Electronic Journal of Differential Equations, Vol. 2004(2004), No. 147, pp. 1–18.
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu
ftp ejde.math.txstate.edu (login: ftp)
GENERALIZED SCALAR CURVATURE TYPE EQUATION ON
COMPLETE RIEMANNIAN MANIFOLDS
MOHAMMED BENALILI, YOUSSEF MALIKI
Abstract. In this work, we investigate positive solutions for a quasilinear
elliptic equation on complete manifold M . This equation extends to the pLaplacian the equation of the prescribed scalar curvature. A minimizing sequence is constructed which converges to a non trivial solution belonging to
C 1,α (K) for any compact set K ⊂ M and some α ∈ (0, 1).
1. Introduction
Let (M, g) be a complete Riemannian manifold of dimension n ≥ 3, with bounded
geometry, R(x) its scalar curvature and p ∈ (1, n). Let H1p (M ) be the standard
Sobolev space endowed with the norm
kukH1p (M ) = k∇ukLp (M ) + kukLp (M ) .
p
In this paper, we seek for a positive solution u ∈ H1,loc
(M ) to the equation
∆p u + a(x)up−1 = f (x)up∗−1 .
(1.1)
∗
p−2
pn
n−p .
where ∆p u = −div(|∇u| ∇u) is the p-Laplacian of u on M and p =
Our results extend those of Druet [2] obtained in the case of compact manifolds.
On complete Riemannian manifold conditions at infinity on f must be added.
n−2
In the case p = 2 and the function a(x) = n(n−1)
R(x), where R(x) is the scalar
curvature of the manifold M , the problem of the existence of a positive solution of
the equation (1.1) is originated from the study of pointwise conformal deformation
of Riemannian metric with prescribed scalar curvature. If in case p = 2, u is
a positive solution of (1.1) on (M, g), then the scalar curvature of the pointwise
4
conformal metric g 0 = u n−2 g is 4(n−1)
n−2 f (cf. [2]). The equation (1.1) is referred as
the generalized scalar curvature type equation.
Our main result in this paper is as follows.
Theorem 1.1. Let (M, g) be a complete non-compact Riemannian n-manifold with
n ≥ 3 , 1 < p < n such that p2 < n. Let a, f ∈ C ∞ (M ) be real valued function on
M . Suppose that operator Lp u = ∆p u + a(x)up−1 is coercive. Under the following
assumptions:
(1) At a point xo where f is maximal, we are in one of the following cases
2000 Mathematics Subject Classification. 58J05.
Key words and phrases. Complete manifolds; p-Laplacian; critical Sobolev exponent.
c
2004
Texas State University - San Marcos.
Submitted November 26, 2004. Published December 13, 2004.
1
2
M. BENALILI, Y. MALIKI
(i) p < 2, n > 3p − 2 and a(xo ) < 0
8(n−1)
(xo )
(ii) p = 2 and (n−2)(n−4)
a(xo ) < −∆f
f (xo ) +
EJDE-2004/147
2R(xo )
n−4
(xo )
(iii) p > 2 and ( n−3p+2
) ∆f
p
f (xo ) < R(xo ).
(2) (M, g) is of bounded geometry, that is: Ricci > −c, where c ≥ 0 is a
constant, and the injectivity radius is strictly positive.
(3) There exists a constant C > 0 such that
Z
Z
Z
n
f dvg < ∞ and
f p/p∗ dvg < ∞.
|∇f | ≤ Cf, |∇2 f | ≤ Cf,
|a| p dvg ≤ C
M
M
M
(4) The functions a and f are bounded and f is strictly positive.
p
Then, there exists a positive solution u ∈ H1,loc
(M ) of (1.1) such that u ∈ C
on any compact set K of M for some α ∈ (0, 1).
1,α
(K)
This article is organized as follows: in the second section we construct a sequence
of minimizing weak solutions, in the third section we give sufficient geometric conditions to guarantee the strong convergence of the minimizing sequence. Using the
Aubin’s test functions, we show in the last section that these geometric conditions
are satisfied.
2. Convergence of the minimizing sequence
In this section, we construct a sequence of weak solutions for (1.1). The following
theorem has been proved in [2].
Theorem 2.1. Let (M, g) be a Riemannian compact manifold 1 < p < n such
that p2 < n, and let a, f ∈ C ∞ (M ) be real functions on M . We assume that the
operator Lp u = ∆p u + a(x)up−1 is coercive. If at a point xo where f is maximal,
we have one of the following cases
(i) p < 2, n > 3p − 2 and a(xo ) < 0
(p)
2R(xo )
8(n−1)
(ii) p = 2 and (n−2)(n−4)
a(xo ) < −∆f
f (p) + n−4
(xo )
(iii) p > 2 and ( n−3p+2
) ∆f
p
f (xo ) < R(xo ).
Then, there exists a positive solution u ∈ H1p (M ) of (1.1) such that u ∈ C
for some α ∈ (0, 1).
1,α
(M )
Let Ωj be an exhaustion of the complete manifold M by compact manifolds
o
with smooth boundary such that Ωj ⊂ Ωj+1 . Let uj be the minimizer given by
Theorem 2.1 for
∆p uj + a(x)ujp−1 = µ(Ωj )f up∗−1
j
uj > 0
in Ωj
in Ωj
(2.1)
uj = 0 on ∂Ωj .
By the monotone decreasness of µ(Ωj ) and the coercivity of the operator Lp u =
∆p u + a(x)up−1 , we have
1
kukH1p (Ωj ) ≤ µ(Ω1 )
(2.2)
c
where c > 0 is a constant. Since (2.2) implies the boundedness of {ui } in H1p (M ),
we can choose a subsequence of {ui } still denoted {ui } such that ui → u weakly in
H1p (M )
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GENERALIZED SCALAR CURVATURE TYPE EQUATION
3
Proposition 2.2. The sequence {ui } converges weakly on every compact set K of
M to a solution u ∈ C 1,α (K) of
∆p u + a(x)up−1 = f up∗−1
u > 0 in K
u=0
in K
(2.3)
on ∂K
for some α ∈ (0, 1).
To prove the boundedness of {ui }in C 1,α (K), we use propositions from the paper
of Druet [2] which have their origin in Tolksdorf [7] Guedda and Veron [3] and
Vazquez [8].
Proposition 2.3. Let (M, g) be a compact Riemannian n-manifold. Assume that
u ∈ H1p (M ) is a solution of ∆p u + a(x)up−1 = f , where n ≥ 2, 1 < p < n, a(x) ∈
n
n
L p (M ) and f ∈ L p (M ), then u ∈ Lt (M ) for t ∈ [1, ∞).
Proposition 2.4. Let (M, g) be a compact Riemannian n-manifold. Assume n ≥ 2,
1 < p < n, f ∈ Ls (M ) for some s > np and u ∈ H1p (M ) is a solution of ∆p u = f
on M . Then u ∈ L∞ (M ).
Proposition 2.5. Let (M, g) be a compact Riemannian n-manifold and h(x, r) ∈
C o (M × R). Assume n ≥ 2, 1 < p < n and ∀(x, r) ∈ M × R, |h(x, r)| ≤
1,α
C|r|p∗−1 + D. If u ∈ H1p (M ) is a solution of ∆p u + h(x, u) = 0, then u ∈ C (M ).
Moreover kukC 1,α (M ) ≤ c̃, where c̃ is a constant depending only on kukL∞ (M ) and
kh(x, r)kL∞ (M ) .
Proof of Proposition 2.2. First we show that the sequence {uj } is bounded in Lt (K)
for any t ∈ [1, +∞). Involving Proposition 2.3, we have only to check that the
n
sequence {a(x) − f up∗−p
} is bounded in L p (K). We have
j
Z
Z
n
n
n
n
p dv ≤ 2 p −1
|a(x) − f up∗−p
|
(|a(x)| p + |f | p up∗
g
j )dvg
j
K
K
n
n
K n
K p∗
p
n ) p + (kf k
= 2 p −1 (kakK
∞ ) (kuj kp∗ )
p
1/p
R
where kukK
|u|p dvg
. Since by the relation(3) the sequence{uj } is
p =
K
bounded in Lp (K), so is in Lp∗ (K), we have the desired conclusion.
Next, we show that {uj } is bounded in L∞ (K). According to Proposition
2.4, we have to show that the sequence {gj } given by gj (x) = −a(x)uj (x)p−1 +
f (x)uj (x)p∗−1 , is bounded in Ls (K) for some s > np . But this fact is a consequence
of proposition 2.3.
Finally, we take h(x, uj ) = a(x)up−1
− f (x)up∗−1
, and since by assumption the
j
j
functions a and f are bounded on the manifold M , one has the boundness of the
sequence {h(x, uj (x)} in the compact set K.
By proposition 2.5, uj ∈ C 1,α (K) and kuj k1,α
K ≤ c(p, n, K, kgj kL∞ (K) ). The
boundedness of {uj } in L∞ (K) implies that {gj } and C(p, n, K, kgj kL∞ (K) ) are
bounded. Consequently, {uj } is bounded in C 1,α (K). So by Arzela-Ascoli theorem
{uj } converges uniformly towards a weak solution u of (2.3) on each compact
set.
4
M. BENALILI, Y. MALIKI
EJDE-2004/147
3. Strong convergence
In this section, we have to show that the solution u is not trivial. To achieve
this task, we give sufficient conditions that guarantee the strong convergence of
minimizers constructed in the previous section. Let K be any compact set of the
complete manifold M , 2K a compact set containing K and η ∈ C ∞ (M ) be the
function
(
0 on K
η(x) =
1 on M − 2K .
Let k > 1 and {uq } be the sequence of minimizers given by Proposition 2.2 and
1
k+p−1
k.kp be the Lp (M )-norm. we are going to estimate the ratio k∇(ηf p∗ uq p )kp .
Letting {Ωk } be the exhaustion, of the complete
R manifold M , considered in the
previous section. Denote by Λk = {u ∈ H1p (Ωk ) : Ωk f |u|p∗ dvg = 1} and Ik (u) the
R
functional Ik (u) = Ωk (|∇u|p + |u|p )dvg .
Proposition 3.1. Under the conditions (2), (3), (4), of Theorem 1.1 and
( sup f (x))p/p∗ inf Ik (u) < K(n, p)−p ,
u∈Λk
M −K
1
k+p−1
p
the ratio k∇(ηf p∗ uq
)kp is bounded.
Proof. For p ≥ 2, using Simon’s inequality [5], that is to say: for any vector fields
X and Y on the manifold M ,
|X + Y |p ≤ Cp |X|p−2 X + |Y |p−2 Y, X + Y
where Cp is a constant depending on p and h., .i denoting the metric. We get
1
k+p−1
k∇(ηf p∗ uq p )kpp
Z
k+p−1
1
=
|∇(ηf p∗ uq p )|p dvg
ZM
k+p−1
k−1
1
1
k+p−1
=
|(uq p ∇(ηf p∗ ) +
(ηf p∗ )uq P ∇uq )|p dvg
p
M
Z h k+p−1
1
1
( p )(p−1)
≤ Cp
uq
|∇(ηf p∗ )|p−2 ∇(ηf p∗ )
M
( k−1
p )(p−1)
1
)p−1 (ηf p∗ )p−1 uq
+ ( k+p−1
p
|∇uq |p−2 ∇uq
i
k−1
k+p−1
1
1
p∗ )p−1 u p ∇u
× uq p ∇(ηf p∗ ) + k+p−1
(ηf
q
q dvg
p
Z
hZ
1
1
p
= Cp
uqk+p−1 |∇(ηf p∗ )|p dvg + ( k+p−1
)
(ηf p∗ )p uk−1
|∇uq |p dvg
q
p
M
M
Z
1
1
1
k+p−1
k+p−2
p−2
p∗
p∗
× p
ηf uq
|∇(ηf )| h∇(ηf p∗ ), ∇uq idvg
M
Z
1
1
k+p−1 p−1
(ηf p∗ )p−1 ukq |∇uq |p−2 h∇(ηf p∗ ), ∇uq idvg ].
+( p )
M
On the other hand,
Z
Z
p p/p∗ k
η f
uq ∆p uq dvg = k
M
M
η p f p/p∗ uk−1
|∇uq |p dvg
q
EJDE-2004/147
GENERALIZED SCALAR CURVATURE TYPE EQUATION
Z
+p
5
1
1
(ηf p∗ )p−1 ukq |∇uq |p−2 h∇uq , ∇(ηf p∗ )idvg
M
and
Z
1
1
∆p (ηf p∗ )dvg
ηf p∗ uk+p−1
q
M
Z
1
=
uk+p−1
|∇(ηf p∗ )|p dvg
q
M
Z
1
1
1
+ (k + p − 1)
|∇(ηf p∗ )|p−2 h∇uq , ∇ηf p∗ idvg
(ηf p∗ )uk+p−2
q
M
so
1
k+p−1
k∇(ηf p∗ uq p )kpp
Z
hZ
1
1
1 k+p−1 p
p∗ )dv +
≤ Cp
ηf p∗ uk+p−1
∆
(ηf
(
)
η p f p/p∗ ukq ∆p uq dvg
p
g
q
k
p
M
M
Z
1
1
1
p−1
− p (k + p − 1)
ηf p∗ uqk+p−2 |∇(ηf p∗ )|p−2 h∇uq , ∇(ηf p∗ )idvg
ZM
i
1
1
k+p−1 p−1 p−1
−( p )
(ηf p∗ )p−1 ukq |∇uq |p−2 h∇uq , ∇(ηf p∗ )idvg .
k
M
1
Multiplying (1.1) by (ηf p∗ )p ukq and integrating over M , we get
Z
1
(ηf p∗ )p ukq ∆p uq dvg
M
Z
Z
1
1
p k+p−1
p∗
=−
a(x)(ηf ) uq
dvg + µ(Ωq )
(ηf p∗ )p f uk+p∗−1
dvg .
q
M
(3.1)
M
Using Hölder inequality, we obtain
Z
1
(ηf p∗ )p f uk+p∗−1
dvg
q
M
Z
p
Z
1− p∗
p/p∗
k+p−1
1
≤ ( sup f )p/p∗
f up∗
dv
(ηf p∗ uq p )p∗ dvg
.
g
q
M −K
M −K
M
The first term of the right-hand side of (3.1) is estimated as
Z
1
a(x)(ηf p∗ )p uqk+p−1 dvg
M
Z
np Z
p/p∗
k+p−1
n
1
≤
|a(x)| p dvg
(ηf p∗ uq p )p∗ dvg
.
M −K
M −K
Since by assumption
Z
n
|a(x)| p dvg
p/n
Z
f dvg < ∞,
<C
M
M
we choose the compact set K so that
Z
f dvg <
M −K
ε
.
C
(3.2)
6
M. BENALILI, Y. MALIKI
EJDE-2004/147
Then
Z
1
(ηf p∗ )p ukq ∆p uq dvg
M
Z
Z
p
1− p∗
≤ ( sup f )p/p∗
f up∗
)
+
ε
q
M −K
M
1
k+p−1
p
ηf p∗ uq
p∗
dvg
p/p∗
(3.3)
.
M
On the other hand we have
1
1
∇(ηf p∗ ) = f p∗ ∇η +
1
1
ηf p∗ −1 ∇f
p∗
and since by assumption |∇f | ≤ C, we obtain
1
1
|∇ηf p∗ | ≤ f p∗ |∇η| +
1
1
ηc p∗
f ≤ Cf p∗
p∗
where C is a universal constant. So
Z
1
1
1
ηf p∗ uk+p−2
|∇(ηf p∗ )|p−2 h∇(ηf p∗ ), ∇uq idvg
q
M
Z
p
≤C
f p∗ uk+p−2
|∇uq |dvg .
q
(3.4)
M −K
Using Hölder inequality we obtain that the right-hand side of this inequality is
bounded above by
Z
f p/p∗ uk+p−2
|∇uq |dvg
q
M −K
Z
≤
p1 Z
p
|∇uq | dvg
M −K
(f p/p∗ uk+p−2
q
p
p−1
(3.5)
1
1− p
dvg )
.
M −K
Applying Hölder’s inequality again, we get
Z
p
(f p/p∗ uqk+p−2 ) p−1 dvg
M −K
≤
Z
up∗
q dvg
p(k+p−2)
Z
p∗(p−1)
M −K
≤ ( sup f )
p(p−1)
f p∗(p−1)−p(k+p−2) dvg
1− p(k+p−2)
p∗(p−1)
(3.6)
M −K
p(k+p−2)
p∗(p−1)
Z
M −K
up∗
q dvg
p(k+p−2)
Z
p∗(p−1)
M −K
f
p/p∗
dvg
1− p(k+p−2)
p∗(p−1)
M −K
As above, we get
Z
1
1
(ηf p∗ )p−1 ukq |∇uq |p−2 h∇uq , ∇(ηf p∗ )idvg
M
Z
≤C
f p/p∗ ukq |∇uq |p−1 dvg
M −K
≤C
Z
M −K
f p/p∗ up∗
q
k Z
p∗
M −K
f p/p∗ |∇uq |
p∗(p−1)
p∗−k
k
1− p∗
.
.
EJDE-2004/147
GENERALIZED SCALAR CURVATURE TYPE EQUATION
p∗
Since α = (p − 1) p∗−k
, we have p − α = p(p∗−k)−p∗(p−1)
= p∗−pk
p∗−k
p∗−k > 0 and
Z
f p/p∗ |∇uq |α dvg
M −K
1− αp
Z
αp Z
p2
≤
|∇uq |p dvg
f p∗(p−α) dvg
M −K
M −K
Z
αp Z
1− αp
p(p−1)
≤ ( sup f ) p∗−pk
|∇uq |p dvg
f p/p∗ dvg
.
M −K
M −K
7
(3.7)
M −K
On the other hand,
1
1
1
∆p (ηf p∗ ) = − div(|∇(ηf p∗ )|p−2 ∇(ηf p∗ ))
1
1
1
1
= |∇ηf p∗ |p−2 ∆(ηf p∗ ) − trace ∇|∇(ηf p∗ )|p−2 ⊗ ∇(ηf p∗ )
and
1
1
1
1
∆(ηf p∗ ) = f p∗ ∆η + η∆f p∗ − trace(∇η ⊗ ∇f p∗ )
1
1
1
1
1
1
1 1
≤ f p∗ ∆η + (1 − )ηf p∗ −2 |∇f |2 + ηf p∗ −1 ∆f + f p∗ −1 |∇f |
p∗
p∗
p∗
p∗
then
1
1
|∆(ηf p∗ )| ≤ Cf p∗
and
p−1
1
1
|∇(ηf p∗ )|p−2 |∆(ηf p∗ )| ≤ Cf p∗ .
From
1
1
1
|∇|∇(ηf p∗ )|p−2 | = (p − 2)|∇(ηf p∗ )|p−3 |∇|∇(ηf p∗ )||
and Kato’s inequality, we deduce that
1
1
1
∇|∇(ηf p∗
)|p−2 ≤ (p − 2)|∇(ηf p∗ )|p−3 |∇2 (ηf p∗ )|.
Now, since
1
∇2 (ηf p∗ )
1
= f p∗ ∇2 η +
1
1
2 p∗
1
1
1
f −1 ∇η ⊗ ∇f + (1 − )ηf p∗ −2 ∇f ⊗ ∇f + ∇2 f ,
p∗
p∗
p∗
p∗
we obtain
p−1
1
∇|∇(ηf p∗
)|p−2 ≤ Cf p∗ .
Finally, we get
1
|∆p (ηf p∗ )| ≤ Cf
and
Z
1
p−1
p∗
1
ηf p∗ uk+p−1
∆p (ηf p∗ )dvg
q
M
Z
≤C
f p/p∗ uk+p−1
q
(3.8)
M −K
≤C
Z
up∗
q dvg
M −K
Sobolev’s inequality leads to
Z
p/p∗
up∗
q dvg
M −K
k+p−1 Z
p∗
M −K
p∗
f p∗k−p+1 dvg
1− k+p−1
p∗
.
8
M. BENALILI, Y. MALIKI
≤ (K(n, p)p + ε)
Z
|∇uq |p dvg + A
EJDE-2004/147
Z
M −K
upq dvg
M −K
Z
≤ (K(n, p) + ε)
A
|∇uq | dvg +
K(n,
p)p + ε
M −K
p
p
Z
upq dvg .
M −K
p−2
From the coercivity of the operator Lp u = −∆p u − a(x)|u| u, we get
Z
p/p∗
up∗
q dvg
M −K
Z
A
1
≤ (K(n, p)p + ε) max 1,
(|∇uq |p + upq )dvg
c
K(n, p)p + ε M −K
Z
≤ C̃
(|∇uq |p + upq )dvg ,
M
1
p
c (K(n, p)
A
+ ε) max 1, K(n,p)
and by construction of the sequence
where C̃ =
p +ε
{uq }, which has a compact support in Ωq , we have
Z
(|∇uq |p + a(x)upq )dvg = λq
M
hence
Z
up∗
q dvg
p/p∗
≤ C̃λq .
M −K
Since by assumption the Lagrange multipliers satisfy
1
λq <
,
K(n, p)p (supM −K f )p/p∗
we have
Z
up∗
q dvg
p/p∗
≤ C.
(3.9)
M −K
Combining inequalities (5) to (13) we obtain
Z
k+p−1
p
p/p∗
1− p∗
1
p
p
p∗
k∇(ηf uq
)kp ≤ λq
sup f
f up∗
+ε
q dvg
M −K
M −K
Z
p/p∗
k+p−1 1
p∗
×
ηf p∗ uq p
dvg
+C.
M −K
Using Sobolev’s inequality, this expression is bounded by
Z
p
p/p∗
1− p∗
λq
sup f
f up∗
+ε
q dvg
M −K
M −K
1
p
k+p−1
p
× (K(n, p) + ε)k∇(ηf p∗ uq
1
k+p−1
p
)kpp + Akηf p∗ uq
(3.10)
kpp + C ,
where K(n, p) is the best constant in the Sobolev’s inequality. For the last term in
(3.10), we write
Z
f p/p∗ uk+p−1
dvg
q
M −K
≤
Z
up∗
q dvg
κ+p−1
Z
p∗
M −K
≤
sup f
M −K
p
f p∗−k−p+1 dvg
1− κ+p−1
p∗
M −K
p(1−p−k)
2
p∗
Z
up∗
q dvg
κ+p−1
Z
p∗
f p/p∗ dvg
1− κ+p−1
p∗
< ∞.
EJDE-2004/147
GENERALIZED SCALAR CURVATURE TYPE EQUATION
1
9
k+p−1
From the assumption on the Lagrange multipliers, k∇(ηf p∗ uq p )kpp is bounded.
In the case 1 < p < 2 , the Simon’s inequality writes
p
p
|X + Y |p ≤ Cp |X|p−2 X + |Y |p−2 Y, X + Y 2 (|X|p + |Y |p )1− 2
where X, Y are any vector fields on the manifold M.
Putting
k+p−1
p
X = uq
Y =
1
∇(ηf p∗ ),
k−1
k−1 k + p − 1
1
1
k+p−1
(ηf p∗ )uq P ∇uq
(ηf p∗ )uq P
p
p
we get
k+p−1
p
k−1
1
1
k+p−1
(ηf p∗ )uq P ∇uq dvg
uq p ∇(ηf p∗ ) +
p
h ( k+p−1 )(p−1)
1
1
≤ Cp uq p
|∇(ηf p∗ )|p−2 ∇(ηf p∗ )
1
( k−1 )(p−1)
+ ( k+p−1
)p−1 (ηf p∗ )p−1 uq p
|∇uq |p−2 ∇uq
p
k+p−1
ip/2
k−1
1
1
k+p−1
× uq p ∇(ηf p∗ ) +
(ηf p∗ )uq P ∇uq
p
h
i1− p2
1
1
k+p−1 p
p∗ )|p + (
p∗ )p uk−1 |∇u |p
× uk+p−1
|∇(ηf
)
(ηf
.
q
q
q
p
Then
k+p−1 1
∇ ηf p∗
p
uq p
p
Z h
1
1
k+p−1 p
≤ Cp
uk+p−1
|∇(ηf p∗ )|p + (
) (ηf p∗ )p uk−1
|∇uq |p
q
q
p
M
1
1
1
k+p−1
(ηf p∗ )uk+p−2
|∇(ηf p∗ )|p−2 h∇uq , ∇(ηf p∗ )i
+
q
p
ip/2
1
1
k + p − 1 p−1
+(
) (ηf p∗ )p−1 ukq |∇uq |p−2 h∇uq , ∇(ηf p∗ )i
p
h
i1− p2
1
1
k+p−1 p
p
× uqk+p−1 |∇(ηf p∗ )|p + (
) (ηf p∗ )p uk−1
|∇u
|
dvg .
q
q
p
And by Hölder’s inequality, the above expression is less than or equal to
Z h
1
1
k+p−1 p
Cp
uqk+p−1 |∇(ηf p∗ )|p + (
) (ηf p∗ )p uk−1
|∇uq |p
q
p
M
1
1
1
k+p−1
+
(ηf p∗ )uk+p−2
|∇(ηf p∗ )|p−2 h∇uq , ∇(ηf p∗ )i
q
p
i 2/p
1
1
k + p − 1 p−1
+(
) (ηf p∗ )p−1 ukq |∇uq |p−2 h∇uq , ∇(ηf p∗ )i dvg
p
Z
1− p2
1
1
k+p−1 p
p∗ )|p + (
p∗ )p uk−1 |∇u |p )dv
×
.
(uk+p−1
|∇(ηf
)
(ηf
q
g
q
q
p
M
1
k+p−1
p
Arguing as in the case p ≥ 2, we obtain that k∇(ηf p∗ uq
)kpp is bounded.
10
M. BENALILI, Y. MALIKI
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4. Generic Theorem
Letting K be any compact set of the complete manifold M , we formulate in this
section a generic theorem. First, we establish
Lemma 4.1. Assume that every subsequence of {uq } which converges in Lp (M )
with p > 1, converges to 0. Also assume there exists a constant C > 0, independent
k+p−1
p
1
of q such that k∇(ηf p∗ uq
)kpp ≤ C, k > 1. Then
Z
1
lim sup
(ηf p∗ uq )p∗ dvg = 0.
q→∞
M
Proof. Suppose that limq→∞ sup
obtain
Z
1
(ηf p∗ uq )p∗ dvg
1
(ηf p∗ uq )p∗ > 0. Using Hölder’s inequality we
R
M
M
≤ sup f
M −K
Z
ηf
k+p−1
p
1
p∗
uq
n(p−1)+p p∗ n(k+p−1)
Z
M
n(k+p−1)
nk−p
uq
dvg
nk−p
n(k+p−1)
M
then
Z
n(k+p−1)
nk−p
lim sup
uq
q→∞
dvg > 0.
M
A contradiction with the fact that every subsequence of uq converging in Lp (M ),
p > 1, converges to 0.
As a consequence of the above
R lemma, we obtain the following generic theorem.
Denote by RΛ = {u ∈ H1p (M ) : M f |u|p∗ dvg = 1} and I(u) is the functional given
by I(u) = M (|∇u|p + |u|p )dvg where M is a complete Riemannian manifold.
Theorem 4.2. Let (M, g) be a complete Riemannian manifold of bounded geometry, 1 < p < n, and let a, f ∈ C ∞ (M ) be real functions on M with f > 0. We
assume that:
(i) The operator Lp u = ∆p u + a(x)up−1 is coercive
(ii) Conditions (3) and (4) of Theorem 1.1 at infinity on f are satisfied
p
(iii) (supM f ) p∗ inf u∈Λ I(u) < K(n, p)−p .
Then (1.1) possesses a positive solution u ∈ C 1,α (K) for any compact set K ⊂ M
and some α ∈ (0, 1).
Proof. Suppose that
p/p∗
µf (x))
Z
p
f up∗
q dvg < 1
K(n, p) lim sup
q→∞
B(xo ,δ)
then by Lemma8, we get that
Z
f up∗
q dvtg = 0
lim sup
q→∞
B(xo ,δ)
which contradicts the fact that
Z
f up∗
q dvg = 1.
(4.1)
M
In fact
Z
M
f up∗
q dvg
Z
=
∪∞
i=1 B(xi ,δ)
f up∗
q dvg
≤
∞ Z
X
i=1
B(xi ,δ)
f up∗
q dvg
(4.2)
EJDE-2004/147
GENERALIZED SCALAR CURVATURE TYPE EQUATION
11
where M = ∪∞
i=1 B(xi , δ).
So for sufficient large q the last term in (4.2) is strictly smaller that 1. Consequently
Z
µf (x))p/p∗ K(n, p)p lim sup
f up∗
q dvg ≥ 1
q→∞
B(xo ,δ)
p/p∗
K(n, p)p < 1, we obtain
Z
lim sup
f up∗
q dvg > 1.
and since by assumption µf (x))
q→∞
B(xo ,δ)
which is a contradiction with (4.1).
Then the condition that every subsequence of the sequence of minimizers {uq }
which converges has 0 as a limit is false and the theorem is proved.
Examples of functions satisfying the conditions of Theorem 4.2. The conditions at infinity in Theorem 4.2 are satisfied, for example by functions decreasing
like power functions: f ∼ r−q , ∇f ∼ ρ−q−1 and ∇2 f ∼ r−q−2 with q > n p∗
p . Since
R
1
p/p∗
is
integrable.
f
dv
<
+∞
implies
that
1
g
M
r (1− n )q+1−n
R
n
−q
p
RIf the function a decays at infinity as r , then the condition that M f dvg ≤
C M f dvg < +∞ implies that the decay rate q satisfies q > p.
5. Test functions
In this section we give the proof of our main result (Theorem 1.1). For this
task we check that the condition (iii) of the generic theorem proved in section 4 is
satisfied.
Let K be any compact set of the manifold M and xo ∈ M − K be the maximum
on of the function f as given in Theorem 1.1. Let r = d(xo , x) the distance function
from xo to any point x in the manifold M − K.
Let δ > 0 be smaller than the injectivity radius; for > 0, we consider the test
function
(
p
p
n
n
(ε + r p−1 )1− p − (ε + δ p−1 )1− p if r < δ
uε (x) =
0
if r ≥ δ .
Note that the function uε was introduced by Aubin in [1]. We have
(
p
p
n
p
p−1 )− p r ( p−1 )
( n−p
if r < δ
p
p−1 ) (ε + r
|∇uε (x)| =
0
if r ≥ δ
so
n−p p
|∇uε (x)| dvg = (
)
p−1
B(xo ,δ)
Z
p
Z
δ
(ε + r
p
p−1
1
−n n+ p−1
)
r
Z
dr
dΩ.
(5.1)
S n−1 (r)
0
n−1
where dΩ denotes
(r).
R the element volume on the sphere S
Let S(r) = S n−1 (r) dΩ. Taking into account the expansion of the determinant
in a system of geodesic coordinates at a point xo , we get
√
g = 1 − Rij xi xj + o(r2 ) .
A computation in [1] gives us
S(r) = ωn−1 (1 −
R 2
r + o(r2 ))
6n
12
M. BENALILI, Y. MALIKI
EJDE-2004/147
where wn−1 is the volume of the standard unit sphere S n−1 in Rn . The integral
(5.1) becomes
Z
|∇uε (x)|p dvg
B(xo ,δ)
=(
n−p p
) ωn−1
p−1
Letting s = rε
1−p
p
δ
Z
p
1
(ε + r 1−p )−n rn+ p−1 1 −
0
R 2
r + o(r2 ) dr.
6n
, we get
Z δε 1−p
p
p −n
1
n−p p
1− n
sn+ p−1
) ωn−1 ε p
1 + s 1−p
|∇uε (x)| dvg = (
p−1
0
B(xo ,δ)
2(p−1) R 2 2(p−1)
× 1−
s ε p + o(s2 ε p ) ds,
6n
Z
p
(5.2)
set
Ipq
∞
Z
tq−1 (1 + t)−p dt
=
with p − q − 1 > 0,
0
1
Z
tp−1 (1 − t)q−1 dt
B(p, q) =
with p > 0, q > 0 .
0
p
Put t = s p−1 , then the integral (5.2) becomes
Z
|∇uε (x)|p dvg
B(xo ,δ)
p
h Z δ p−1 ε−1
1
p−1 n−p p
1− n
=
(
) ωn−1 ε p
(1 + t)−n tn(1− p ) dt
p
p−1
0
p
Z δ p−1
ε−1
i
2(p−1)
1 R(xo )
1
− ε2(1− p )
(1 + t)−n t(n+2)(1− p ) dt + o(ε p )
6n
0
hZ ∞
1
p−1 n−p p
1− n
p
(
) ωn−1 ε
(1 + t)−n tn(1− p ) dt
=
p
p−1
0
Z ∞
Z ∞
1 R(xo )
1
1
2(1− p
)
)
−n (n+2)(1− p
−ε
(1 + t) t
dt −
(1 + t)−n tn(1− p ) dt
p
6n
−1
p−1
δ
ε
Z0 ∞
i
2(p−1)
1 R(xo )
1
2(1− p
)
)
−n (n+2)(1− p
p
+ε
(1
+
t)
t
dt
+
o(ε
)
.
p
6n
δ p−1 ε−1
We have
Z
lim
ε→0
∞
p
δ p−1
1
(1 + t)−n tn(1− p ) dt = 0
ε−1
and if n + 2 > 3p, then
Z
lim
ε→0
So
Z
∞
p
δ p−1
1
(1 + t)−n t(n+2)(1− p ) dt = 0 .
ε−1
|∇uε (x)|p dvg
B(xo ,δ)
=
h n(1− 1 )
i
n
1 R(xo ) (n+2)(1− 1 )
1
p−1 n−p p
p
p
(
) ωn−1 ε1− p In
− ε2(1− p )
In
+ o(ε2(1− p ) ) .
p
p−1
6n
EJDE-2004/147
GENERALIZED SCALAR CURVATURE TYPE EQUATION
13
On the other hand, a simple computation, for (p > q + 1), gives the following
formula
Γ(q + 1)Γ(p − q − 1)
Ipq = B(q + 1, p − q − 1) =
,
Γ(p)
where Γ denotes the Euler function from which we obtain the following relation
1
)
(n+2)(1− p
In
Γ((n + 2)(1 − p1 ) + 1)Γ( n+2
p − 3)
=
Γ(n(1 − p1 ) + 1)Γ( np − 1)
1
)
n(1− p
In
1
n(1− p
)
= a(n, p)In
.
Finally the equality (5.2) becomes
Z
|∇uε (x)|p dvg
B(xo ,δ)
n n(1− 1 )
1
1
p−1 n−p p
R(xo )
p
(
) ωn−1 ε1− p In
[1 − ε2(1− p ) a(n, p)
+ o(ε2(1− p ) )].
p
p−1
6n
R
The expansion of B(xo ,δ) aupε dvg is computed in the same way as above,
Z
aupε dvg
(5.3)
=
B(xo ,δ)
δ
Z
upε rn−1 dr
=
0
δ
Z
√
a gdΩ
Z
S n−1 (r)
p
p
n
(ε + r p−1 )1− p − ν rn−1 dr
=
0
Z
1
1
a(xo ) + ∇ij a(xo )xi xj + o(r2 ) 1 − Rij (xo )xi xj + o(r2 ) dΩ
2
6
S n−1 (r)
Z δ
h
i
p
p
p
n
n
= ωn−1 a(xo )
rn−1 (ε + r p−1 )p−n 1 − pν(ε + r p−1 ) p −1 + o (ε + r p−1 ) p −1
×
0
i
R(xo )
∆a(xo ) 2
× 1−
+
r + o(r2 ) dr .
6n
2nR(xo )
h
1−p
Putting s = r p , we get
Z
aupε dvg
B(xo ,δ)
= ωn−1 a(xo )ε
p− n
p
δε
Z
1−p
p
h
p
p
n
sn−1 (1 + s p−1 )p−n 1 − pνε p −1 (1 + s p−1 )
0
+ o(1 + s
p
p−1
)
n
p −1
ih
i
n
1
R(xo )
∆a(xo ) 2 2(1− p1 )
ε p −1 1 −
+
s ε
+ ε2(1− p ) o(s2 ) ds.
6n
2na(xo )
p
Letting t = s p−1 , we get
Z
aupε dvg
B(xo ,δ)
p
Z δ p−1
ε−1
1
p−1
p− n
=
ωn−1 a(xo )ε p
tn(1− p )−1 (1 + t)p−n
p
0
h
i
n
n
−1
× 1 − pνε p (1 + t) + o(1 + t)ε p −1
14
M. BENALILI, Y. MALIKI
EJDE-2004/147
i
1
1
∆a(xo ) 2(1− p1 ) 2(1− p1 )
R(xo )
+
t
ε
+ ε2(1− p ) o(t2(1− p ) ) dt
6n
2na(xo )
p
Z δ p−1
ε−1
n
1
p−1
p− n
=
ωn−1 a(xo )ε p
tn(1− p )−1 (1 + t)p−n dt + o(εp− p ) .
p
0
× [1 −
Since p < n2 , we have
∞
Z
lim
1
p
δ p−1
ε→0
tn(1− p )−1 (1 + t)p−n dt = 0 ;
ε−1
therefore,
Z
aupε dvg =
n n(1− 1 )−1
n
p−1
ωn−1 a(xo )εp− p In−p p
+ o(εp− p ).
p
From the formulae
n(1− 1 )−1
In−p p
Γ n(1 − p1 ) Γ( np − p)
=
,
Γ(n − p)
n(1 − p1 )Γ n(1 − p1 ) Γ( np − 1)
Γ n(1 − p1 ) + 1 Γ( np − 1)
=
=
Γ(n)
Γ(n)
1
n(1− p
)
In
we deduce that
1
n(1− p
)−1
In−p
Finally, we get
Z
=
Γ(n)Γ( np − p)
n(1 −
1
p )Γ(n
n
aupε dvg = εp− p
B(xo ,δ)
−
p)Γ( np
1
n(1− p
)
− 1)
In
1
n(1− p
)
= b(p, n)In
.
1
n
n(1− p
)
p−1
+ o(εp− p ).
ωn−1 a(xo )b(p, n)In
p
(5.4)
Now, we compute the term
Z
f up∗
ε dvg
B(xo ,δ)
δ
Z
=
p
p−1
−ν
p∗
rn−1 (ε + r p−1 )1− p − ν
p∗
r
n−1
(ε + r
1− n
p
)
dr
S n−1 (r)
0
δ
Z
√
f gdΩ
Z
p
=
n
dr
0
Z
1
1
f (xo ) + ∇ij f (xo )xi xj + o(r2 ) 1 − Rij (xo )xi xj + o(r2 ) dΩ
2
6
×
S n−1 (r)
δ
Z
p
n
rn−1 (ε + r p−1 )1− p − ν
=
p∗
dr
0
Z
×
h
f (xo ) +
S n−1 (r)
Z
δ
i
1
Rij (xo ) i j
∇ij f (xo ) − f (xo )
x x + o(r2 ) dΩ
2
6
p
0
Z
= ωn−1 f (xo )
0
R(xo )
∆f (xo ) 2
+
r + o(r2 ) dr
6n
2nf (xo )
R(xo )
∆f (xo ) 2
+
r + o(r2 ) dr .
1−
6n
2nf (xo )
n
rn−1 (ε + r p−1 )1− p − ν
= ωn−1 f (xo )
δ
p
rn−1 (ε + r p−1 )−n
p∗
1−
EJDE-2004/147
GENERALIZED SCALAR CURVATURE TYPE EQUATION
Letting s = rε
Z
1−p
p
, we get
f up∗
ε dvg
B(xo ,δ)
Z δε 1−p
p
p
−n
p−1
p
=
ωn−1 f (xo )ε
sn−1 (1 + s p−1 )−n
p
0
1 R(xo )
∆f (xo ) 2 2(1− p1 )
× 1−
+
s ε
+ o(s2 )ε2(1− p ) ds .
6n
2nf (xo p)
p
Setting t = s p−1 , we obtain
Z
f up∗
ε dvg
B(xo ,δ)
p
Z δ p−1
ε−1
−n
1
p−1
=
ωn−1 f (xo )ε p
tn(1− p )−1 (1 + t)−n
p
0
1
1 R(xo )
∆f (xo ) 2(1− p1 ) 2(1− p1 )
× 1−
+
t
ε
+ o(t2(1− p ) )ε2(1− p ) dt.
6n
2nf (xo )
Since
Z
lim
ε→0
and
Z
lim
ε→0
∞
p
δ p−1
1
tn(1− p )−1 (1 + t)−n dt = 0
ε−1
∞
p
δ p−1
1
t(n+2)(1− p )−1 (1 + t)−n dt = 0
ε−1
provided that p < 1 + n2 , which is the case if p2 < n, we deduce that
Z
h n(1− 1 )−1
−n
p−1
p
p
f up∗
dv
=
ω
f
(x
)ε
In
g
n−1
o
ε
p
B(xo ,δ)
i
1
∆f (xo ) (n+2)(1− p1 )−1 2(1− p1 )
R(xo )
+
In
ε
−
+ o(ε2(1− p ) ) .
6n
2nf (xo )
Hence, by putting
Γ (n + 2)(1 − p1 ) Γ n−2p+1
p
c(n, p) =
Γ n(1 − p1 ) Γ( np )
one has
Z
f up∗
ε dvg
B(xo ,δ)
1
1
n(1− p
)−1 n(1− p
)
p−1
R(xo )
∆f (xo ) ωn−1 f (xo )In
1−
+
c(p, n)In
p
6n
2nf (xo )
1
1 + ε2(1− p ) + o ε2(1− p )
=ε
−n
p
and since
1
n(1− p
)−1
In
it follows that
Z
=
Γ( np )
n(n − p1 )Γ( np − 1)
1
n(1− p
)
In
f up∗
ε dvg
B(xo ,δ)
=ε
−n
p
1
n(1− p
)
p−1
ωn−1 f (xo )d(n, p)In
p
1
n(1− p
)
= d(n, p)In
15
16
M. BENALILI, Y. MALIKI
EJDE-2004/147
1
1 ∆f (xo ) R(xo )
× 1−
+
c(n, p)ε2(1− p ) + o ε2(1− p ) .
6n
2nf (xo )
From equalities (5.3) and (5.4), we get
Z
(|∇upε | + aupε )dvg
B(xo ,δ)
=
1
p−1
n − p p 1− np n(1− p1 ) a(n, p)
ωn−1 (
) ε
In
1−
R(xo )ε2(1− p )
p
p−1
6n
1 n−p p
+ b(n, p)(
) a(xo )εp−1 + o ε2(1− p ) + o εp−1 .
p−1
(5.5)
Since
Z
f up∗
ε dvg
p
p∗
B(xo ,δ)
1 n(1− p
) p/p∗
p−1
ωn−1 f (xo )d(n, p)In
p
1
1 R(xo )
∆f (xo ) × 1−
+
c(n, p)ε2(1− p ) + o ε2(1− p ) ]p/p∗
6n
2nf (xo )
n
= ε1− p
we get
R
(|∇upε | + aupε )dvg
p/p∗
R
f up∗
ε dvg
B(xo ,δ)
B(xo ,δ)
1 p
− p
n(1− p
) n n−p p
p−1
ωn−1 f (xo )In
(
) d(n, p)f (xo ) p∗
p
p−1
1
1 a(n, p)
× 1−
a(xo )ε2(1− p ) − b(n, p)a(xo )εp−1 + o ε2(1− p ) + o εp−1
6n
1
1 p R(xo ) ∆f (xo ) × 1+
+
c(n, p)ε2(1− p ) + o ε2(1− p ) .
p∗ 6n
2nf (p)
(5.6)
Now, since the function
p 1− n
p
φ = 1 + r p−1
=
realizes the best constant in the Sobolev’s imbedding H1p (Rn ) ⊂ Lp∗ (Rn ), that is
Z
Z
p/p∗
φp∗ dx
= K(n, p)p
|φ|p dx ,
Rn
Rn
we get
Z ∞
p
1
n−p p
) ωn−1
(1 + r p−1 )−n rn+ p−1 dr
p−1
0
Z ∞
p/p∗
p
= K(n, p)−p ωn−1
(1 + r p−1 )−n rn−1 dr
.
(
0
By letting t = r
p
p−1
, we have
Z ∞
1
p−1 n−p p
(
) ωn−1
(1 + t)−n tn(1− p ) dt
p
p−1
0
Z
p/p∗
1
p
−1 ∞
−p
= K(n, p)
ωn−1
(1 + t)−n tn(1− p )−1 dr
.
p
0
EJDE-2004/147
GENERALIZED SCALAR CURVATURE TYPE EQUATION
17
Therefore,
1
1 n(1− p
)
n(1− p
) p/p∗
p−1
p−1 n−p p
(
) ωn−1 In
= K(n, p)−p
ωn−1 d(n, p)In
p
p−1
p
which implies
1 p
p
) n
n(1− p
n−p p p−1
d(n, p)− p∗ .
K(n, p)−p = (
)
ωn−1 In
p−1
p
Then the equality (5.6) becomes
R
(|∇upε | + aupε )dvg
B(xo ,δ)
p
p∗
R
f up∗
ε dvg
B(xo ,δ)
h
p
1
a(n, p)
= K(n, p)−p f (xo )− p∗ 1 −
R(xo )ε2(1− p )
6n
i
1 p−1 p
−(
) b(n, p)R(xo )εp−1 + o ε2(1− p ) + o(εp−1 )
n−p
1
1
∆f (xo ) p R(xo )
(5.7)
+
c(n, p)ε2(1− p ) + o(ε2(1− p ) )
× 1+
p∗ 6n
2nf (xo )
h
n a(n, p)
p
p c(n, p) 2(1− p1 ) = K(n, p)−p f (xo )− p∗ 1 −
−
ε
R(xo )
6n
p∗ 6n
o
1
p−1 p
p ∆f (xo )
−(
) b(n, p)a(xo )εp−1 −
c(n, p)ε2(1− p )
n−p
p∗ 2nf (xo )
i
1 2(1− p
)
p−1
+o ε
+o ε
.
If 1 < p < 2 and n + 2 > 3p, the bracket in the equality (5.7) is equivalent to
p−1 p
1+(
) b(n, p)a(xo )εp−1 .
n−p
Then, if a(xo ) < 0, we get
p−1 p
1+(
) b(n, p)a(xo )εp−1 < 1.
(5.8)
n−p
If p = 2, the bracket reads
a(n, 2) n − 2 c(n, 2) 1 2
n − 2 ∆f (xo )
R(xo ) − (
) b(n, 2)a(xo ) −
c(n, 2)
−
6n
n
6n
n−2
n 2nf (xo )
where the quantities a(n, 2), b(n, 2) and c(n, 2) are replaced by their respective
expressions. The condition
a(n, 2)
n − 2
1 2 b(n, 2)
n − 2 ∆f (xo )
−
R(xo ) − (
)
a(xo ) −
>0
2
6nc(n, 2)
6n
n − 2 c(n, 2)
n 2nf (xo )
implies
(n + 2)(n − 2) n − 2 −
R(xo )
6n2 (n − 4)
6n2
1 2 4(n − 1)(n − 2)2
n − 2 ∆f (xo )
−(
)
a(xo ) −
> 0;
2
n−2
n (n − 4)
n 2nf (xo )
that is,
∆f (xo )
2
8(n − 1)
−
a(xo ) +
R(xo ) < 0.
(5.9)
f (xo )
n−4
(n − 2)(n − 4)
18
M. BENALILI, Y. MALIKI
Now for, p > 2, the bracket in question is equivalent to
a(n, p)
n − p
n − p ∆f (xo )
−
R(xo ) −
.
6nc(n, p)
6n2
n 2nf (xo )
The condition
(n + 2)(n − p)
n − p
n − p ∆f (xo )
−
R(xo ) −
>0
6n2 (n − 3p + 2)
6n2
2n2 f (xo )
becomes
∆f (xo )
p
<
R(xo ).
f (xo )
n − 3p + 2
Each of the conditions (5.8), (5.9) and (5.10) assures that
R
(|∇upε | + aupε )dvg
B(xo ,δ)
−p
−p/p∗
p/p∗ < K(n, p) f (xo )
R
p∗
f
u
dv
ε
g
B(xo ,δ)
EJDE-2004/147
(5.10)
and a fortiori the condition (iii) of the generic theorem is satisfied. Therefore, our
main theorem (Theorem 1.1) is proved.
References
[1] T. Aubin; Equations différentielles non liné aires et probl ème de Yamabé concernant la
courbure scalaire, Journal des Mathématiques Pures et Applications 53 (1976), 269-296.
[2] O. Druet; Generalized scalar curvature type equations on compact Riemannian manifolds,
Proc. Roy. Soc. Edinburgh Sect. A 130 (2000), no. 4, 767–788.
[3] M. Guedda and L. Veron; Quasilinear elliptic equations involving critical Sobolev exponents,
Non linear analysis, Theory, Methods and Applications, 13 (1989), 879-902.
[4] D. Holcman; Prescribed scalar curvature problem on complete manifolds, J. Math. Pures Appl.
(9) 80 (2001), no. 2, 223–244.
[5] S. Simon; Regularité de la solution d’une é quation non linéaire dans Rn . Ph. Benilan and J.
Robent eds. Lecture notes in Math 665, Spinger Verlag, Berlin (1978), 205-227.
[6] P. Tolksdorf; On the Dirichlet problem for quasilinear equations in domain with canonical
boundary points, Communications in Partial Differential Equations, 8 (1983), 773-817.
[7] P. Tolksdorf, Regularity for a more general class of quasilinear elliptic equations. Journal of
Differential Equations, 51 (1984), 126-150.
[8] S. L. Vasquez; A strong maximum principle for some quasilinear elliptic equation. Applied
Mathematics and Optimisation 12 (1989), 191-202.
Mohammed Benalili
Faculté des Sciences, Dept. de Mathématiques, B. P. 119, Université Aboubekr Belkaı̈d,
Tlemcen, Algerie
E-mail address: m benalili@mail.univ-tlemcen.dz
Youssef Maliki
Faculté des Sciences, Dept. de Mathématiques, B. P. 119, Université Aboubekr Belkaı̈d,
Tlemcen, Algerie
E-mail address: m-youssef@mail.univ-tlemcen.dz