This work deals with the system (−Δ)mu = a(x) vp, (−Δ)mv = b(x) uq with Dirichlet boundary condition in a domain Ω⊂Rn , where Ω is a ball if n ≥ 3 or a smooth perturbation of a ball when n = 2. We prove that, under appropriate conditions on the parameters (a, b, p, q, m, n), any nonnegative solution (u, v) of the system is bounded by a constant independent of (u, v). Moreover, we prove that the conditions are sharp in the sense that, up to some border case, the relation on the parameters are also necessary. The case m = 1 was considered by Souplet (Nonlinear Partial Differ Equ Appl 20:464–479, 2004). Our paper generalize to m ≥ 1 the results of that paper.Facultad de Ciencias ExactasConsejo Nacional de Investigaciones Científicas y Técnica

Durán, Ricardo Guillermo

Sanmartino, Marcela

Toschi, Marisa

Annali di Matematica Pura ed Applicata (1923 -)

English

SEDICI - Repositorio de la UNLP

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ON THE EXISTENCE OF BOUNDED SOLUTIONS FOR A NONLINEAR
ELLIPTIC SYSTEM
RICARDO G. DURA´N, MARCELA SANMARTINO, AND MARISA TOSCHI
Abstract. This work deals with the system (−∆)mu = a(x) vp, (−∆)mv = b(x)uq with
Dirichlet boundary condition in a domain Ω ⊂ IRn, where Ω is a ball if n ≥ 3 or a smooth
perturbation of a ball when n = 2.
We prove that, under appropriate conditions on the parameters (a, b, p, q,m, n), any non-
negative solution (u, v) of the system is bounded by a constant independent of (u, v). More-
over, we prove that the conditions are sharp in the sense that, up to some border case, the
relation on the parameters are also necessary.
The case m = 1 was considered by Souplet in [7]. Our paper generalize to m ≥ 1 the
results of that paper.
1. Introduction
In this paper we consider the nonlinear problem


(−∆)mu = a(x) vp in Ω
(−∆)mv = b(x)uq in Ω(
∂
∂ν
)j
u =
(
∂
∂ν
)j
v = 0 on ∂Ω 0 ≤ j ≤ m− 1,
(1.1)
where Ω is the unit ball, namely, Ω = B = {x ∈ Rn : |x| < 1} when n ≥ 3, and B or
some perturbations of B for the case n = 2 (see [4] for details of this perturbation), ∂∂ν is
the normal derivative, p, q > 0, pq > 1, and a, b are nonnegative bounded functions. Let us
remark that the restriction on the domains is due to the fact that we will use that the Green
function of the corresponding linear problem is positive.
For the case m = 1, a priori bounds for non-negative solutions of (1.1) in a C2 bounded
domain Ω were obtained by P. Souplet in [7]. To recall the results in that paper we introduce
α =
2(p + 1)
pq − 1
and β =
2(q + 1)
pq − 1
.
Souplet proved that, if max{α, β} > n− 1, then
‖u‖L∞(Ω), ‖v‖L∞(Ω) ≤ C, (1.2)
where the constant C depends only on p, q, a, b, and Ω.
Moreover, he proved that the result is sharp in the sense that, if max{α, β} < n− 1, then
there exist a non-negative solutions of (1.1) which are not bounded.
Our goal is to obtain similar results for non-negative solutions of (1.1) for a general m.
A key tool used in [7] are some weighted a priori estimates for the associated linear problem
given by {
−∆u = f in Ω
u = 0 on ∂Ω.
Supported by ANPCyT (PICT 01307), by Universidad de Buenos Aires (grant X070), by Universidad
Nacional de La Plata (grant X500), and by CONICET (PIP 11220090100625). The first author is a member
of CONICET, Argentina.
1
2Then, in order to generalize the a priori estimates for the case m ≥ 2 we will need to extend
the weighted estimates to higher order linear problems. Non trivial technical modifications
are needed to prove those estimates. Moreover, since we need to use positivity of the Green
function, we have to restrict the domain Ω as mentioned above. Indeed, for m ≥ 2 and
general regions the Green function is not necessarily positive.
2. Weighted a priori estimates for the linear problem
We will denote by d(x) the distance from x to the boundary of Ω and we will work with
the Banach space Lpdm(Ω) where the norm is given by
‖u‖Lp
dm
(Ω) =
(∫
Ω
|u|p dm dx
)1/p
for 1 ≤ p <∞ and ‖u‖L∞
dm
(Ω) := ‖u‖L∞(Ω).
In our arguments we will use some results given in [3] for the linear problem{
(−∆)mu = f in Ω(
∂
∂ν
)j
u = 0 on ∂Ω 0 ≤ j ≤ m− 1.
(2.1)
We recall those results in the following lemma. Let us remark that these results, and conse-
quently our proposition below, are valid in more general domains than those considered here.
Indeed, the hypotheses used are that Ω is a bounded domain with C6m+4 boundary for n = 2
and C5m+2 boundary for n > 2.
Lemma 2.1. Let u ∈ C2m(Ω) and f ∈ C(Ω) satisfy (2.1).
• If 2m > n, then there exists C > 0 such that for all θ ∈ [0, 1]
‖u d−m+θn‖L∞(Ω) ≤ C ‖f d
m−(1−θ)n‖L1(Ω).
• Let 1 ≤ p ≤ q ≤ ∞. If 1p −
1
q < min{
2m
n , 1}, then taking α ∈ (
1
p −
1
q ,min{
2m
n , 1}] there
exists C > 0 such that for all θ ∈ [0, 1]
‖u d−m+θnα‖Lq(Ω) ≤ C ‖f d
m−(1−θ)nα‖Lp(Ω).
Proof : See Proposition 4.2 in [3].
We have the following a priori estimates for solutions of problem (2.1).
Proposition 2.2. Let 1 ≤ p ≤ q ≤ ∞. Let f ∈ Lpdm(Ω) and let u be a weak solution of (2.1).
We have
(1) if n ≤ m, then u ∈ L∞(Ω) and there exists C > 0 such that
‖u‖L∞(Ω) ≤ C ‖f‖L1
dm
(Ω)
(2) if 1p −
1
q <
2m
n+m , then u ∈ L
p
dm(Ω) and there exists C > 0 such that
‖u‖Lq
dm
(Ω) ≤ C ‖f‖Lp
dm
(Ω).
Proof : From Lemma 2.1 we have that, for 2m > n and θ ∈ [0, 1],
‖u d−m+θn‖L∞(Ω) ≤ C ‖f d
m−(1−θ)n‖L1(Ω). (2.2)
Then taking θ = 1 and using that −m+ n < 0 and d(x) ≤ diam(Ω) we obtain
‖u‖L∞(Ω) ≤ C ‖u d
−m+n‖L∞(Ω) ≤ C ‖f d
m‖L1(Ω)
and so (1) is proved.
3On the other hand, using again Lemma 2.1, we have that, if there exists α ∈ (1p −
1
q ,min{1,
2m
n }] and θ ∈ [0, 1] such that{
−m+ θ nα = mq
m− (1− θ)nα = mp
(2.3)
we obtain
‖u‖Lqdm (Ω)
≤ C ‖f‖Lpdm (Ω)
for 1p −
1
q < min{1,
2m
n }.
Solving system (2.3) we obtain
α = (2 +
1
q
−
1
p
)
m
n
and θ = (
1
q
+ 1) (2 −
1
p
+
1
q
)−1.
We are going to show that α and θ satisfy the required conditions if 2m−nm ≤
1
p −
1
q <
2m
n+m .
Since 1 ≤ p we have θ ∈ [0, 1]. On the other hand, from the definition of α, it is easy to see
that the condition 1p −
1
q < α is equivalent to
1
p −
1
q <
2m
n+m , which is one of our hypothesis.
Finally we have to see that α ≤ min{1, 2mn }. Since p ≤ q we have α ≤
2m
n . Therefore, it
only remains to consider the case 2mn > 1. But α ≤ 1 is equivalent to
2m−n
m ≤
1
p −
1
q and so
the proposition is proved under this restriction.
Suppose now that 1p −
1
q <
2m−n
m . In this case, for 2m > n, using again the first part of
Lemma 2.1, for all θ˜ ∈ [0, 1] we have
‖u d−m+θ˜n‖L∞(Ω) ≤ C ‖f d
m−(1−θ˜)n‖L1(Ω).
Moreover, if θ˜ ≤ mnq +
m
n , it follows that
‖u‖Lq
dm
(Ω) ≤ ‖u d
−m+θ˜n‖L∞(Ω).
Analogously, if 1− mn +
m
np ≤ θ˜,
‖f dm−(1−θ˜)n‖L1(Ω) ≤ C‖f‖Lp
dm
(Ω).
Therefore, if we can choose θ˜ satisfying 1− mn +
m
np ≤ θ˜ ≤
m
nq +
m
n we have
‖u‖Lq
dm
(Ω) ≤ C ‖f‖Lp
dm
(Ω),
but, such a θ˜ exists because 1p −
1
q ≤
2m−n
m and the proposition is proved. 
Remark 2.3. The condition in (2) is almost optimal, i.e., if 1p −
1
q >
2m
n+m then the a priori
estimate does not hold in general. We postpone the proof of this observation to the end of
the paper because we will use the same technique as in the proof of our second main theorem.
In the proof of the following proposition we will denote with λ1,m the first eigenvalue of the
operator (−∆)m and with φ1,m > 0 a corresponding eigenfunction normalized by
∫
Ω φ1,m = 1.
We will use that there exist two positive constants c1 and c2 such that, in Ω,
c1 d
m ≤ φ1,m ≤ c2 d
m, (2.4)
see [2].
Proposition 2.4. Let 1 ≤ k < n+mn−m . If u is a solution of (2.1) with f ∈ L
1
dm(Ω) and f > 0,
then there exists C > 0 such that
‖u‖Lkdm (Ω)
≤ C ‖u‖L1
dm
(Ω). (2.5)
4Proof : Taking p = 1 in the previous proposition we obtain for 1 ≤ k < n+mn−m
‖u‖Lk
dm
(Ω) ≤ C ‖f‖L1dm (Ω)
.
Using integration by parts and that f > 0 we have
‖f‖L1
dm
=
∫
Ω
(−∆)mu dm dx ≤
∫
Ω
(−∆)muφ1,m dx
≤ C
∫
Ω
u (−∆)mφ1,m dx = C λ1,m
∫
Ω
uφ1,m dx
≤ C
∫
Ω
u dm dx ≤ C ‖u‖L1
dm
.

3. Main results
We consider problem (1.1) and define the exponents
α =
2m(p + 1)
pq − 1
and β =
2m(q + 1)
pq − 1
.
Then, the natural extension of the results in [7] is given by the following
Theorem 3.1. If
max(α, β) > n−m, (3.1)
then, any non-negative solution of (1.1) satisfies
‖u‖L∞(Ω), ‖v‖L∞(Ω) ≤ C (3.2)
where C is a positive constant which depends only on a, b, p, q, m, and Ω.
We also prove, in the following theorem, that condition (3.1) is almost optimal. We cannot
say optimal because we do not know what happens in the case max(α, β) = n−m.
Theorem 3.2. If
max(α, β) < n−m, (3.3)
then, there exist nonnegative bounded functions a and b, such that (1.1) have some non-
negative solution (u, v), with u and v unbounded functions.
Once we have the results of the previous section, the proofs of both theorems follows the
lines of the case m = 1 proved in [7]. A key point in the arguments given in that paper are
the estimates ∫
Ω
uφ1,m ,
∫
Ω
v φ1,m ≤ C. (3.4)
A straightforward extension of the arguments given in [8], to prove these estimates in the case
m = 1, is not possible. Indeed, the proof given in that paper is based on a lemma of [1] which
uses the maximum principle in subsets of Ω. An analogous maximum principle is not valid in
the case m ≥ 2. We will give a different proof of this lemma using pointwise estimates for the
Green function Gm of problem (2.1) given below. This is why we have to restrict Ω in order
to have that the Green function is positive, i.e., we assume that Ω = B = {x ∈ Rn : |x| < 1}
when n ≥ 3, and Ω = B or some perturbations of B for the case n = 2 (see [4] for details of
this perturbation). We have: for 2m < n,
Gm(x, y) ≥ C |x− y|
2m−n min
{
1,
d(x)m d(y)m
|x− y|2m
}
, (3.5)
for 2m = n,
Gm(x, y) ≥ C log
(
1 +
d(x)m d(y)m
|x− y|2m
)
≥ C log
(
2 +
d(y)
|x− y|
)
min
{
1,
d(x)m d(y)m
|x− y|2m
}
, (3.6)
5and for 2m > n,
Gm(x, y) ≥ C d(x)
m−n/2 d(y)m−n/2 min
{
1,
d(x)n/2 d(y)n/2
|x− y|n
}
. (3.7)
The proofs of these estimates can be found in [4] for the case of m = n = 2 and in [6] for
the rest of the cases.
Lemma 3.3. Assume h ≥ 0, h ∈ L1dm(Ω) and v a solution of{
(−∆)mv = h in Ω(
∂
∂ν
)j
v = 0 on ∂Ω 0 ≤ j ≤ m− 1.
(3.8)
Then there exists C > 0, depending only on Ω and m, such that for all x ∈ Ω
v(x)
dm(x)
≥ C
∫
Ω
hdm. (3.9)
Proof : By the representation formula
v(x) =
∫
Ω
Gm(x, y)h(y) dy
it is enough to prove that
Gm(x, y) ≥ C d(x)
m d(y)m.
Consider, for example, the case 2m < n and suppose that d(x)
m d(y)m
|x−y|2m
≥ 1. Then, it follows
from (3.5), that
Gm(x, y) ≥ C |x− y|
2m−n ≥ d(x)m−n/2d(y)m−n/2 ≥ Cd(x)md(y)m
where in the last step we have used that Ω is bounded. On the other hand, if the minimum
on the right hand side of (3.5) is attained in d(x)
m d(y)m
|x−y|2m we have
Gm(x, y) ≥ C |x− y|
−nd(x)md(y)m ≥ Cd(x)md(y)m.
The proofs for the cases 2m = n and 2m > n are analogous, using now (3.6) and (3.7)
respectively. 
Proof of Theorem 3.1:
Step 1: From (3.4) and (2.4) it follows immediately,
‖u‖L1dm
+ ‖v‖L1dm
≤ C, (3.10)
and therefore, for n ≤ m,
‖u‖L∞(Ω), ‖v‖L∞(Ω) ≤ C
is an immediate consequence of (1) in Proposition 2.2.
On the other hand, if n > m, it follows from Proposition 2.4, that
‖u‖Lkdm
+ ‖v‖Lkdm
≤ C(k) (3.11)
for 1 ≤ k < n+mn−m .
Clearly we may assume q ≥ p and β > n−m. Then, it is easy to check that p < n+mn−m and
so, there exists some k such that
k ≥ p and k ≥
n+m
n−m
− ǫ, (3.12)
with ǫ to be chosen below, for which (3.11) holds.
Step 2: Assume now that we can choose k1 ∈ (k, ∞] such that
1
k1
>
p
k
−
2m
n+m
. (3.13)
6Then, using Proposition 2.2 we have
‖u‖
L
k1
dm
≤ C ‖(−∆)mu‖
L
k/p
dm
≤ C ‖vp‖
L
k/p
dm
= C ‖v‖p
Lk
dm
, (3.14)
which is finite because 1 ≤ k < n+mn−m .
Observe that, if k > (n+m)pq2m(q+1) , we can take k1 >
(n+m)q
2m satisfying (3.13).
Step 3: Assume
k1 > q (3.15)
and let k2 ∈ (k1, ∞] be such that
1
k2
>
q
k1
−
2m
n+m
. (3.16)
From Proposition 2.2 we have
‖v‖
L
k2
dm
≤ C ‖(−∆)mv‖
L
k1/q
dm
≤ C ‖uq‖
L
k1/q
dm
= C ‖u‖q
L
k1
dm
which is finite by step 2.
Step 4: We can see that conditions (3.13), (3.15), (3.16) and min{k1, k2} >
k
ρ for ρ ∈ (0, 1),
to be chosen below, are equivalent to
A :=
p
k
−
2m
n+m
<
1
k1
< min
{
ρ
k
,
1
q
}
(3.17)
and
q
k1
−
2m
n+m
<
1
k2
<
ρ
k
. (3.18)
Observe now that, if
k ≤
(n +m) pq
2m(q + 1)
, (3.19)
we have A > 0. Therefore (3.17) can be solved for k1 ∈ [1,+∞) and with
1
k1
arbitrarily closed
to A whenever
p− ρ
k
<
2m
n+m
(3.20)
and
p
k
−
2m
n+m
<
1
q
. (3.21)
But, (3.20) holds if ρ satisfies
n−m
n+m
p < ρ < 1, (3.22)
and such a ρ exists because p < n+mn−m .
On the other hand, since β = 2m (q+1)pq−1 > n−m, we have
1
q >
p (n−m)
n+m −
2m
n+m . Then, since
k < n−mn+m we can choose ǫ such that (3.21) holds.
Let us now see that condition (3.18) can be fulfilled. Indeed, it is enough to see that all
our parameters can be chosen such that
q
k1
−
2m
n+m
<
ρ
k
. (3.23)
Taking 1k1 in (3.17) closed enough to A we have that (3.23) is equivalent to
ρ > 1− η, (3.24)
where η := 2mn+m (q + 1) k − (pq − 1).
Indeed, if 1k1 is closed to A =
p
k −
2m
n+m , then
q
k1
− 2mn+m is closed to
qp
k −
2mq
n+m −
2m
n+m .
Now, ρ < 1 is equivalent to
k >
n+m
β
, (3.25)
7but since β > n−m it is possible to take ǫ small enough in (3.12) such that (3.25) is satisfied.
Finally we can take ρ ∈ (0, 1) closed enough to one such that que (3.22) and (3.24) hold.
Step 5: It follows from step 4 that if (3.11) holds for some k satisfying (3.12) and (3.19),
then (3.11) is true with k/ρ (as a consequence of (3.17) and (3.18)).
Iterating the procedure we can reach, after a finite number of steps, some value k¯ > (n+m)pq2m(q+1) .
Then, it follows from the comment at the end of step 2 that there exists k¯1 >
(n+m)q
2m ≥
(n+m)p
2m
such that ‖u‖
L
k¯1
dm
≤ C.
Taking now k1 = k¯1, we can take k2 = ∞ in step 3 to conclude that ‖v‖L∞(Ω) ≤ C.
Analogously, by step 2 we obtain ‖u‖L∞(Ω) ≤ C. 
4. Existence of singular solutions.
In order to prove Theorem 3.2 we follow the ideas of [7]. First we will construct a function
f ∈ L1dm(Ω) such that the corresponding solution of the linear problem (2.1) is not bounded.
Recall that our domain Ω is a ball when n ≥ 3, and smooth perturbations of a ball in the
case n = 2. In any case, given x0 ∈ ∂Ω, there exist r > 0 and a revolution cone Σ1 with
vertex x0 such that Σ := Σ1 ∩B2r(x0) ⊂ Ω. Now, for 0 < α < n−m we define
f(x) = |x− x0|
−(α+2m)χΣ,
where χΣ denotes the characteristic function of Σ. Then, it is easy to see that f ∈ L
1
dm(Ω).
Let u > 0 be the solution of (2.1) with f as right-hand side. Then, we have
u(x) =
∫
Ω
Gm(x, y) |y − x0|
−(α+2m)χΣ(y) dy.
Using this representation formula together with the estimates of the Green function (3.5),
(3.6) and (3.7) it is not difficult to see that, for x ∈ Ω,
u(x) ≥ C |x− x0|
−αχΣ(x). (4.1)
Proof of Theorem 3.2: Recall that α = 2m(p+1)pq−1 and β =
2m(q+1)
pq−1 , and we are assuming
0 < α, β < n−m. We define
φ(x) = |x− x0|
−(α+2m) χΣ(x) and ψ(x) = |x− x0|
−(β+2m) χΣ(x).
Let u and v be non-negative and such that

(−∆)mu = φ in Ω
(−∆)mv = ψ in Ω(
∂
∂ν
)j
u =
(
∂
∂ν
)j
v = 0 on ∂Ω 0 ≤ j ≤ m− 1.
Then, it follows from (4.1) that u /∈ L∞(Ω), v /∈ L∞(Ω),
v(x)p ≥
(
C |x− x0|
−βχΣ(x)
)p
= C |x− x0|
−(α+2m) χΣ(x) = C φ(x)
and
u(x)q ≥
(
C |x− x0|
−αχΣ(x)
)q
= C |x− x0|
−(β+2m) χΣ(x) = C ψ(x).
Therefore, defining a = φ/vp and b = ψ/uq we have that a and b are nonnegative bounded
functions, and (u, v) solves
(−∆)mu = a(x) vp and (−∆)mv = b(x)uq.

We end the paper by proving the observation given in Remark 2.3 concerning the optimality
of condition (2) in Proposition 2.2.
8Proposition 4.1. Assume 1 ≤ p ≤ q ≤ ∞ and 1p −
1
q >
2m
n−m . Then there exists f ∈ L
p
dm(Ω)
such that u /∈ Lqdm(Ω), where u is the unique solution of (2.1).
Proof : Let 0 < α < n−m and we define, as above, f(x) = |x− x0|
−(α+2m)χΣ(x). Then
we have
‖f‖p
Lp
dm
(Ω)
=
∫
Σ
|x− x0|
−(α+2m)p d(x)m dx ≤
∫
Σ
|x− x0|
−(α+2m)p+m dx,
and then, since p < n+mα+2m , f ∈ L
p
dm(Ω).
But, for x ∈ Σ there exists a positive constant C such that d(x) ≥ C|x−x0|, and therefore,
it follows from (4.1) that for q ≥ n+mα , u /∈ L
q
dm(Ω). To conclude the proof we observe that,
since 1p −
1
q >
2m
n−m , we can choose α ∈ (0, n −m) such that
n+m
q < α <
n+m
p−2m . 
Finally let us mention that, to our knowledge, it is not known what happens in general in
the limit case 1p −
1
q =
2m
n−m . In the case p > m+ 1 we have proved in [5] that
‖u‖Lq
dm
(Ω) ≤ C ‖f‖Lp
dm
(Ω).
References
[1] Ha¨ım Brezis and Xavier Cabre´, Some simple nonlinear PDE’s without solutions, Boll. Unione Mat. Ital.
Sez. B Artic. Ric. Mat. (8) 1 (1998), no. 2, 223–262.
[2] Philippe Cle´ment and Guido Sweers, Uniform anti-maximum principle for polyharmonic boundary value
problems, Proc. Amer. Math. Soc. 129 (2001), no. 2, 467–474.
[3] Anna Dall’Acqua and Guido Sweers, Estimates for Green function and Poisson kernels of higher-order
Dirichlet boundary value problems, J. Differential Equations 205 (2004), no. 2, 466–487.
[4] , On domains for which the clamped plate system is positivity preserving, Partial differential equa-
tions and inverse problems, Contemp. Math., vol. 362, Amer. Math. Soc., Providence, RI, 2004, pp. 133–
144.
[5] R. G. Dura´n, M. Sanmartino, and M. Toschi, Weighted a priori estimates for solutions of (−△)mu = f
with homogeneous dirichlet conditions poisson equation, arxiv: 1006.1337v1 (2010).
[6] Hans-Christoph Grunau and Guido Sweers, Positivity for equations involving polyharmonic operators with
Dirichlet boundary conditions, Math. Ann. 307 (1997), no. 4, 589–626.
[7] Philippe Souplet, A survey on Lpδ spaces and their applications to nonlinear elliptic and parabolic problems,
Nonlinear partial differential equations and their applications, GAKUTO Internat. Ser. Math. Sci. Appl.,
vol. 20, Gakko¯tosho, Tokyo, 2004, pp. 464–479.
[8] , Optimal regularity conditions for elliptic problems via Lpδ -spaces, Duke Math. J. 127 (2005),
no. 1, 175–192.
Departamento de Matema´tica, Facultad de Ciencias Exactas y Naturales, Universidad de
Buenos Aires, 1428 Buenos Aires, Argentina
E-mail address: rduran@dm.uba.ar
Departamento de Matema´tica, Facultad de Ciencias Exactas, Universidad Nacional de La
Plata, 1900 La Plata (Buenos Aires), Argentina
E-mail address: tatu@mate.unlp.edu.ar
Departamento de Matema´tica, Facultad de Ciencias Exactas, Universidad Nacional de La
Plata, 1900 La Plata (Buenos Aires), Argentina
E-mail address: mtoschi@mate.unlp.edu.ar


On the existence of bounded solutions for a nonlinear elliptic system

Servicio de Difusión de la Creación Intelectual

arXiv:1011.0412v1  [math.AP]  1 Nov 2010ON THE EXISTENCE OF BOUNDED SOLUTIONS FOR A NONLINEARELLIPTIC SYSTEMRICARDO G. DURA´N, MARCELA SANMARTINO, AND MARISA TOSCHIAbstract. This work deals with the system (−∆)mu = a(x) vp, (−∆)mv = b(x)uq withDirichlet boundary condition in a domain Ω ⊂ IRn, where Ω is a ball if n ≥ 3 or a smoothperturbation of a ball when n = 2.We prove that, under appropriate conditions on the parameters (a, b, p, q,m, n), any non-negative solution (u, v) of the system is bounded by a constant independent of (u, v). More-over, we prove that the conditions are sharp in the sense that, up to some border case, therelation on the parameters are also necessary.The case m = 1 was considered by Souplet in [7]. Our paper generalize to m ≥ 1 theresults of that paper.1. IntroductionIn this paper we consider the nonlinear problem(−∆)mu = a(x) vp in Ω(−∆)mv = b(x)uq in Ω(∂∂ν)ju =(∂∂ν)jv = 0 on ∂Ω 0 ≤ j ≤ m− 1,(1.1)where Ω is the unit ball, namely, Ω = B = {x ∈ Rn : |x| < 1} when n ≥ 3, and B orsome perturbations of B for the case n = 2 (see [4] for details of this perturbation), ∂∂ν isthe normal derivative, p, q > 0, pq > 1, and a, b are nonnegative bounded functions. Let usremark that the restriction on the domains is due to the fact that we will use that the Greenfunction of the corresponding linear problem is positive.For the case m = 1, a priori bounds for non-negative solutions of (1.1) in a C2 boundeddomain Ω were obtained by P. Souplet in [7]. To recall the results in that paper we introduceα =2(p + 1)pq − 1and β =2(q + 1)pq − 1.Souplet proved that, if max{α, β} > n− 1, then‖u‖L∞(Ω), ‖v‖L∞(Ω) ≤ C, (1.2)where the constant C depends only on p, q, a, b, and Ω.Moreover, he proved that the result is sharp in the sense that, if max{α, β} < n− 1, thenthere exist a non-negative solutions of (1.1) which are not bounded.Our goal is to obtain similar results for non-negative solutions of (1.1) for a general m.A key tool used in [7] are some weighted a priori estimates for the associated linear problemgiven by {−∆u = f in Ωu = 0 on ∂Ω.Supported by ANPCyT (PICT 01307), by Universidad de Buenos Aires (grant X070), by UniversidadNacional de La Plata (grant X500), and by CONICET (PIP 11220090100625). The first author is a memberof CONICET, Argentina.12Then, in order to generalize the a priori estimates for the case m ≥ 2 we will need to extendthe weighted estimates to higher order linear problems. Non trivial technical modificationsare needed to prove those estimates. Moreover, since we need to use positivity of the Greenfunction, we have to restrict the domain Ω as mentioned above. Indeed, for m ≥ 2 andgeneral regions the Green function is not necessarily positive.2. Weighted a priori estimates for the linear problemWe will denote by d(x) the distance from x to the boundary of Ω and we will work withthe Banach space Lpdm(Ω) where the norm is given by‖u‖Lpdm(Ω) =(∫Ω|u|p dm dx)1/pfor 1 ≤ p <∞ and ‖u‖L∞dm(Ω) := ‖u‖L∞(Ω).In our arguments we will use some results given in [3] for the linear problem{(−∆)mu = f in Ω(∂∂ν)ju = 0 on ∂Ω 0 ≤ j ≤ m− 1.(2.1)We recall those results in the following lemma. Let us remark that these results, and conse-quently our proposition below, are valid in more general domains than those considered here.Indeed, the hypotheses used are that Ω is a bounded domain with C6m+4 boundary for n = 2and C5m+2 boundary for n > 2.Lemma 2.1. Let u ∈ C2m(Ω) and f ∈ C(Ω) satisfy (2.1).• If 2m > n, then there exists C > 0 such that for all θ ∈ [0, 1]‖u d−m+θn‖L∞(Ω) ≤ C ‖f dm−(1−θ)n‖L1(Ω).• Let 1 ≤ p ≤ q ≤ ∞. If 1p −1q < min{2mn , 1}, then taking α ∈ (1p −1q ,min{2mn , 1}] thereexists C > 0 such that for all θ ∈ [0, 1]‖u d−m+θnα‖Lq(Ω) ≤ C ‖f dm−(1−θ)nα‖Lp(Ω).Proof : See Proposition 4.2 in [3].We have the following a priori estimates for solutions of problem (2.1).Proposition 2.2. Let 1 ≤ p ≤ q ≤ ∞. Let f ∈ Lpdm(Ω) and let u be a weak solution of (2.1).We have(1) if n ≤ m, then u ∈ L∞(Ω) and there exists C > 0 such that‖u‖L∞(Ω) ≤ C ‖f‖L1dm(Ω)(2) if 1p −1q <2mn+m , then u ∈ Lpdm(Ω) and there exists C > 0 such that‖u‖Lqdm(Ω) ≤ C ‖f‖Lpdm(Ω).Proof : From Lemma 2.1 we have that, for 2m > n and θ ∈ [0, 1],‖u d−m+θn‖L∞(Ω) ≤ C ‖f dm−(1−θ)n‖L1(Ω). (2.2)Then taking θ = 1 and using that −m+ n < 0 and d(x) ≤ diam(Ω) we obtain‖u‖L∞(Ω) ≤ C ‖u d−m+n‖L∞(Ω) ≤ C ‖f dm‖L1(Ω)and so (1) is proved.3On the other hand, using again Lemma 2.1, we have that, if there exists α ∈ (1p −1q ,min{1,2mn }] and θ ∈ [0, 1] such that{−m+ θ nα = mqm− (1− θ)nα = mp(2.3)we obtain‖u‖Lqdm (Ω)≤ C ‖f‖Lpdm (Ω)for 1p −1q < min{1,2mn }.Solving system (2.3) we obtainα = (2 +1q−1p)mnand θ = (1q+ 1) (2 −1p+1q)−1.We are going to show that α and θ satisfy the required conditions if 2m−nm ≤1p −1q <2mn+m .Since 1 ≤ p we have θ ∈ [0, 1]. On the other hand, from the definition of α, it is easy to seethat the condition 1p −1q < α is equivalent to1p −1q <2mn+m , which is one of our hypothesis.Finally we have to see that α ≤ min{1, 2mn }. Since p ≤ q we have α ≤2mn . Therefore, itonly remains to consider the case 2mn > 1. But α ≤ 1 is equivalent to2m−nm ≤1p −1q and sothe proposition is proved under this restriction.Suppose now that 1p −1q <2m−nm . In this case, for 2m > n, using again the first part ofLemma 2.1, for all θ˜ ∈ [0, 1] we have‖u d−m+θ˜n‖L∞(Ω) ≤ C ‖f dm−(1−θ˜)n‖L1(Ω).Moreover, if θ˜ ≤ mnq +mn , it follows that‖u‖Lqdm(Ω) ≤ ‖u d−m+θ˜n‖L∞(Ω).Analogously, if 1− mn +mnp ≤ θ˜,‖f dm−(1−θ˜)n‖L1(Ω) ≤ C‖f‖Lpdm(Ω).Therefore, if we can choose θ˜ satisfying 1− mn +mnp ≤ θ˜ ≤mnq +mn we have‖u‖Lqdm(Ω) ≤ C ‖f‖Lpdm(Ω),but, such a θ˜ exists because 1p −1q ≤2m−nm and the proposition is proved. Remark 2.3. The condition in (2) is almost optimal, i.e., if 1p −1q >2mn+m then the a prioriestimate does not hold in general. We postpone the proof of this observation to the end ofthe paper because we will use the same technique as in the proof of our second main theorem.In the proof of the following proposition we will denote with λ1,m the first eigenvalue of theoperator (−∆)m and with φ1,m > 0 a corresponding eigenfunction normalized by∫Ω φ1,m = 1.We will use that there exist two positive constants c1 and c2 such that, in Ω,c1 dm ≤ φ1,m ≤ c2 dm, (2.4)see [2].Proposition 2.4. Let 1 ≤ k < n+mn−m . If u is a solution of (2.1) with f ∈ L1dm(Ω) and f > 0,then there exists C > 0 such that‖u‖Lkdm (Ω)≤ C ‖u‖L1dm(Ω). (2.5)4Proof : Taking p = 1 in the previous proposition we obtain for 1 ≤ k < n+mn−m‖u‖Lkdm(Ω) ≤ C ‖f‖L1dm (Ω).Using integration by parts and that f > 0 we have‖f‖L1dm=∫Ω(−∆)mu dm dx ≤∫Ω(−∆)muφ1,m dx≤ C∫Ωu (−∆)mφ1,m dx = C λ1,m∫Ωuφ1,m dx≤ C∫Ωu dm dx ≤ C ‖u‖L1dm.3. Main resultsWe consider problem (1.1) and define the exponentsα =2m(p + 1)pq − 1and β =2m(q + 1)pq − 1.Then, the natural extension of the results in [7] is given by the followingTheorem 3.1. Ifmax(α, β) > n−m, (3.1)then, any non-negative solution of (1.1) satisfies‖u‖L∞(Ω), ‖v‖L∞(Ω) ≤ C (3.2)where C is a positive constant which depends only on a, b, p, q, m, and Ω.We also prove, in the following theorem, that condition (3.1) is almost optimal. We cannotsay optimal because we do not know what happens in the case max(α, β) = n−m.Theorem 3.2. Ifmax(α, β) < n−m, (3.3)then, there exist nonnegative bounded functions a and b, such that (1.1) have some non-negative solution (u, v), with u and v unbounded functions.Once we have the results of the previous section, the proofs of both theorems follows thelines of the case m = 1 proved in [7]. A key point in the arguments given in that paper arethe estimates ∫Ωuφ1,m ,∫Ωv φ1,m ≤ C. (3.4)A straightforward extension of the arguments given in [8], to prove these estimates in the casem = 1, is not possible. Indeed, the proof given in that paper is based on a lemma of [1] whichuses the maximum principle in subsets of Ω. An analogous maximum principle is not valid inthe case m ≥ 2. We will give a different proof of this lemma using pointwise estimates for theGreen function Gm of problem (2.1) given below. This is why we have to restrict Ω in orderto have that the Green function is positive, i.e., we assume that Ω = B = {x ∈ Rn : |x| < 1}when n ≥ 3, and Ω = B or some perturbations of B for the case n = 2 (see [4] for details ofthis perturbation). We have: for 2m < n,Gm(x, y) ≥ C |x− y|2m−n min{1,d(x)m d(y)m|x− y|2m}, (3.5)for 2m = n,Gm(x, y) ≥ C log(1 +d(x)m d(y)m|x− y|2m)≥ C log(2 +d(y)|x− y|)min{1,d(x)m d(y)m|x− y|2m}, (3.6)5and for 2m > n,Gm(x, y) ≥ C d(x)m−n/2 d(y)m−n/2 min{1,d(x)n/2 d(y)n/2|x− y|n}. (3.7)The proofs of these estimates can be found in [4] for the case of m = n = 2 and in [6] forthe rest of the cases.Lemma 3.3. Assume h ≥ 0, h ∈ L1dm(Ω) and v a solution of{(−∆)mv = h in Ω(∂∂ν)jv = 0 on ∂Ω 0 ≤ j ≤ m− 1.(3.8)Then there exists C > 0, depending only on Ω and m, such that for all x ∈ Ωv(x)dm(x)≥ C∫Ωhdm. (3.9)Proof : By the representation formulav(x) =∫ΩGm(x, y)h(y) dyit is enough to prove thatGm(x, y) ≥ C d(x)m d(y)m.Consider, for example, the case 2m < n and suppose that d(x)m d(y)m|x−y|2m≥ 1. Then, it followsfrom (3.5), thatGm(x, y) ≥ C |x− y|2m−n ≥ d(x)m−n/2d(y)m−n/2 ≥ Cd(x)md(y)mwhere in the last step we have used that Ω is bounded. On the other hand, if the minimumon the right hand side of (3.5) is attained in d(x)m d(y)m|x−y|2m we haveGm(x, y) ≥ C |x− y|−nd(x)md(y)m ≥ Cd(x)md(y)m.The proofs for the cases 2m = n and 2m > n are analogous, using now (3.6) and (3.7)respectively. Proof of Theorem 3.1:Step 1: From (3.4) and (2.4) it follows immediately,‖u‖L1dm+ ‖v‖L1dm≤ C, (3.10)and therefore, for n ≤ m,‖u‖L∞(Ω), ‖v‖L∞(Ω) ≤ Cis an immediate consequence of (1) in Proposition 2.2.On the other hand, if n > m, it follows from Proposition 2.4, that‖u‖Lkdm+ ‖v‖Lkdm≤ C(k) (3.11)for 1 ≤ k < n+mn−m .Clearly we may assume q ≥ p and β > n−m. Then, it is easy to check that p < n+mn−m andso, there exists some k such thatk ≥ p and k ≥n+mn−m− ǫ, (3.12)with ǫ to be chosen below, for which (3.11) holds.Step 2: Assume now that we can choose k1 ∈ (k, ∞] such that1k1>pk−2mn+m. (3.13)6Then, using Proposition 2.2 we have‖u‖Lk1dm≤ C ‖(−∆)mu‖Lk/pdm≤ C ‖vp‖Lk/pdm= C ‖v‖pLkdm, (3.14)which is finite because 1 ≤ k < n+mn−m .Observe that, if k > (n+m)pq2m(q+1) , we can take k1 >(n+m)q2m satisfying (3.13).Step 3: Assumek1 > q (3.15)and let k2 ∈ (k1, ∞] be such that1k2>qk1−2mn+m. (3.16)From Proposition 2.2 we have‖v‖Lk2dm≤ C ‖(−∆)mv‖Lk1/qdm≤ C ‖uq‖Lk1/qdm= C ‖u‖qLk1dmwhich is finite by step 2.Step 4: We can see that conditions (3.13), (3.15), (3.16) and min{k1, k2} >kρ for ρ ∈ (0, 1),to be chosen below, are equivalent toA :=pk−2mn+m<1k1< min{ρk,1q}(3.17)andqk1−2mn+m<1k2<ρk. (3.18)Observe now that, ifk ≤(n +m) pq2m(q + 1), (3.19)we have A > 0. Therefore (3.17) can be solved for k1 ∈ [1,+∞) and with1k1arbitrarily closedto A wheneverp− ρk<2mn+m(3.20)andpk−2mn+m<1q. (3.21)But, (3.20) holds if ρ satisfiesn−mn+mp < ρ < 1, (3.22)and such a ρ exists because p < n+mn−m .On the other hand, since β = 2m (q+1)pq−1 > n−m, we have1q >p (n−m)n+m −2mn+m . Then, sincek < n−mn+m we can choose ǫ such that (3.21) holds.Let us now see that condition (3.18) can be fulfilled. Indeed, it is enough to see that allour parameters can be chosen such thatqk1−2mn+m<ρk. (3.23)Taking 1k1 in (3.17) closed enough to A we have that (3.23) is equivalent toρ > 1− η, (3.24)where η := 2mn+m (q + 1) k − (pq − 1).Indeed, if 1k1 is closed to A =pk −2mn+m , thenqk1− 2mn+m is closed toqpk −2mqn+m −2mn+m .Now, ρ < 1 is equivalent tok >n+mβ, (3.25)7but since β > n−m it is possible to take ǫ small enough in (3.12) such that (3.25) is satisfied.Finally we can take ρ ∈ (0, 1) closed enough to one such that que (3.22) and (3.24) hold.Step 5: It follows from step 4 that if (3.11) holds for some k satisfying (3.12) and (3.19),then (3.11) is true with k/ρ (as a consequence of (3.17) and (3.18)).Iterating the procedure we can reach, after a finite number of steps, some value k¯ > (n+m)pq2m(q+1) .Then, it follows from the comment at the end of step 2 that there exists k¯1 >(n+m)q2m ≥(n+m)p2msuch that ‖u‖Lk¯1dm≤ C.Taking now k1 = k¯1, we can take k2 = ∞ in step 3 to conclude that ‖v‖L∞(Ω) ≤ C.Analogously, by step 2 we obtain ‖u‖L∞(Ω) ≤ C. 4. Existence of singular solutions.In order to prove Theorem 3.2 we follow the ideas of [7]. First we will construct a functionf ∈ L1dm(Ω) such that the corresponding solution of the linear problem (2.1) is not bounded.Recall that our domain Ω is a ball when n ≥ 3, and smooth perturbations of a ball in thecase n = 2. In any case, given x0 ∈ ∂Ω, there exist r > 0 and a revolution cone Σ1 withvertex x0 such that Σ := Σ1 ∩B2r(x0) ⊂ Ω. Now, for 0 < α < n−m we definef(x) = |x− x0|−(α+2m)χΣ,where χΣ denotes the characteristic function of Σ. Then, it is easy to see that f ∈ L1dm(Ω).Let u > 0 be the solution of (2.1) with f as right-hand side. Then, we haveu(x) =∫ΩGm(x, y) |y − x0|−(α+2m)χΣ(y) dy.Using this representation formula together with the estimates of the Green function (3.5),(3.6) and (3.7) it is not difficult to see that, for x ∈ Ω,u(x) ≥ C |x− x0|−αχΣ(x). (4.1)Proof of Theorem 3.2: Recall that α = 2m(p+1)pq−1 and β =2m(q+1)pq−1 , and we are assuming0 < α, β < n−m. We defineφ(x) = |x− x0|−(α+2m) χΣ(x) and ψ(x) = |x− x0|−(β+2m) χΣ(x).Let u and v be non-negative and such that(−∆)mu = φ in Ω(−∆)mv = ψ in Ω(∂∂ν)ju =(∂∂ν)jv = 0 on ∂Ω 0 ≤ j ≤ m− 1.Then, it follows from (4.1) that u /∈ L∞(Ω), v /∈ L∞(Ω),v(x)p ≥(C |x− x0|−βχΣ(x))p= C |x− x0|−(α+2m) χΣ(x) = C φ(x)andu(x)q ≥(C |x− x0|−αχΣ(x))q= C |x− x0|−(β+2m) χΣ(x) = C ψ(x).Therefore, defining a = φ/vp and b = ψ/uq we have that a and b are nonnegative boundedfunctions, and (u, v) solves(−∆)mu = a(x) vp and (−∆)mv = b(x)uq.We end the paper by proving the observation given in Remark 2.3 concerning the optimalityof condition (2) in Proposition 2.2.8Proposition 4.1. Assume 1 ≤ p ≤ q ≤ ∞ and 1p −1q >2mn−m . Then there exists f ∈ Lpdm(Ω)such that u /∈ Lqdm(Ω), where u is the unique solution of (2.1).Proof : Let 0 < α < n−m and we define, as above, f(x) = |x− x0|−(α+2m)χΣ(x). Thenwe have‖f‖pLpdm(Ω)=∫Σ|x− x0|−(α+2m)p d(x)m dx ≤∫Σ|x− x0|−(α+2m)p+m dx,and then, since p < n+mα+2m , f ∈ Lpdm(Ω).But, for x ∈ Σ there exists a positive constant C such that d(x) ≥ C|x−x0|, and therefore,it follows from (4.1) that for q ≥ n+mα , u /∈ Lqdm(Ω). To conclude the proof we observe that,since 1p −1q >2mn−m , we can choose α ∈ (0, n −m) such thatn+mq < α <n+mp−2m . Finally let us mention that, to our knowledge, it is not known what happens in general inthe limit case 1p −1q =2mn−m . In the case p > m+ 1 we have proved in [5] that‖u‖Lqdm(Ω) ≤ C ‖f‖Lpdm(Ω).References[1] Ha¨ım Brezis and Xavier Cabre´, Some simple nonlinear PDE’s without solutions, Boll. Unione Mat. Ital.Sez. B Artic. Ric. Mat. (8) 1 (1998), no. 2, 223–262.[2] Philippe Cle´ment and Guido Sweers, Uniform anti-maximum principle for polyharmonic boundary valueproblems, Proc. Amer. Math. Soc. 129 (2001), no. 2, 467–474.[3] Anna Dall’Acqua and Guido Sweers, Estimates for Green function and Poisson kernels of higher-orderDirichlet boundary value problems, J. Differential Equations 205 (2004), no. 2, 466–487.[4] , On domains for which the clamped plate system is positivity preserving, Partial differential equa-tions and inverse problems, Contemp. Math., vol. 362, Amer. Math. Soc., Providence, RI, 2004, pp. 133–144.[5] R. G. Dura´n, M. Sanmartino, and M. Toschi, Weighted a priori estimates for solutions of (−△)mu = fwith homogeneous dirichlet conditions poisson equation, arxiv: 1006.1337v1 (2010).[6] Hans-Christoph Grunau and Guido Sweers, Positivity for equations involving polyharmonic operators withDirichlet boundary conditions, Math. Ann. 307 (1997), no. 4, 589–626.[7] Philippe Souplet, A survey on Lpδ spaces and their applications to nonlinear elliptic and parabolic problems,Nonlinear partial differential equations and their applications, GAKUTO Internat. Ser. Math. Sci. Appl.,vol. 20, Gakko¯tosho, Tokyo, 2004, pp. 464–479.[8] , Optimal regularity conditions for elliptic problems via Lpδ -spaces, Duke Math. J. 127 (2005),no. 1, 175–192.Departamento de Matema´tica, Facultad de Ciencias Exactas y Naturales, Universidad deBuenos Aires, 1428 Buenos Aires, ArgentinaE-mail address: rduran@dm.uba.arDepartamento de Matema´tica, Facultad de Ciencias Exactas, Universidad Nacional de LaPlata, 1900 La Plata (Buenos Aires), ArgentinaE-mail address: tatu@mate.unlp.edu.arDepartamento de Matema´tica, Facultad de Ciencias Exactas, Universidad Nacional de LaPlata, 1900 La Plata (Buenos Aires), ArgentinaE-mail address: mtoschi@mate.unlp.edu.ar

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On the existence of bounded solutions for a nonlinear elliptic system

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