1 Introduction
In this paper, dedicated to Laurent Veron, we study the so-called Maximum Principle for some quasilinear elliptic equations with lower order terms having natural (i.e., quadratic) growth with respect to the gradient.
The subject is strongly related to the papers by Laurent Veron and his coauthors [4, 5] and to a previous paper dedicated to Laurent for his sixtieth birthday (see [6]).
In an elliptic boundary value problem like
(1.1)
{
A
(
u
)
=
f
(
x
)
in
Ω
,
u
=
0
on
∂
Ω
,
where Ω is a bounded open set in ℝN, A is a second order elliptic operator in divergence form and f(x)≥0 (of course not zero a.e.), the Weak Maximum Principle states that u(x)≥0, and it is zero at most in a zero measure set; whereas the Strong Maximum Principle states that the set where u(x)=0 is even “smaller than a zero measure set” (e.g., empty).
In (1.1) the presence of a lower order term can destroy the Maximum Principle property (see [14] and the introduction of [12]). Nevertheless, we will prove the Strong Maximum Principle for some quasilinear elliptic equations with lower order terms having natural (i.e., quadratic) growth with respect to the gradient. One of the main motivations for the interest in quasilinear elliptic equations with lower order terms having quadratic growth with respect to the gradient comes from the Calculus of Variations (in the study of integral functionals). Indeed, consider the following examples:
J
(
v
)
=
1
2
∫
Ω
A
(
x
,
v
)
|
∇
v
|
2
+
λ
2
∫
Ω
v
2
-
∫
Ω
f
v
,
where Ω is a bounded open subset of ℝN, N>2, and λ>0. Then the Euler–Lagrange equation for J is (at least formally) the quasilinear elliptic problem
(1.2)
{
-
div
(
A
(
x
,
u
)
∇
u
)
+
1
2
A
′
(
x
,
u
)
|
∇
u
|
2
+
λ
u
=
f
in
Ω
,
u
=
0
on
∂
Ω
.
Similar problems (not necessarily Euler–Lagrange equations of some functional) have been studied in the literature (see, for example, [9, 10, 3, 7]), and existence of solutions has been proved under different assumptions on the datum f and on the function A. For example, if A′(x,s) is a bounded function, and f belongs to Lm(Ω), with m>N2, then there exist (see [9, 10]) solutions in W01,2(Ω)∩L∞(Ω) (the presence of the lower, zero-th order term is fundamental in order to prove existence). If one assumes a sign condition on A′(x,s), that is,
(1.3)
A
′
(
x
,
s
)
sgn
(
s
)
≥
ν
>
0
,
then one can prove existence of solutions in W01,2(Ω) under the assumption that f only belongs to L1(Ω) (see, for example, [3, 7, 8]).
An example of function A(x,s) satisfying (1.3) is
A
(
x
,
s
)
=
a
(
x
)
+
|
s
|
r
,
where 0<α≤a(x)≤β and r>1. In this case, problem (1.2) has been studied in [6], and becomes
(1.4)
{
-
div
(
[
a
(
x
)
+
|
u
|
r
]
∇
u
)
+
r
2
u
|
u
|
r
-
2
|
∇
u
|
2
=
f
in
Ω
,
u
=
0
on
∂
Ω
.
In this paper, we will prove that if f≥0 (and, of course, is not identically zero), then the Strong Maximum Principle holds for equations similar to (1.2) and (1.4), not necessarily Euler–Lagrange equations of some functional of the Calculus of Variations, assuming in Section 2 that the sign condition (1.3) does not hold (and we will consider “regular” data and bounded solutions), or in Section 3 that it holds (assuming L1(Ω) data and unbounded solutions). In the final Section 4, we will prove existence of solutions, and validity of the Strong Maximum Principle, for a quasilinear elliptic equation with a “control” between the datum f and the lower order term λ(x)u.
2 General Case and Bounded Solutions
In this section we will deal with a problem similar to (1.2), under no sign condition on the lower order, quadratic gradient term, and under the assumption that f≥0 is sufficiently “regular” in order to have bounded solutions.
More precisely, let Ω be a bounded, open subset of ℝN, N≥2, and let a(x), b(x) be measurable functions such that
(2.1)
0
<
α
≤
a
(
x
)
≤
β
and
(2.2)
|
b
(
x
)
|
≤
γ
,
where α, β, γ are positive real numbers. Let also f be a function such that
(2.3)
f
∈
L
m
(
Ω
)
,
m
>
N
2
.
In [10], existence of weak bounded solutions for general quasilinear Dirichlet problems with lower order terms having natural growth with respect to the gradient is proved. Here, in order to simplify our discussion, we confine ourselves to the simpler, but important case
(2.4)
u
∈
W
0
1
,
2
(
Ω
)
:
-
div
(
[
a
(
x
)
+
|
u
|
q
]
∇
u
)
+
λ
u
+
b
(
x
)
|
u
|
p
|
∇
u
|
2
=
f
,
under the assumptions λ>0, p,q≥0, (2.1), (2.2), and (2.3).
A weak solution u in W01,2(Ω)∩L∞(Ω) of (2.4) is a function such that
(2.5)
∫
Ω
[
a
(
x
)
+
|
u
|
q
]
∇
u
∇
ϕ
+
λ
∫
Ω
u
ϕ
+
∫
Ω
b
(
x
)
|
u
|
p
|
∇
u
|
2
ϕ
=
∫
Ω
f
ϕ
,
for every ϕ∈W01,2(Ω)∩L∞(Ω), and in [10] it is proved the existence of such a solution.
If, in addition, we suppose that f≥0, then u≥0.
Theorem 2.1 (Weak Maximum Principle).
If f≥0, then the weak solution u given by [10] is such that u≥0 almost everywhere in Ω.
Proof.
We rewrite (2.4) as
(2.6)
-
div
(
[
a
(
x
)
+
|
u
|
q
]
∇
u
)
+
λ
u
+
b
(
x
)
+
|
u
|
p
|
∇
u
|
2
=
y
(
x
)
,
where we have defined y(x)=f(x)+b(x)-|u|p|∇u|2. In the weak formulation of (2.6), we use as test function g(u), where the real function g(t) is defined as
g
(
t
)
=
{
0
if
t
≥
0
,
1
-
e
-
A
t
if
t
<
0
,
with A>0 to be chosen later. Note that g(u) is an admissible test function, since g(u) belongs to W01,2(Ω)∩L∞(Ω). Then
∫
Ω
[
a
(
x
)
+
|
u
|
q
]
|
∇
u
|
2
g
′
(
u
)
+
λ
∫
Ω
u
g
(
u
)
+
∫
Ω
b
(
x
)
+
|
u
|
p
|
∇
u
|
2
g
(
u
)
=
∫
Ω
y
(
x
)
g
(
u
)
.
Since g(u)=0, where u≥0, we have that
∫
{
u
<
0
}
[
a
(
x
)
+
|
u
|
q
]
|
∇
u
|
2
g
′
(
u
)
+
λ
∫
{
u
<
0
}
u
g
(
u
)
+
∫
{
u
<
0
}
b
(
x
)
+
|
u
|
p
|
∇
u
|
2
g
(
u
)
=
∫
{
u
<
0
}
y
(
x
)
g
(
u
)
,
that is, if M=∥u∥L∞(Ω),
∫
{
u
<
0
}
|
∇
u
|
2
[
α
g
′
(
u
)
]
+
λ
∫
{
u
<
0
}
u
g
(
u
)
≤
∫
{
u
<
0
}
y
(
x
)
g
(
u
)
+
∫
{
u
<
0
}
|
∇
u
|
2
[
-
γ
M
p
g
(
u
)
]
.
The choice A=γMpα implies αg′(u)+γMpg(u)=0, so that we have
λ
∫
{
u
<
0
}
u
g
(
u
)
≤
∫
{
u
<
0
}
y
(
x
)
g
(
u
)
≤
0
,
which can be rewritten as
λ
∫
{
u
<
0
}
u
e
A
u
-
1
e
A
u
≤
0
.
On the other hand, on the set {u<0}, u[eAu-1] is positive; this implies that the set {u<0} must have zero measure. ∎
If f≥0 (and not identically zero), then the solution u is not only positive, but the Strong Maximum Principle holds.
Theorem 2.2 (Strong Maximum Principle).
If f≥0 (and not almost everywhere equal to zero), then for every set ω⊂⊂Ω, there exists mω>0 such that u(x)≥mω almost everywhere in ω.
Proof.
By the previous result, u≥0, so it is a solution of
-
div
(
[
a
(
x
)
+
u
q
]
∇
u
)
+
b
(
x
)
u
p
|
∇
u
|
p
+
λ
u
=
f
.
Let 0≤ϕ∈W01,2(Ω)∩L∞(Ω) and let h(t) be the real decreasing function h(t)=e-γMpαt, where M=∥u∥L∞(Ω). In (2.5) (the weak formulation of (2.4)), we use ϕh(u) as test function to obtain
∫
Ω
[
a
(
x
)
+
u
q
]
|
∇
u
|
2
h
′
(
u
)
ϕ
+
∫
Ω
b
(
x
)
u
p
|
∇
u
|
2
h
(
u
)
ϕ
+
∫
Ω
[
a
(
x
)
+
u
q
]
h
(
u
)
∇
u
∇
ϕ
+
λ
∫
Ω
u
ϕ
h
(
u
)
=
∫
Ω
f
ϕ
h
(
u
)
.
Therefore (since h(t) is decreasing), using (2.1) we have
∫
Ω
|
∇
u
|
2
[
α
h
′
(
u
)
]
ϕ
+
∫
Ω
|
∇
u
|
2
[
γ
M
p
h
(
u
)
]
ϕ
+
∫
Ω
[
a
(
x
)
+
u
q
]
h
(
u
)
∇
u
∇
ϕ
+
λ
∫
Ω
u
h
(
u
)
ϕ
≥
∫
Ω
f
h
(
u
)
ϕ
.
Note that αh′(u)+γMph(u)=0, so that we have
∫
Ω
a
(
x
)
+
u
q
e
γ
M
p
α
u
∇
u
∇
ϕ
+
λ
∫
Ω
u
1
e
γ
M
p
α
u
ϕ
≥
∫
Ω
f
1
e
γ
M
p
α
u
ϕ
≥
∫
Ω
f
1
e
γ
M
p
+
1
α
ϕ
.
Observe that
0
≤
1
e
γ
M
p
α
u
≤
1
,
so that we have that u is a positive, bounded function such that
(2.7)
∫
Ω
a
(
x
)
+
u
q
e
γ
M
p
α
u
∇
u
∇
ϕ
+
λ
∫
Ω
u
1
e
γ
M
p
α
u
ϕ
≥
∫
Ω
f
1
e
γ
M
p
+
1
α
ϕ
,
for every ϕ≥0 in W01,2(Ω)∩L∞(Ω). Define
A
(
x
)
=
a
(
x
)
+
u
q
e
γ
M
p
α
u
,
and consider the solution w in W01,2(Ω)∩L∞(Ω) of the linear Dirichlet problem
(2.8)
∫
Ω
A
(
x
)
∇
w
∇
v
+
λ
∫
Ω
w
v
=
∫
Ω
f
e
γ
M
p
+
1
α
v
for all
v
∈
W
0
1
,
2
(
Ω
)
.
Since
α
0
=
α
e
γ
M
p
+
1
α
≤
a
(
x
)
+
u
q
e
γ
M
p
α
u
≤
β
+
M
q
=
β
0
,
the weak solution w exists, is bounded (see [13]) and satisfies the Strong Maximum Principle (see [11, 13, 14]), that is, there exists mω>0 such that
w
(
x
)
≥
m
ω
almost everywhere in
ω
.
Now we subtract the equation satisfied by w from inequality (2.7) satisfied by u, to have that
∫
Ω
A
(
x
)
∇
(
u
-
w
)
∇
ϕ
+
λ
∫
Ω
(
u
-
w
)
ϕ
≥
0
,
for every ϕ≥0 in W01,2(Ω)∩L∞(Ω). The choice ϕ=(u-w)-, and the fact that A(x)≥α0>0, implies that (u-w)-=0, that is, u≥w. Then, for every ω⊂⊂Ω, there exists a strictly positive constant mω such that u(x)≥mω>0 almost everywhere in ω, and the proof is complete. ∎
Remark 2.3.
Note that, even though the equation in (2.4) is nonlinear, the proof of the Strong Maximum Principle uses the fact that it holds for the linear equation (2.8) satisfied by the function w.
Remark 2.4.
Suppose now to have a positive, bounded supersolution of (2.4), that is, a function w≥0 in W01,2(Ω)∩L∞(Ω) such that
∫
Ω
[
a
(
x
)
+
w
q
]
∇
w
∇
ϕ
+
∫
Ω
b
(
x
)
w
p
|
∇
w
|
2
ϕ
+
λ
∫
Ω
w
ϕ
=
∫
Ω
g
ϕ
,
for every ϕ in W01,2(Ω)∩L∞(Ω), with g≥0 in Lm(Ω), m>N2; then the same proof of Theorem 2.2 yields that w satisfies the Strong Maximum Principle.
3 Lower Order Terms with Sign Condition and Unbounded Solutions
In this section we will assume a sign condition on the lower order, quadratic gradient term, and will suppose that f is a positive function belonging to L1(Ω). For k≥0 and t in ℝ, define
T
k
(
s
)
=
max
(
-
k
,
min
(
s
,
k
)
)
.
In [6], the existence of weak solutions for the Dirichlet problem
(3.1)
u
∈
W
0
1
,
2
(
Ω
)
:
-
div
(
[
a
(
x
)
+
|
u
|
q
]
∇
u
)
+
b
(
x
)
u
|
u
|
p
-
1
|
∇
u
|
2
=
f
,
under various assumptions on the datum f with respect to the two positive parameters p and q, was proved. More precisely, assuming (2.1) and that 0<ν≤b(x)≤γ, if f belongs to Lm(Ω), then we have the following:
or if 2N(q+1)2N+p(N-2)+4q≤m≤N2, and 2q≥p≥q-1,
or if 2NN+2≤m≤N2, and q≥1, 2p≥q-1≥p,
then there exists a weak solution of (3.1), that is, a function u in W01,2(Ω) such that
b
(
x
)
|
u
|
p
|
∇
u
|
2
∈
L
1
(
Ω
)
,
and
(3.2)
∫
Ω
[
a
(
x
)
+
u
q
]
∇
u
∇
φ
+
∫
Ω
b
(
x
)
u
p
|
∇
u
|
2
φ
=
∫
Ω
f
φ
,
for every φ∈W01,2(Ω)∩L∞(Ω).
As for problem (2.4), if f≥0, then u≥0.
Theorem 3.1 (Weak Maximum Principle).
If f≥0, then the solution u of (3.1) given by [6] is such that u≥0.
Proof.
In (3.2), take -T1(u-) as test function. Then
-
∫
Ω
[
a
(
x
)
+
u
q
]
∇
u
∇
T
1
(
u
-
)
-
∫
Ω
b
(
x
)
u
|
u
|
p
-
1
|
∇
u
|
2
T
1
(
u
-
)
=
-
∫
Ω
f
T
1
(
u
-
)
,
that is,
∫
Ω
[
a
(
x
)
+
|
u
|
q
]
∇
T
1
(
u
-
)
∇
T
1
(
u
-
)
+
∫
Ω
b
(
x
)
|
u
-
|
p
|
∇
u
-
|
2
T
1
(
u
-
)
≤
0
,
which implies T1(u-)≤0, for every k>0; then u-=0, and so u≥0. ∎
As in Section 2, the Strong Maximum Principle holds.
Theorem 3.2 (Strong Maximum Principle).
If f≥0 (and not almost everywhere equal to zero), then for every set ω⊂⊂Ω, there exists mω>0 such that u(x)≥mω almost everywhere in ω.
Proof.
Since u is positive by the previous result, we have that u is a solution of
∫
Ω
[
a
(
x
)
+
u
q
]
∇
u
∇
φ
+
∫
Ω
b
(
x
)
u
p
|
∇
u
|
2
φ
=
∫
Ω
f
φ
,
for every φ in W01,2(Ω)∩L∞(Ω). Let ε>0, and let, for t≥0,
Let also ϕ≥0 be a function in W01,2(Ω)∩L∞(Ω), and choose φ=ρε(u)ϕ as test function in the equation solved by u. We obtain, since ρε′(t)=-1ε for 1<t<1+ε and zero otherwise,
-
1
ε
∫
{
1
≤
u
≤
1
+
ε
}
[
a
(
x
)
+
u
q
]
|
∇
u
|
2
ϕ
+
∫
Ω
[
a
(
x
)
+
u
q
]
∇
u
∇
ϕ
ρ
ε
(
u
)
+
∫
Ω
b
(
x
)
u
p
|
∇
u
|
2
ρ
ε
(
u
)
ϕ
=
∫
Ω
f
ρ
ε
(
u
)
ϕ
.
We observe now that the first term is negative, so that we have
∫
Ω
[
a
(
x
)
+
u
q
]
∇
u
∇
ϕ
ρ
ε
(
u
)
+
∫
Ω
b
(
x
)
u
p
|
∇
u
|
2
ρ
ε
(
u
)
ϕ
≥
∫
Ω
f
ρ
ε
(
u
)
ϕ
≥
∫
Ω
T
1
(
f
)
ρ
ε
(
u
)
ϕ
.
Letting ε tend to zero, and observing that ρε(t) converges to the characteristic function of the set {0≤t≤1}, we thus have, by Lebesgue’s theorem,
∫
Ω
[
a
(
x
)
+
u
q
]
∇
u
∇
ϕ
χ
{
0
≤
u
≤
1
}
+
∫
Ω
b
(
x
)
u
p
|
∇
u
|
2
ϕ
χ
{
0
≤
u
≤
1
}
≥
∫
Ω
T
1
(
f
)
ϕ
χ
{
0
≤
u
≤
1
}
.
Since ∇T1(u)=∇uχ{0≤u≤1}, we can rewrite the above inequality as
∫
Ω
[
a
(
x
)
+
T
1
(
u
)
q
]
∇
T
1
(
u
)
∇
ϕ
+
∫
Ω
b
(
x
)
T
1
(
u
)
p
|
∇
T
1
(
u
)
|
2
ϕ
≥
∫
Ω
g
ϕ
,
where we have defined g=T1(f)χ{0≤u≤1}. Thus, we have proved that w=T1(u) is a bounded and positive supersolution of the equation
-
div
(
[
a
(
x
)
+
w
q
]
∇
w
)
+
b
(
x
)
w
p
|
∇
w
|
2
=
g
.
Thanks to Remark 2.4, w satisfies the Strong Maximum Principle. Therefore, for every ω⊂⊂Ω, there exists a constant mω>0 such that w(x)≥mω almost everywhere in ω. Recalling that w=T1(u), we thus have that
u
(
x
)
≥
T
1
(
u
(
x
)
)
=
w
(
x
)
≥
m
ω
>
0
almost everywhere in
ω
,
and the proof is complete. ∎
4 “Controlled” Lower Order Terms
In this section (following [1] and [2]) we will consider a slightly different equation from (2.4) or (3.1). Namely, we will consider the equation
(4.1)
-
div
(
[
a
(
x
)
+
u
q
]
∇
u
)
+
b
(
x
)
u
|
u
|
p
-
1
|
∇
u
|
2
+
λ
(
x
)
u
=
f
,
with a such that (2.1) holds, p,q≥0, b a measurable function such that
(4.2)
0
<
ν
≤
b
(
x
)
≤
γ
,
λ
(
x
)
a function in L1(Ω), and f a function such that
(4.3)
0
≤
f
(
x
)
≤
Q
λ
(
x
)
,
for some Q>0. Our result is the following.
Theorem 4.1.
Under assumptions (2.1), (4.2) and (4.3), there exists a solution u in W01,2(Ω)∩L∞(Ω) of (4.1). Furthermore, u satisfies the Strong Maximum Principle.
Proof.
For every n in ℕ there exists (see [7]) a solution un≥0 in W01,2(Ω) of
-
div
(
[
a
(
x
)
+
T
n
(
u
n
)
q
]
∇
u
n
)
+
b
(
x
)
u
n
p
|
∇
u
n
|
p
+
λ
(
x
)
u
n
=
f
,
that is, one has
∫
Ω
[
a
(
x
)
+
T
n
(
u
n
)
q
]
∇
u
n
∇
ϕ
+
∫
Ω
b
(
x
)
u
n
p
|
∇
u
n
|
2
ϕ
+
∫
Ω
λ
(
x
)
u
n
ϕ
=
∫
Ω
f
ϕ
,
for every ϕ in W01,2(Ω)∩L∞(Ω). Choosing ϕ=T1((un-Q)+) (which is admissible since ϕ belongs to W01,2(Ω)∩L∞(Ω)), we obtain
∫
{
Q
≤
u
n
≤
Q
+
1
}
[
a
(
x
)
+
T
n
(
u
n
)
q
]
|
∇
u
n
|
2
+
∫
Ω
b
(
x
)
u
n
p
|
∇
u
n
|
2
T
1
(
(
u
n
-
Q
)
+
)
+
∫
Ω
λ
(
x
)
u
n
T
1
(
(
u
n
-
Q
)
+
)
=
∫
Ω
f
T
1
(
(
u
n
-
Q
)
+
)
.
Since the first and the second term are positive, we thus have that
∫
Ω
λ
(
x
)
u
n
T
1
(
(
u
n
-
Q
)
+
)
≤
∫
Ω
f
T
1
(
(
u
n
-
Q
)
+
)
≤
Q
∫
Ω
λ
(
x
)
T
1
(
(
u
n
-
Q
)
+
)
,
which can be rewritten as
∫
Ω
λ
(
x
)
(
u
n
-
Q
)
T
1
(
(
u
n
-
Q
)
+
)
≤
0
.
Since (un-Q)T1((un-Q)+)≥0 almost everywhere, we have that
∫
Ω
λ
(
x
)
(
u
n
-
Q
)
T
1
(
(
u
n
-
Q
)
+
)
=
0
,
which then implies that 0≤un≤Q almost everywhere; that is, un belongs to L∞(Ω) and ∥un∥L∞(Ω)≤Q. We now choose n>Q and define u=un. Since 0≤u≤Q, we have Tn(u)=u, so that u is a solution in W01,2(Ω)∩L∞(Ω) of
-
div
(
[
a
(
x
)
+
u
q
]
∇
u
)
+
b
(
x
)
u
p
|
∇
u
|
p
+
λ
(
x
)
u
=
f
.
Since u is positive and bounded, we can repeat the same proof of Theorem 3.2 to prove that u satisfies the Strong Maximum Principle; that is, for every ω⊂⊂Ω, there exists mω>0 such that
u
(
x
)
≥
m
ω
almost everywhere in
ω
.
∎
Acknowledgements
The authors would like to thank Alessandro of the Campus Cafe for the endless supply of cappuccinos during the writing of the present paper.
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