Doklady Mathematics, Vol. 59, No. 3, 1999, pp. 351–355. Translated from Doklady Akademii Nauk, Vol. 366, No. 1, 1999, pp. 13–17.
Original Russian Text Copyright © 1999 by Mitidieri, Pohozaev.
English Translation Copyright © 1999 by åÄàä “ç‡Û͇ /Interperiodica” (Russia).
MATHEMATICS
Nonexistence of Positive Solutions for a System of Quasilinear
Elliptic Equations and Inequalities in RN
E. Mitidieri* and Corresponding Member of the RAS S. I. Pohozaev**
Received November 24, 1998
1. INTRODUCTION
Much attention has been given recently to nonexistence theorems for systems of quasilinear elliptic equations and inequalities. Nonexistence problems were
examined in [1] for the general class of quasilinear
elliptic inequalities of the form
In nonexistence theorems, we always assume that
the feasible solution of the inequality belongs to a local
function space for which the integrals of relevant products make sense. This fact is marked in the notation of
spaces by a subscript α.
N
– div ( A ( x, u, Du )Du ) ≥ f ( x, u, Du ) in R ,
N
2. NONEXISTENCE THEOREM
(1.1)
u ≥ 0, u ò 0 in R .
Here, N ≥ 1, D is the gradient of the function u: RN → R,
and A and f are given functions satisfying suitable
growth and regularity conditions. The interested reader
is referred to [2], where exact formulations can be
found, and references to recent studies concerning the
nonexistence of solutions are given.
The method employed in [1] is based on the concept
of a nonlinear capacity and was developed in [2] as
applied to the theory of solution nonexistence. This
method assigns an index to any pair (A, f), where A is a
differential operator in conservative form and f is a
given scalar function. If this index satisfies a suitable
inequality that involves the dimension N of RN, then the
problem
The main result of this paper is the following statement.
Theorem 2.1. Let N > p, q, q > 1 and
(i) q – 1 < q1,
p – 1 < p1.
If
 p ( q – 1 ) + qq 1
N–p
- – -------------,
max  ------------------------------------------------- p1 q1 – ( p – 1 ) ( q – 1 ) p – 1
(2.1)
q ( p – 1 ) + p p1
N –q
-------------------------------------------------- – ------------- ≥ 0,
p 1 q 1 – ( p – 1 ) ( q – 1 ) q – 1 
then the problem
N
A ( u ) ≥ f ( u ) in R ,
N
(1.2)
u ≥ 0, u ò 0 in R
has no solution belonging to the local space depending
on A and f.
This approach can be extended in several directions.
The goal of this paper is to show that the same approach
can be applied to some vector-valued problems.
The main result is a theorem of nonexistence of positive solutions to a system of quasilinear elliptic (possibly degenerating) inequalities. In the radial case, this
result was proved in [3].
* Dipartimento di Scienze Matematiche Universita’ degli
Studi di Trieste, Piazzale Europe 1, Trieste, 34127 Italy
e-mail: [email protected]
** Steklov Institute of Mathematics, Russian Academy of
Sciences, ul. Gubkina 8, Moscow, 117966 Russia
e-mail: [email protected]
– div ( Du
p–2
– div ( D v
q–2
Du ) ≥ v
q1
Dv ) ≥ u
p1
N
in R ,
(2.2)
N
u ≥ 0, u ò 0 in R ,
v ≥ 0, v ò 0 in R
N
1, p
1, q
has no solutions (u, v) ∈ W α, loc (RN) × W α, loc (RN).
1, p
1, q
Proof. Suppose that (u, v) ∈ W loc (RN) × W loc (RN)
is a solution to problem (2.2). Without loss of generality, we can assume that u > 0 and v > 0 almost everywhere in RN.
∞
Let ϕ ∈ C 0 (RN) be a standard nonnegative cut-off
function and α < 0 be sufficiently small. Multiplying
the first and second inequalities in (2.2) by uαϕ and
vαϕ, respectively, and integrating by parts, we find (see
also [1])
351
352
MITIDIERI, POHOZAEV
∫u
p1
(α + p – 1)a = p1,
α
q α–1
v ϕ d x + c ε'' D v v ϕ d x
∫
(2.3)
q
Dϕ α + q – 1
-v
≤ c ε' -----------d x,
q–1
ϕ
∫
∫
q1 α
∫v
p α–1
∫
v u ϕ d x + d ε'' Du u
ϕ dx
(2.4)
p
Dϕ α + p – 1
≤ d ε' -------------u
d x,
p–1
ϕ
∫
where c ε' , c ε'' , d ε' , d ε'' > 0 are constants depending only
on α, p, q, and ε > 0.
Multiplying (2.2) by ϕ, we obtain
∫
∫u
∫
p
p1
∫
ϕ dx ≤ ( D v v
q
α–1
ϕ dx )
∫
q–1
----------q
p
Dϕ α + p – 1 
v ϕ d x ≤ D ε  -------------u
dx
p–1

ϕ
∫
q1
(2.6)
(2.7)
Dϕ ( 1 – α ) ( p – 1 )  p
dx ,
×  -------------u
 ϕp – 1

∫
p1
q
Dϕ α + q – 1 
-v
ϕ d x ≤ E ε  -----------dx
 ϕq – 1

∫
q–1
----------q
(2.8)
Dϕ ( 1 – α ) ( p – 1 )  q
-v
×  -----------dx ,
 ϕq – 1

where Eε and Dε depend only on p, q, α, and ε > 0.
Applying the Hölder inequality with parameters a
and a' to the first integral on the right-hand side of (2.7),
we obtain
∫
p
Dϕ α + p – 1 
 -------------u
dx
 ϕp – 1

p–1
-----------p
( α + p – 1 )a
≤ u
ϕ d x


∫
pa'
Dϕ
- d x
×  -------------- ϕ pa' – 1 
∫
1 1
where --- + ---- = 1. Choosing
a a'
p–1
-----------pa'
p–1
-----------pa
(2.9)
,
(2.10)
1
---p
∫
1
p
∫
p y'
(2.11)
1
--y'
Dϕ
- d x ,
×  -------------- ϕ p y' – 1 
∫
∫
p1
v ϕ d x ≤ D ε  u ϕ d x
∫
q1
1
----- py 
p–1
-----------pa 
pa'
Dϕ
---------------- d x
 ϕ pa' – 1 
∫
p–1
-----------pa'
(2.12)
1
-------
p y'
p y'
Dϕ
---------------- d x ;
×  u ϕ dx

  ϕ p y' – 1 
∫
p1
∫
i.e.,
∫
1
---
q
∫
p–1
-----------pa'
--Dϕ ( 1 – α ) ( p – 1 )
 u ( 1 – α ) ( p – 1 )y ϕ d x y
-------------u
d
x
≤
p–1


ϕ
p–1
-----------p
1
----
p
pa'
Dϕ
---------------- d x
 ϕ pa' – 1 
1 1
where --- + --- = 1.
y y'
Choosing (1 – α)(p – 1)y = p1 in (2.11) and taking
into account (2.10), we find
1
--q
q
p–1
-----------pa 
p
∫
(2.5)
1
---p
Dϕ ( 1 – α ) ( p – 1 ) 
-v
×  -----------dx .
 ϕq – 1

Using (2.3) and (2.4), we obtain from these estimates
∫
∫
p1
ϕ dx ≤ Dε ( u ϕ dx )
p α–1
v ϕ d x ≤  Du u ϕ d x
q1
∫
∫u
q1
Dϕ ( 1 – α ) ( p – 1 ) 
dx .
×  -------------u
 ϕp – 1

Performing similar manipulations with parameters y,
y1 > 1 for the third integral in (2.10), we obtain
p–1
----------- p
Dϕ ( 1 – α ) ( p – 1 ) 
×  -------------u
dx ,
 ϕp – 1

∫
we deduce from (2.7) the inequality
p1
v ϕ d x ≤ D ε  u ϕ d x
∫
q1
pa'
p–1
-----------pa'
p–1 1
 ----------- + ----- pa
py
p y'
1
------p y'
(2.13)
Dϕ
Dϕ
 --------------- d x
- d x ,
×  -------------- ϕ pa' – 1 
 ϕ p y' – 1 
where the parameters a and y are chosen so that
∫
∫
1 1
--- + --- = 1,
( α + p – 1 )a = p 1 ,
y y'
(2.14)
1 1
( 1 – α ) ( p – 1 )y = p 1 .
--- + ---- = 1,
a a'
Note that this choice of a and y is possible by virtue of
assumption (i), if α is small.
We define the new parameters b and κ as
1 1
--- + ---- = 1,
( α + q – 1 )b = q 1 ,
κ κ'
(2.15)
( 1 – α ) ( q – 1 )κ = q 1
1 1
--- + ---- = 1,
b b'
and estimate the left-hand side of (2.8) as described
above. As a result, we obtain
DOKLADY MATHEMATICS
Vol. 59
No. 3
1999
NONEXISTENCE OF POSITIVE SOLUTIONS FOR A SYSTEM
∫
q1
u ϕ d x ≤ E ε  v ϕ d x


∫
p1
qb'
Dϕ
- d x
×  -------------- ϕ qb' – 1 
∫
q–1
----------qb'
Dϕ
 --------------- d x
 ϕ qκ' – 1 
∫
 N – pa' N – py' 
σ 2 = m  ------------------ + ------------------ 
py' 
 pa'
–1 1
 q----------+ ----- qb
qκ
qκ'
1
------qκ'
(2.16)
∫
1 – mn
n
-------
qκ'
≤
n
Dε Eε 

pa'
qκ'
Dϕ
Dϕ
- d x  ---------------- d x
×  -------------- ϕ qκ' – 1   ϕ pa' – 1 
∫
 u ϕ d x


∫
p1
p y'
Dϕ
- d x
×  -------------- ϕ p y' – 1 
∫
∫
1 – mn
m
------p y'
≤
m
Eε Dε 

qb'
Dϕ
 --------------- d x
 ϕ qb' – 1 
∫
∫
Using the chosen parameters (2.14) and (2.15), we can
find σ1 and σ2 in explicit form:
∫
p y'
(2.17)
1
-------
p y'
Dϕ
 --------------- d x ,
 ϕ p y' – 1 
∫
m
-------
pa'
pa'
Dϕ
--------------- d x
pa' – 1

ϕ
q–1
----------qb'
σ 1 = σv : = N – p
n
-------
qb'
qb'
Dϕ
--------------- d x
qb' – 1

ϕ
p–1
-----------pa'
qκ'
Dϕ
 --------------- d x
 ϕ qκ' – 1 
∫
(2.18)
1
------qκ'
,
where
p–1 1
n := ------------ + ------ ,
py
pa
q–1 1
m := ----------- + ------.
qκ
qb
(2.19)
Taking into account (2.14) and (2.15), it is easy to see
that
q–1
m = -----------,
q1
p–1
n = ------------ .
p1
(2.20)
Note that, by our assumption, the exponent appearing
on the right-hand sides of (2.17) and (2.18) satisfies the
inequality
p1 q1 – ( p – 1 ) ( q – 1 )
- > 0.
1 – mn = -------------------------------------------------p1 q1
Now, changing the variables and choosing a suitable ϕ
(see [1] for more detail), we obtain
∫v
B
σ1
dx ≤ Fε R ,
(2.21)
R
∫u
B
q1
p1
σ2
dx ≤ Gε R ,
(2.22)
R
where Fε and Gε are positive constants independent of
R, and
 N – qb' N – qκ' 
σ 1 = n  ----------------- + ------------------ 
qκ' 
 qb'
Vol. 59
No. 3
p ( q – 1 ) + qq 1 

- ,
– ( p – 1 )  -------------------------------------------------q
p
 1 1 – ( p – 1)(q – 1) 
σ 2 = σu : = N – q
Now, assume that
 p ( q – 1 ) + qq 1
N–p
max  -------------------------------------------------- – -------------,
 p1 q1 – ( p – 1 ) ( q – 1 ) p – 1
q( p – 1) + p p1
N –q
-------------------------------------------------- – ------------- 
p1 q1 – ( p – 1 ) ( q – 1 ) q – 1 
(2.24)
p ( q – 1 ) + qq 1
N–p
- – ------------- > 0.
= -------------------------------------------------p1 q1 – ( p – 1 ) ( q – 1 ) p – 1
Then, it follows from (2.21) that
∫v
R
q1
d x = 0,
N
i.e., v ; 0. This contradicts our assumption that v > 0
in RN, which completes the proof of the theorem in the
case of (2.24). The case when the maximum in (2.24) is
equal to
q ( p – 1 ) + p p1
N–q
-------------------------------------------------- – ------------- > 0
p1 q1 – ( p – 1 ) ( q – 1 ) q – 1
is similar.
Consider the critical case
 p ( q – 1 ) + qq 1
N–p
max  -------------------------------------------------- – -------------,
p–1
q
–
(
p
–
1
)
(
q
–
1
)
p
 1 1
q ( p – 1 ) + pp 1
N –q
-------------------------------------------------- – ------------- 
p1 q1 – ( p – 1 ) ( q – 1 ) q – 1 
(the case
1999
(2.23)
 q ( p – 1 ) + p p1 
- .
– ( q – 1 )  ------------------------------------------------- p1 q1 – ( p – 1 ) ( q – 1 ) 
p ( q – 1 ) + qq 1
N–p
- – ------------- = 0
= -------------------------------------------------p1 q1 – ( p – 1 ) ( q – 1 ) p – 1
N – pa'
N – py'
+ ------------------ ( p – 1 ) + ------------------ ,
pa'
py'
DOKLADY MATHEMATICS
N – qb'
N – qκ'
+ ----------------- ( q – 1 ) + ------------------ .
qb'
qκ'
.
Combining (2.13) with (2.16), we finally obtain
 v q1 ϕ d x


353
(2,25)
354
MITIDIERI, POHOZAEV
1, p
 p1 + 1 q1 + 1  N – 2
, -------------------  ≥ ------------- ,
max  ------------------2
 p1 q1 – 1 p1 q1 – 1 
q ( p – 1 ) + pp 1
N –q
-------------------------------------------------- – ------------- 
p1 q1 – ( p – 1 ) ( q – 1 ) q – 1 
then the problem
q( p – 1) + p p1
N–q
- – ------------- = 0
= -------------------------------------------------p1 q1 – ( p – 1 ) ( q – 1 ) q – 1
q1
Du
–div  -------------------------- ≥ v


α
1 + Du
is analogous).
∞
p1
Dv
– div  --------------------------- ≥ u


β
1 + Dv
By virtue of our choice of ϕ ∈ C 0 (RN), we have
∫v
q1
dx ≤
BR
∫
B2 R

v ϕ dx ≤ 
R <

×
R <
q1
p–1
------------
p α–1
∫
Du u
x < 2R
1
----
∫
x
 p
ϕ d x

(2.26)
N
u ≥ 0, u ò 0 in R ,
v ≥ 0, v ò 0 in R
∫
BR

v d x

< 2R
∫
x

p ( q – 1 ) + qq 1
N–p
- – -------------,
max  ------------------------------------------------- p1 q1 – ( p – 1 ) ( q – 1 ) p – 1
mn
σv
q1
(2.27)
R .
q( p – 1) + p p1
N –q
-------------------------------------------------- – -------------  < 0,
p1 q1 – ( p – 1 ) ( q – 1 ) q – 1 
Since σv = 0, (2.21) implies
∫v
R
q1
d x < ∞.
N
This means that the right-hand side of (2.27) tends to
zero as R → ∞; therefore,
∫v
R
q1
then problem (2.2) has a positive solution.
To verify this, we consider, for simplicity, the case
q = p. For ε > 0 and x ∈ RN, define
ε
u ( x ) : = -------------------------------------------------------------------------- ,
d x = 0.
1 + x

N
This contradiction completes the proof.
Remark. It is easy to see that the technique used to
prove Theorem 2.1 can be applied to other quasilinear
systems. As an example, we formulate the following
statement.
Theorem 2.2. Let N > p1 > 1 and
(i) p – 1 < p1, q – 1 < q1.
Suppose that A, B: RN × RN × RN × R+ × R+ →
R+\{0} is a bounded Caratheodory function.
If (2.1) holds, then the problem
– div ( A ( x, u, v , Du, D v ) Du
p–2
– div ( B ( x, u, v , Du, D v ) D v
q–2
N
3. EXACTNESS OF THEOREM 2.1
Theorem 2.1 is exact. In other words, if
Following a line of reasoning similar to the first part of
the proof, we deduce

v d x ≤ c̃ 
R <
N
in R ,
has no solution (u, v) ∈ C2(RN) × C2(RN).
p
Dϕ ( 1 – α ) ( p – 1 )  p
-------------u
d x .
p–1

ϕ
< 2R
q1
1, q
has no solution (u, v) ∈ W α, loc (RN) × W α, loc (RN).
Corollary 2.3. Let N > 2; p1, q1 > 1; and α, β ≥ 0. If

p ( q – 1 ) + qq 1
N–p
- – -------------,
max  ------------------------------------------------- p1 q1 – ( p – 1 ) ( q – 1 ) p – 1
Du ) ≥ v
q1
Dv ) ≥ u
p1
N
u ≥ 0, u ò 0 in R ,
v ≥ 0, v ò 0 in R
N
N
in R ,
p/ ( p – 1 )

 q1 + p – 1 
( p – 1 )  ------------------------------------2-
 p1 q1 – ( p – 1 ) 
ε
-.
v ( x ) : = ------------------------------------------------------------------------
+ p–1 
1 + x

p/ ( p – 1 )

p1
( p – 1 )  ------------------------------------2-
 p1 q1 – ( p – 1 ) 
If
p–1
p–1
p
----------------------------------------------q1 – p + 1
q1 – p + 1
q1 – p + 1
 ----------------------ε < min  p
( q1 + p – 1 )
( p – 1)

×
1
-----------------------q1 – p + 1
σv
,
p
p–1
-----------------------p1 – p + 1
× ( p – 1)
( q1 + p – 1 )
p
-----------------------p1 – p + 1
p–1
-----------------------q1 – p + 1
1
-----------------------p1 – p + 1 
σu
,

then (u, v) is a positive solution to problem (2.2).
DOKLADY MATHEMATICS
Vol. 59
No. 3
1999
NONEXISTENCE OF POSITIVE SOLUTIONS FOR A SYSTEM
355
The definition of the functions (u, v) can be modified so that they are a solution to problem (2.2) for p ≠ q.
This proves that Theorem 2.1 is also exact in this case.
(project no. 96-1060), and the Russian Foundation for
Basic Research (project no. 96-01-00097).
ACKNOWLEDGMENTS
1. Mitidieri, E. and Pohozaev, S.I., Dokl. Akad. Nauk, 1998,
vol. 359, no. 4, pp. 456–460.
2. Pohozaev, S.I., Dokl. Akad. Nauk, 1997, vol. 357, no. 5,
pp. 592–594.
3. Clement, Ph., Manasevich, R., and Mitidieri, E., Commun. Partial Differ. Equations, 1993, vol. 18, pp. 2071–
2106.
REFERENCES
Mitidieri acknowledges the support of the Italian
Foundation MURST, and Pohozaev acknowledges the
support of the Department of Mathematics of the University of Trieste, the international association INTAS
DOKLADY MATHEMATICS
Vol. 59
No. 3
1999