Existence results for nonproper elliptic equations involving the Pucci operator

TLDR: In this paper, the authors study the problem in general smooth bounded domains and show that it possesses nontrivial solutions provided that: f is sublinear, or f is superlinear and the equation admits a priori bounds.

Abstract: We study the equation in general smooth bounded domain Ω, and show it possesses nontrivial solutions provided that: f is sublinear, or f is superlinear and the equation admits a priori bounds. The existence result in the superlinear case is based on a new Liouville type theorem for − ℳλ,Λ +(D 2 u) = u p in a half-space.

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Existence results for nonproper elliptic equations
involving the Pucci operator
Alexander Quaas, Boyan Sirakov
To cite this version:
Alexander Quaas, Boyan Sirakov. Existence results for nonproper elliptic equations involving the Pucci
operator. Communications in Partial Dierential Equations, Taylor & Francis, 2006, 31, pp.987-1003.
�hal-00004768�

Existence results for nonproper elliptic
equations involving the Pucci operator
Alexander QUAAS
1
and Boyan SIRAKOV
Abstract
We study the equation
½
−M
+
λ,Λ
(D
2
u) = f(x, u) in Ω,
u = 0 on ∂Ω,
in general smooth bounded domain Ω and show it possesses nontrivial
solutions provided
• f is sublinear, or
• f is superlinear and the equation admits a priori bounds.
The existence result in the superlinear case is based on a new Liouville
type theorem for −M
+
λ,Λ
(D
2
u) = u
p
in a half-space.
MSC Numbers : 35J60, 35J65, 37C25
1 Introduction
This paper is a contribution to the study of uniformly elliptic fully nonlinear
elliptic equations of the form
F (x, u, Du, D
2
u) = 0. (1.1)
The existence of solutions of (1.1) has been extensively investigated for coer-
cive (or proper) uniformly elliptic operators F , mainly through adaptations
of Perron’s Method - see for example [12] and [11]. However, relatively little
is known when the assumption of coercivity (that is, monotonicity in u) is
dropped. On the other hand, when the second order operator is linear or in
divergence form, a vast number of existence results are known.
In this paper, we focus on the model problem



−M
+
λ,Λ
(D
2
u) = f(x, u) in Ω,
u = 0 on ∂Ω,
(1.2)
1
Supported by FONDECYT, Grant N. 1040794, and ECOS grant C02E08.
1

where Ω is a bounded regular domain in R
N
, and M
+
λ,Λ
is the extremal Pucci
operator ([29]), with parameters 0 < λ ≤ Λ, defined by
M
+
λ,Λ
(M) = Λ
X
e
i
>0
e
i
+ λ
X
e
i
<0
e
i
, (1.3)
for any symmetric N ×N matrix M; here e
i
= e
i
(M), i = 1, ..., N, denote the
eigenvalues of M. All results we obtain can be restated for equation (1.2),
with M
+
λ,Λ
replaced by M
−
λ,Λ
(M
−
λ,Λ
is defined by exchanging the places of
λ and Λ in (1.3)), see also the remark at the end of this section. Pucci’s
operators are extremal in the sense that
M
+
λ,Λ
(M) = sup
A∈A
tr(AM) , M
−
λ,Λ
(M) = inf
A∈A
tr(AM), (1.4)
where A denotes the set of all symmetric matrices whose eigenvalues lie in
the interval [λ, Λ]. For more details on these operators we refer for example
to the monograph of Cabr´e and Caffarelli [10]. Notice that M
+
λ,Λ
is not in
divergence form.
Pucci’s extremal operators appear for example in the context of stochastic
control when the diffusion coefficient is a control variable, see for example
the book of A. Bensoussan and J.L. Lions [4].
The study of (1.2) has been taken up only very recently in [18] and [31],
where some results about existence of solutions in a ball or in a convex domain
are proved (see Remark 2 after Theorem 1.6).
When λ = Λ = 1, M
±
λ,Λ
coincide with the Laplace operator, so that (1.2)
becomes the classical equation
½
−∆u = f(x, u) in Ω,
u = 0 on ∂Ω.
(1.5)
For this equation and, in general, for equations involving divergence form
operators existence results can be obtained by variational methods - see for
example the survey papers [25], [30]. Another successful approach for study-
ing existence of solutions of (1.5) are topological methods. General references
on this topic are the book [15] and the survey paper [27].
Our approach to study the existence problem for (1.2) falls into the group
of topological methods and is based on the degree theory for compact op-
erators in positive cones (Kransnoselskii’s index). This approach has been
successfully applied by many authors to a variety of problems. Of special
interest to us is the work of de Figueiredo, Lions and Nussbaum [19], where
appears an abstract existence theorem on which we base our arguments (see
Theorem 4.1 in Section 4).
2

Next we list our results. A standing assumption on the nonlinearity
f(x, u) will be the following condition.
(f
0
) f is a H¨older continuous function on Ω × [0, ∞), such that f(x, 0) = 0
and f(x, s) ≥ −γs for some γ ≥ 0 and all s ≥ 0, x ∈
Ω.
First, we show that (1.2) has a positive solution provided the problem is
sublinear, in the sense that
(H
0
) lim sup
u→∞
f(x, u)
u
< µ
+
1
< lim inf
u→0
f(x, u)
u
≤ ∞, uniformly in x ∈
Ω.
Here µ
+
1
> 0 denotes the first eigenvalue of the Pucci operator M
+
λ,Λ
, associ-
ated to a positive eigenfunction. The existence of µ
+
1
is studied by Felmer and
Quaas in [18] when Ω is ball, and by Quaas in [31] for any regular bounded
domain. More properties of µ
+
1
are estabished in the recent paper by Busca,
Esteban and Quaas [7]. In Section 2 we quote the results from these papers
that we need.
Theorem 1.1 Suppose (f
0
) and (H
0
) hold. Then problem (1.2) has a posi-
tive classical solution.
Remark 1. A typical nonlinearity which satisfies (f
0
) and (H
0
) is the
function f(x, u) = a(x)u
p
, where 0 < p < 1 and a(x) is bounded between
two positive constants.
Remark 2. Theorem (1.1) seems to be the first result in the literature
which concerns sublinear equations involving the Pucci operator.
Next we turn to superlinear equations, that is, equations in which the
nonlinearity satisfies
(H
0
) lim sup
u→0
f(x, u)
u
< µ
+
1
< lim inf
u→∞
f(x, u)
u
≤ ∞, uniformly in x ∈
Ω.
In order to state the existence theorem, we consider the family of problems
obtained from (1.2) by replacing f(x, u) with f(x, u + t), for t ≥ 0. Let A
t
denote the set of nonnegative classical solutions for any such problem and
let S
t
= ∪
0≤s≤t
A
s
.
Theorem 1.2 Suppose (f
0
) and (H
0
) hold. Suppose in addition that for
each t ≥ 0 there exist a constant C depending only on t, Ω and f such that
kuk
L
∞
(Ω)
≤ C for all u ∈ S
t
. (1.6)
Then problem (1.2) has a positive classical solution.
3

Remark. In the sequel we shall consider nonlinearities with power-like
growth at infinity, in which case it is enough to have condition (1.6) only
for t = 0, that is, to assume equation (1.2) admits a priori bounds.
Theorem 1.2 settles the existence question provided a priori bounds exist.
Consequently, we next concentrate on getting such bounds for (1.2). We use
the blow-up method of Gidas and Spruck, which has turned to be the most
powerful tool for obtaining a priori bounds in more classical situations. The
most important drawback of this method is that it depends on availability of
non-existence results (we shall refer to these as Liouville type theorems) for
equations of type (1.2), when Ω is the whole space or a half-space, and such
results are often difficult to get. We note that ever since the fundamental
work of Gidas and Spruck [21] there has been a multitude of Liouville type
results for equations of the type −∆u + f(u) = 0.
Let us recall the recent progress in proving Liouville type theorems for
equations involving Pucci’s operator, a very interesting question by itself.
First, Cutri and Leoni [13] studied the problem
M
+
λ,Λ
(D
2
u) + u
p
= 0 in R
N
u ≥ 0 in R
N
,
(1.7)
where p > 1. They obtained the following Liouville type theorem.
Theorem 1.3 (Cutri-Leoni) Suppose N ≥ 3 and set
p
+
:=
˜
N
˜
N − 2
, with
˜
N =
Λ
λ
(N − 1) + 1.
If 1 < p ≤ p
+
then the only viscosity supersolution of (1.7) is u ≡ 0.
Next, in the radial case a Liouville type theorem for a larger range of p
can be obtained for solutions (as opposed to just supersolutions) of (1.7).
Felmer and Quaas proved the following theorem in [17].
Theorem 1.4 (Felmer-Quaas) Let N ≥ 3. Then there exist a number
p
+
∗
> p
+
> 1 such that if 1 < p < p
+
∗
then (1.7) does not have a non trivial
radially symmetric classical solution.
When the parameters λ and Λ are equal, one gets p
+
∗
= p
N
, where p
N
=
(N + 2)/(N − 2) is the usual Sobolev critical exponent. Note that in the case
λ < Λ we have p
+
∗
> max{p
N
, p
+
}, so there is a gap between the exponents
of Theorem 1.3 and Theorem 1.4. It is an open problem to show that (1.7)
has no solutions in the range p
+
< p < p
+
∗
(the result of Gidas and Spruck
states that this is the case when λ = Λ).
4

Frequently asked questions

Q1. What contributions have the authors mentioned in the paper "Existence results for nonproper elliptic equations involving the pucci operator" ?

The authors study the equation { −M+λ, Λ ( D 2u ) = f ( x, u ) in Ω, u = 0 on ∂Ω, in general smooth bounded domain Ω and show it possesses nontrivial solutions provided • f is sublinear, or • f is superlinear and the equation admits a priori bounds. 

Q2. What is the powerful method for obtaining a priori bounds?

The authors use the blow-up method of Gidas and Spruck, which has turned to be the most powerful tool for obtaining a priori bounds in more classical situations. 

Q3. What is the important drawback of the blow-up method?

The most important drawback of this method is that it depends on availability of non-existence results (we shall refer to these as Liouville type theorems) for equations of type (1.2), when Ω is the whole space or a half-space, and such results are often difficult to get. 

Q4. What is the simplest way to show that a nonlinearity is positive?

Next the authors turn to superlinear equations, that is, equations in which the nonlinearity satisfies(H0) lim sup u→0f(x, u)u < µ+1 < lim inf u→∞f(x, u)u ≤ ∞, uniformly in x ∈ Ω. 

Q5. What is the result of the a priori proof of theorem 1.6?

Theorem 1.6 Assume N ≥ 3, f does not depend on x, satisfies the hypotheses (f0), (H 0), and(f1) there exist p ∈ (1, p +] and a constant C∗ > 0 such thatlim s→+∞f(s)sp = C∗. 

Q6. What is the boundary lemma for a elliptic operator?

Suppose un and gn are sequences of continuous functions such that un is a solution (or subsolution, or supersolution) of the equation−M+λ,Λ(D 2un) = gn(x)in a domain Ω. Suppose un and gn converge uniformly on compact subsets of Ω to functions u and g. 

Q7. what is the gn if vn satisfies?

Then vn satisfies−M+λ,Λ(D 2vn) = gn(x),wheregn = βn f(x, un)un vn + γ(βn − 1)vn,so|gn(x)| ≤ (βn(µ + 1 − ε) + γ(βn − 1))vn + kβn ‖un‖ ,so gn is bounded. 

Q8. What is the main purpose of this paper?

In this paper, the authors focus on the model problem −M+λ,Λ(D 2u) = f(x, u) in Ω,u = 0 on ∂Ω, (1.2)1Supported by FONDECYT, Grant N. 1040794, and ECOS grant C02E08. 

Q9. What is the first possibility of the problem?

Under the hypotheses of Theorem 3.1, if the problemM+λ,Λ(D 2u) + f(u) = 0 (3.18)has a nontrivial nonnegative bounded solution in RN+ such that u = 0 on ∂RN+ , then the same problem has a positive solution in R N−1. 

Q10. What is the simplest way to prove the existence of a linear operator?

For a fixed function v ∈ W 2,N loc (Ω) there exists a symmetric measurable matrix A(x) ∈ A (A is defined in (1.4)), such thatM+λ,Λ(D 2v) = LAv,where LA is the second order linear elliptic operator associated to A, that is LA = ∑ aij(x)∂ij = tr(AD 2(·)). 

Q11. What is the proof of the Lemma 3.1?

By Lemma 2.1 and Theorem 2.2 the authors can fix ε0 > 0 such that M − λ,Λ(D2·) + cβ(x) satisfies the maximum principle in the domain Σβ∗+ε0 \\ Σβ∗−ε0 . 

Q12. What is the first possibility excluded by v(0, x0N)?

The first possibility is excluded by v(0, x0N) = 1. Introduce the functionszβ(y, xN) = v(y, 2β − xN) − v(y, xN)defined in Σβ ∩ Q for all β ≤ β ∗ + 1/2. The authors have, by continuity,zβ ∗ ≥ 0 and zβ ∗ (0, x0N) = 0.Since M−λ,Λ(D 2zβ ∗ ) + lzβ ∗ ≤ 0 the strong maximum principle implieszβ ∗= 0 in Σβ∗ ∩ Q. 

Q13. What is the simplest way to get a priori bounds for a classical problem?

in the radial case a Liouville type theorem for a larger range of p can be obtained for solutions (as opposed to just supersolutions) of (1.7). 

Q14. What is the simplest version of the theorem?

The authors are going to prove Theorem 1.5 by using a (simplified) version of the proof of Berestycki, Caffarelli and Nirenberg [6], who showed that solutions of −∆u+f(u) = 0 in a half space which are at most exponential at infinity are necessarily monotone in xN . 

Q15. What is the maximum of the solution to (1.2) away from the boundary?

In this case, the maximum of the solution to (1.2) is away from the boundary so the Liouville type theorem in the half space is not needed to establish the a priori bounds. 

Q16. What is the simplest way to solve the elliptic equations?

It is easy to see, with the help of standard existence results for proper (γ ≥ 0) fully nonlinear elliptic equations, combined with Theorems 2.3 and 2.4, that L is well defined and compact (for details see [31]). 

Q17. What is the simplest way to solve a classical problem?

At denote the set of nonnegative classical solutions for any such problem and let St = ∪0≤s≤t As.Theorem 1.2 Suppose (f0) and (H 0) hold. 

Q18. What is the a priori bounds of the theorem 1.6?

Remark 2. Using an argument based on Theorem 1.4 Felmer and Quaas showed that Theorem 1.6 is valid for p ∈ (1, p+∗ ), provided Ω is a ball, see [17] and [18].