|∇un | dx
E
a(x, un )|∇Tσ (un )| dx +
a(x, un )|∇un | dx.
=
{x∈E:|un (x)|≥σ}
E
Hence, choosing E with |E| small enough and since Tk (un ) → Tk (u) in
W01,2 (Ω), it follows that
a(x, un )|∇Tσ (un )| dx → 0 uniformly in n.
(3.21)
E
Let us analyze the second term on the right hand in the equation (3.21).
a(x, un )|∇un | dx
{x∈E:|un (x)|≥σ}
14
Boumediene Abdellaoui, Lucio Boccardo, Ireneo Peral, and Ana Primo
≤
a (x, un )|∇un | dx +
2α
{x∈E:|un (x)|≥σ}
2
{x∈E:|un (x)|≥σ}
a2(1−α) (x, un ) dx.
Since a(x, s) satisfies hypothesis (H) in (3.20), then using the fact that un →
u strongly in Lp (Ω), ∀p < 2∗ , and thanks to Lemma 3.2, it follows that
choosing E with |E| small enough,
a(x, un )|∇un | dx → 0 uniformly in n.
{x∈E:|un (x)|≥σ}
Applying Vitali’s theorem, we conclude the proof.
Remark 3.9. Let us consider the main example given in (1.8); then, applying the previous Theorem 3.8, we obtain that
m−1 |∇u|2 , ∀|u| ≥ σ,
(1 + |u|m )2α |∇u|2 ≤ C m
2 |u|
(1 + |u|m )2(1−α) ≤ C|u|β , β < 2∗ , ∀|u| ≥ σ.
1,2
∗
Let α = m−1
2m < 1; then if 1 < m < 2 − 1 and u ∈ W0 (Ω) is a solution
to problem (1.8) in the entropy sense, we have that u is a solution in the
distributional sense.
3.1. Regularity of solutions. According to the previous section, there
exists a solution u ∈ W01,2 (Ω) to problem (1.1) for all f ∈ L1 (Ω). In this
section, we report the extra summability of u given by extra summability of
f , under the next stronger hypothesis on h. It is worth pointing out that
the one order term provides an extra summability, beyond of that given by
the classical Stampacchia results. Let consider the problem (1.1) with h
satisfying the condition
|∇φ|2 dx
1
Ω
h(x) ≥ 0, h(x) ∈ L (Ω), λ1 (h) =
inf
> 0. (3.22)
2
φ∈W01,2 (Ω)\{0} Ω h|φ| dx
In particular, notice that condition (3.22) implies condition (2.1). Consider
f ∈ Lm (Ω), and let’s distinguish two cases according to duality of f : m ≥
2N
2N
N +2 and 1 < m < N +2 .
The proofs are similar to those in [1] (see also [11]) and thus we will skip
them.
Theorem 3.10.
(1) Assume that f ∈ Lm (Ω) with m ≥ N2N
+2 and that hypothesis (3.22)
1,2
holds. Let u ∈ W0 (Ω) be a solution to problem (1.1). Then,
|u|q
|u|p |∇u|2 dx < ∞, for all 1 ≤ p < 2, and
dx < ∞, ∀q < 4.
2
Ω
Ω |x|
Quasilinear elliptic equations with natural growth
15
(2) Assume that f ∈ Lm (Ω) with 1 < m < N2N
+2 , and suppose that
1,2
hypothesis (2.1) holds for h. Let u ∈ W0 (Ω) be a solution to problem
(1.1); then
2∗ 2(m − 1)
2N
up |∇u|2 dx < ∞, ∀p < p̄ =
, 2∗ =
.
∗
2m − 2 (m − 1)
N −2
Ω
Theorem 3.11. Assume that f ∈ Lm (Ω) with m ≥
be a solution to problem (1.1) with h(x) = |x|1 2 .
2N
N +2 .
Let u ∈ W01,2 (Ω)
1,2
N
p
i) If
2 , then u ∈ W0 (Ω) for all p > 1; in particular, we have
m≥
up
dx < ∞, ∀p ≥ 1 and α < N.
α
Ω |x|
m(N −2)
N
1 2∗
ii) If N2N
+2 ≤ m < 2 , then by setting r = 2 2∗ + 1 ≡ N −2m ≥ 2 we
m−
have up ∈ W01,2 (Ω), for all p ≤ r.
2
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