Electronic Journal of Differential Equations, Vol. 2003(2003), No. 56, pp. 1–8.
ISSN: 1072-6691. URL: http://ejde.math.swt.edu or http://ejde.math.unt.edu
ftp ejde.math.swt.edu (login: ftp)
EXISTENCE OF SOLUTIONS FOR A CLASS OF ELLIPTIC
SYSTEMS IN RN INVOLVING THE p-LAPLACIAN
ALI DJELLIT & SAADIA TAS
Abstract. Using a variational approach, we study a class of nonlinear elliptic systems derived from a potential and involving the p-Laplacian. Under
suitable assumptions on the nonlinearities, we show the existence of nontrivial
solutions.
1. Introduction
In this paper, we deal with the nonlinear elliptic system
∂F
−∆p u =
(x, u, v) in RN ,
∂u
(1.1)
∂F
N
(x, u, v) in R .
−∆q v =
∂v
The nonlinearities on the right hand side are the gradient of a C 1 -functional F
and 4p is the so-called p-Laplacian operator i.e. ∆p u = div(|∇u|p−2 ∇u); u and
v are unknown real-valued functions defined in RN and belonging to appropriate
function spaces; 1 < p, q < N . Many authors studied the existence of solutions for
such problems (equations or systems) for which explicit solutions generally can not
be given.
We observe that there exists a vast literature on the use of the Mountain Pass
Theorem. Before stating our main theorem, we recall some work about some nonlinear problems: Rabinowitz [11] investigated a class of superlinear Schrödinger
equations of the form −∆u + q(x)u = f (x, u) defined in RN using a variational approach based on a variant of the Mountain Pass Theorem. The nonlinear function
verifies |∂f /∂u| ≤ a|u|p−1 + b; a and b are positive constants; 1 < p < 2∗ − 1 (in
this case, the equation is said superlinear). Yu [14] obtained sufficient conditions on
the nonlinearity for the existence of positive solutions for some nonlinear equations
p−2
of the form − div(|a(x)∇u|
∇u) = b(x)|u|p−2 u + f (x, u) defined on a smooth
exterior domains; a(x) and b(x) are smooth functions. Costa [4] studied a class of
elliptic systems −∆u + a(x)u = f (x, u, v); −∆v + b(x)v = g(x, u, v) in RN ; with
(f, g) = ∇F , the potential F is nonquadratic at infinity. The partial derivatives
satisfy the conditions |∇f (x, U )| + |∇g(x, U )| ≤ c(1 + |U |p−1 ); 1 < p < 2∗ − 1 if
2000 Mathematics Subject Classification. 35P65, 35P30.
Key words and phrases. p-Laplacian, nonlinear elliptic system, mountain pass theorem,
Palais-Smale condition.
c
2003
Southwest Texas State University.
Submitted March 24, 2003. Published Maay 8, 2003.
1
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ALI DJELLIT & SAADIA TAS
EJDE–2003/56
N ≥ 3 (or 1 ≤ p < +∞ if N = 1, 2). Do O [9] considered a non-autonomous perturbed eigenvalue problem involving the p-Laplacian of the form −∆p u = f (x, u)
defined in RN ; the nonlinearity f interacts with the first eigenvalue of a corresponding problem, and also verifies the following estimate |f (x, u)| ≤ ϕ(x)|u|r + ψ(x)|u|s ;
ϕ(x) and ψ(x) are suitable functions; 0 < r ≤ p − 1 ≤ s < p∗ − 1.
In this work, we show the existence of nontrivial solutions for System (1.1)
in homogeneous Sobolev spaces under mixed subcritical growth conditions; the
primitive F being intimately connected with the first eigenvalue of an appropriate
system. Using a weak version of the Palais-Smale condition, due to Cerami [5],
we can apply the Mountain Pass Theorem to System (1.1). Our main goal in this
article is to illustrate how the ideas introduced in [4, 9, 11] can be applied to handle
the problem of existence of nontrivial solutions for System (1.1).
This paper is organized as follows. In section 2, we present some preliminary
results and definitions; we also introduce precise assumptions under which our
problem is studied. We reserve the section 3 for the proof of the main result.
2. Notations and Hypotheses
We first recall some standard definitions and notations. Let Z be a reflexive
Banach space endowed with a norm k · k. Let I ∈ C 1 (Z, R). We say that I satisfies
the Cerami condition, denoted by (C) condition, if every (wn ) ∈ Z such that
|I(wn )| ≤ c and (1 + kwn k)I 0 (wn ) → 0
contains a convergent subsequence in the norm of Z.
m
be the critical Sobolev exponent of m.
For 1 < m < N , let m∗ = NN−m
1,m
N
Let D (R ) be the closure of C0∞ (RN ) with respect to the norm kuk1,m ≡
R
1/m
k∇ukm =
|∇u|m dx
. D1,m (RN ) is a reflexive Banach space and may
RN
∗
be written D1,m (RN ) = {u ∈ Lm (RN ) : ∇u ∈ (Lm (RN ))N }. Moreover, Sobolev
imbedding holds; in fact there exists a positive constant c such that kukm∗ ≤
ckuk1,m for all u ∈ D1,m (RN ) (see [13]).
∗
Now we denote by Z the product space D1,p (RN ) × D1,q (RN ) ; Z designates
the dual space equipped with the dual norm k · k∗ .
For (u, v) in Z, we define the functionals I, J, K by
1
1
J(u, v) = kukp1,p + kvkq1,q ,
p
q
Z
K(u, v) =
F (x, u(x), v(x))dx,
RN
I(u, v) = J(u, v) − K(u, v).
In this article we use the following hypotheses:
(H1) F ∈ C 1 (RN × R2 , R) and F (x, 0, 0) = 0.
(H2) For all U = (u, v) ∈ R2 and for almost every x ∈ RN
∂F
(x, U )| ≤ a1 (x)|U |p1 −1 + a2 (x)|U |p2 −1 ,
∂u
∂F
|
(x, U )| ≤ b1 (x)|U |q1 −1 + b2 (x)|U |q2 −1 .
∂v
|
EJDE–2003/56
EXISTENCE OF SOLUTIONS FOR ELLIPTIC SYSTEMS
3
Here 1 < p1 , q1 < min(p, q), max(p, q) < p2 , q2 < min(p∗ , q ∗ ), ai ∈
Lαi (RN ) ∩ Lβi (RN ), bi ∈ Lγi (RN ) ∩ Lδi (RN ), i = 1, 2.
p∗
,
p∗ − pi
γi =
p∗ q ∗
,
∗
−
i − 1) − q
δi =
αi =
βi =
p∗ q ∗
p∗ (p
q∗
,
q ∗ − qi
p∗ q ∗
p∗ q ∗
.
∗
−
i − 1) − p
q ∗ (q
(H3) U.∇F (x, U ) − F (x, U ) ≤ 0, for all (x, U ) ∈ RN × R2 − {(0, 0)}, where
∂F
∇F = ( ∂F
∂u , ∂v ). This type of condition has been introduced by Costa [4].
(H4) At last we suppose the existence of two positive and bounded functions
a ∈ LN/p (RN ) and b ∈ LN/q (RN ) such that
lim sup
|U |→0
pq|F (x, U )|
pq|F (x, U )|
< λ1 < lim inf
.
p
q
qa(x)|u| + pb(x)|v|
|U |→+∞ qa(x)|u|p + pb(x)|v|q
Putting
Z
Z
o
n
1
1
a(x)|u|p dx +
b(x)|v|q dx = 1 ,
Λ = (u, v) ∈ Z :
p RN
q RN
λ1 = inf Λ J(u, v) is the first eigenvalue of the system
−∆p u = λa(x)|u|p−2 u
in RN ,
−∆q v = λb(x)|v|q−2 v
in RN .
(2.1)
Remark The hypothesis (H4) is related with the interaction of the potential F
and λ1 . Costa [4] was the first to introduce such assumption. A variant of this
condition appeared in Do O [9]. An example of such functions is
F (x, u, v) = −a(x)|u|α |v|β ;
max(p, q) < α, β < min(p∗ , q ∗ ).
It is easy to prove that F satisfies (H1), (H3) and (H4). In order to obtain (H2),
we use Young’s inequality.
3. Existence of solutions
Taking into account the above hypotheses, we have some assertions.
Lemma 3.1. Under Hypotheses (H1) and (H2), the functional K is well defined
and is of class C 1 on Z. Moreover, its derivative is
Z
∂F
∂F
0
K (u, v)(w, z) =
(
(x, u, v)w +
(x, u, v)z)dx ∀(u, v), (w, z) ∈ Z.
∂v
RN ∂u
Proof. K is well defined on Z. Indeed, for all (u, v) ∈ Z and by virtue of (H1) and
(H2), we have
Z u
∂F
F (x, u, v) =
(x, s, v)ds + F (x, 0, v)
∂s
Z0 u
Z v
∂F
∂F
=
(x, s, v)ds +
(x, 0, s)ds + F (x, 0, 0)
∂s
0
0 ∂s
and
F (x, u, v) ≤c1 [a1 (x)(|u|p1 + |v|p1 −1 |u|) + a2 (x)(|u|p2 + |v|p2 −1 |u|)
+ b1 (x)|v|q1 + b2 (x)|v|q2 ].
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ALI DJELLIT & SAADIA TAS
EJDE–2003/56
Using Hölder’s inequality and Sobolev’s imbedding and the fact that ai ∈ Lαi (RN )∩
Lβi (RN ), bi ∈ Lγi (RN ), we get
Z
F (x, u, v)dx
RN
1 −1
2
1
≤ c2 (ka1 kα1 kukp1,p
+ ka1 kβ1 kvkp1,q
kuk1,p + ka2 kα2 kukp1,p
p2 −1
1
2
+ ka2 kβ2 kvk1,q
kuk1,p + kb1 kγ1 kvkq1,q
+ kb2 kγ2 kvkq1,q
) < +∞.
Observe that K 0 (u, v) is also well defined on Z since
Z
∂F
1 −1
1 −1
(x, u, v)wdx ≤c3 ka1 kα1 kukp1,p
+ ka1 kβ1 kvkp1,q
RN ∂u
2 −1
2 −1
+ ka2 kα2 kukp1,p
+ ka2 kβ2 kvkp1,q
kwk1,p < +∞.
and
Z
RN
∂F
1 −1
1 −1
(x, u, v)zdx ≤c4 kb1 kδ1 kukq1,p
+ kb1 kγ1 kvkq1,q
∂v
2 −1
2 −1
+ kb2 kδ2 kukq1,p
+ kb2 kγ2 kvkq1,q
kzk1,q < +∞.
Now, we show that K is differentiable in sense of Fréchet at each point (u, v) of Z
i.e. ∀ε > 0, there exists δ = δ(ε, u, v) > 0 such that (kwk1,p + kzk1,q ) ≤ δ implies
|K(u + w, v + z) − K(u, v) − K 0 (u, v)(w, z)| ≤ ε(kwk1,p + kzk1,q ).
0
Let BR be the ball of radius R, centered at the origin of RN . We put BR
=
N
1,p
1,q
R
−
B
and
we
define
a
functional
K
on
D
(B
)
×
D
(B
)
by
K
(u,
v)
=
R
R
R
R
R
R
F
(x,
u(x),
v(x))dx.
Taking
(H1)
and
(H2)
into
account,
it
is
well-known
that
BR
KR ∈ C 1 (D1,p (BR ) × D1,q (BR )) and for any (w, z) ∈ D1,p (BR ) × D1,q (BR ), we
have
Z
∂F
∂F
0
KR (u, v)(w, z) =
(
(x, u, v)w +
(x, u, v)z)dx.
∂v
BR ∂u
0
Moreover, KR
is compact from Z to Z ∗ (see [9, 10, 12]). On the other hand, for all
(u, v) , (w, z) ∈ Z, we have
K(u + w, v + z) − K(u, v) − K 0 (u, v)(w, z)
0
≤ KR (u + w, v + z) − KR (u, v) − KR
(u, v)(w, z)
Z
∂F
∂F
+
(F (x, u + w, v + z) − F (x, u, v) −
(x, u, v)w −
(x, u, v)z)dx.
0
∂u
∂v
BR
By the Mean-value theorem, we can write
F (x, u + w, v + z) − F (x, u, v) =
∂F
∂F
(x, u + θ1 w, v)w +
(x, u, v + θ2 z)z,
∂u
∂v
for θ1 , θ2 ∈]0, 1[. By the growth condition (H2) and the fact that for i = 1, 2,
kai kLαi (BR0 ) + kai kLβi (BR0 ) → 0,
kbi kLγi (BR0 ) + kbi kLδi (BR0 ) → 0,
(3.1)
EJDE–2003/56
EXISTENCE OF SOLUTIONS FOR ELLIPTIC SYSTEMS
5
as R → ∞, we obtain for R sufficiently large that
Z
∂F
∂F
(F (x, u + w, v + z) − F (x, u, v) −
(x, u, v)w −
(x, u, v)z)dx
0
∂u
∂v
BR
≤ ε kwk1,p + kzk1,q .
We have only to show that K 0 is continuous on Z. Let (un , vn ) → (u, v) in Z. For
(w, z) ∈ Z , we have
|K 0 (un , vn )(w, z) − K 0 (u, v)(w, z)|
0
0
= |KR
(un , vn )(w, z) − KR
(u, v)(w, z)|
Z
∂F
∂F
(x, un , vn ) +
(x, u, v))wdx|
+|
(
0
∂u
∂u
BR
Z
∂F
∂F
+|
(
(x, un , vn ) +
(x, u, v))zdx|.
0
∂v
∂v
BR
0
Then KR
is continuous on D1,p (BR ) × D1,q (BR ) (see [9, 11]). The first expression
on the right hand side of the above equation tends to 0 as n → +∞; we use (H2)
and (3.1) to prove that both the second and the third expressions tend also to 0 as
R sufficiently large.
Remark The functional J is of class C 1 on Z and its derivative is
Z
Z
0
p−2
J (u, v)(w, z) =
|∇u| ∇u∇wdx +
|∇v|q−2 ∇v∇zdx.
RN
RN
Lemma 3.2. Under assumptions (H1) and (H2), K 0 is compact from Z to Z ∗ .
Proof. Let (un , vn ) be a bounded sequence in Z. Then there is a subsequence
denoted again (un , vn ) weakly convergent to (u, v) in Z. As before, we write
|K 0 (un , vn )(w, z) − K 0 (u, v)(w, z)|
0
0
= |KR
(un , vn )(w, z) − KR
(u, v)(w, z)|
Z
∂F
∂F
+|
(
(x, un , vn ) −
(x, u, v))w dx|
0
∂u
∂u
B
Z R
∂F
∂F
+|
(
(x, un , vn ) −
(x, u, v))z dx|.
0
∂v
∂v
BR
Since the restriction operator is continuous, we have (un , vn ) * (u, v) in D1,p (BR )×
0
D1,q (BR ). Because of the compactness of KR
, the first expression on the right hand
side of the equation tends to 0 as n → +∞; as above both the second and the third
expressions tend also to 0 as R sufficiently large.
Lemma 3.3. If (H1), (H2), and (H3) hold then I = J − K satisfies the condition
(C).
Proof. Let (un , vn ) ⊂ Z such that
(i) |I(un , vn )| ≤ c.
∗
(ii) (1 + kun k1,p + kvn k1,q )I 0 (un , vn ) → 0 in Z , as n → +∞.
6
ALI DJELLIT & SAADIA TAS
EJDE–2003/56
From (ii), we have I 0 (un , vn )(w, z) ≤ εn → 0 as n → +∞, ∀ (w, z) ∈ Z. In
particular, for (w, z) = (un , vn ), we get
I 0 (un , vn )(un , vn )
= kun kp1,p + kvn kq1,q −
Z
(
RN
∂F
∂F
(x, un , vn )un +
(x, un , vn )vn )dx ≤ εn .
∂u
∂v
On the other hand,
1
1
kun kp1,p + kvn kq1,q −
p
q
Then, taking (H2) into account, we get
Z
F (x, un , vn )dx ≤ c.
I(un , vn ) =
RN
εn + c ≥ I 0 (un , vn )(un , vn ) − I(un , vn )
1
1
= (1 − )kun kp1,p + (1 − )kvn kq1,q
p
q
Z
∂F
∂F
(x, un , vn )un −
(x, un , vn )vn )dx
+
(F (x, un , vn ) −
∂u
∂v
N
R
1
1
≥ (1 − )kun kp1,p + (1 − )kvn kq1,q .
p
q
Hence, (un , vn ) is bounded in Z. There is a subsequence denoted again (un , vn )
weakly convergent in Z. Since K 0 is compact, K 0 (un , vn ) is a Cauchy’s sequence in
∗
Z . We have J 0 (u, v) = I 0 (u, v) + K 0 (u, v), ∀(u, v) ∈ Z and
J 0 (un , vn ) − J 0 (um , vm ) (un − um , 0)
Z
=
(|∇un |p−2 ∇un − |∇um |p−2 ∇um )(∇un − ∇um )dx.
RN
Observe that for all λ, µ ∈ RN ,
(
(|λ|p−2 λ − |µ|p−2 µ)(λ − µ)
|λ − µ|p ≤ p/2
(|λ|p−2 λ − |µ|p−2 µ)(λ − µ)
(|λ| + |µ|)(2−p)p/2
if p ≥ 2,
if 1 < p < 2,
(see [6, 15]). Substituting λ and µ by ∇un and ∇um respectively and integrating
over RN , we obtain
kun − um kp1,p ≤ (J 0 (un , vn ) − J 0 (um , vm ))(un − um , 0), if p ≥ 2.
and
kun − um k21,p ≤ |(J 0 (un , vn ) − J 0 (um , vm ))(un − um , 0)|(kun kp1,p + kum kp1,p )
2−p
2
,
if 1 < p < 2.
Since (un ) is bounded in D1,p (RN ) and (J 0 (un , vn ) − J 0 (um , vm ))(un − um , 0) →
0 as n, m → +∞, then (un ) is a Cauchy’s sequence in D1,p (RN ). Hence (un )
converges in D1,p (RN ). In the same way, we prove that (vn ) converges in D1,q (RN ).
Moreover the functional I = J − K verifies the geometric conditions of the
Mountain Pass Theorem, summarized in the following lemma.
Lemma 3.4. Under Assumptions (H1), (H2), (H3), and (H4), the functional I
satisfies
(I1) There exist ρ, σ > 0 such that kuk1,p + kvk1,q = ρ implies I(u, v) ≥ σ > 0.
(I2) There exists E ∈ Z such that kEkZ > ρ and I(E) ≤ 0.
EJDE–2003/56
EXISTENCE OF SOLUTIONS FOR ELLIPTIC SYSTEMS
7
Proof. By (H4), there exists ρ > 0 such that
1
1
kuk1,p + kvk1,q = ρ =⇒ F (x, u, v) < λ1 ( a(x)|u|p + b(x)|v|p ).
p
q
The variational characterization of λ1 (see [7]) gives
Z
1
1
F (x, u, v)dx < kukp1,p + kvkq1,q .
p
q
RN
Then there exist ρ, σ > 0 such that kuk1,p + kvk1,q = ρ implies I(u, v) ≥ σ > 0. Let
(ϕ, ψ) be an eigenfunction associated with λ1 . In view of (H4), we get for ε > 0
and t sufficiently large,
1
1
t
t
F (x, t p ϕ, t q ψ) ≥ (λ1 + ε)( a(x)|ϕ|p + b(x)|ψ|p ).
p
q
Hence
Z
Z
Z
1
1
1
1
t
t
I(t p ϕ, t q ψ) =
|∇ϕ|p dx +
|∇ψ|q dx −
F (x, t p ϕ, t q ψ)dx
p RN
q RN
N
R
Z
Z
t
t
p
q
≤
|∇ϕ| dx +
|∇ψ| dx
p RN
q RN
Z
Z
t
t
p
− (λ1 + ε)
a(x)|ϕ| dx +
b(x)|ψ|p dx
p RN
q RN
Z
Z
1
1
≤ −tε(
a(x)|ϕ|p dx +
b(x)|ψ|p dx).
p RN
q RN
1
we deduce that limt→+∞ I(t p ϕ, t1/q ψ) = −∞. So, for t large, I(t1/p ϕ, t1/q ψ) ≤ 0.
Consequently, the functional I has a critical value. Note that the critical points
of I are precisely the weak solutions of System (1.1).
Now, we can state the main theorem.
Theorem 3.5. System (1.1) has at least one nontrivial solution (u, v).
Proof. In view of Lemmas 3.3 and 3.4, we can apply the Mountain-Pass theorem
(see [8, 10, 13]) to conclude that system (1.1) has a nontrivial weak solution.
Acknowledgments. The authors want to express their gratitude to ANDRU for
providing support through the grant CU 39904, and to the anonymous referee for
his/her helpful comments.
References
[1] S. M. Berger, Nonlinearity and Functional Analysis. Academic Press, New York, San Francisco, London, 1977.
[2] P. Bartolo, V. Benci and D. Fortunato, Abstract critical point theorems and applications
to some nonlinear problems with strong resonance at infinity. Nonlinear Analysis TMA 7
(1983), 981-1012.
[3] V. Benci and P. H. Rabinowitz, Critical point theorems for indefinite functionals. Invent.
Math. 52 (1979), 241-273.
[4] D. G. Costa, On a class of elliptic systems in RN . E. J. D. E, Vol. 1994 (1994), No. 07, 1-14.
[5] G. Cerami, Un criterio de esistenza per i punti critici su varietà ilimitate. Istituto Lombardo
di Scienze e Lettere 112 (1978), 332-336.
[6] G. Dinca and P. Jebelean, Some existence results for a class of nonlinear equations involving
a duality mapping. Nonlinear analysis 46 (2001), 347-363.
[7] J. Ffleckinger, R. F. Manásevich, N. M. Stavrakakis, F. de Thélin, Principal eigenvalues for
some quasilinear elliptic equation on RN . V.2, N.6, November 1997,pp.981-1003.
8
ALI DJELLIT & SAADIA TAS
EJDE–2003/56
[8] O. Kavian, Introduction à la théorie des points critiques et applications aux problèmes elliptiques. Springer-Verlag France, Paris, 1993.
[9] J. M. B. do O, Solutions to perturbed eigenvalue problems of the p-Laplacian in RN . Electronic J. Diff. Eqns., Vol. 11 (1997), 1-15.
[10] P. H. Rabinowitz, Minimax Methods in Critical Point Theory with Applications to Differential Equations. CBMS Regional Conf. Ser. in Math. 65, AMS, Providence, RI, 1986.
[11] P. H. Rabinowitz, On a class of Nonlinear Schrödinger Equations. z. angew. Math. Phys. 43
(1992), 270-291.
[12] N. M. Stavrakakis and N. B. Zographopoulos, Existence results for quasilinear elliptic systems
in RN . Electronic J. Diff. Eqns., Vol. 1999 (1999), No. 39, 1-15.
[13] M. Willem, Minimax Theorems, Bikhauser, Boston-Basel-Berlin, 1996.
[14] L. S. Yu, Nonlinear p-Laplacian Problems on Unbounded domains. Proc. Amer. Math. Soc.
115 (1992) No. 4, 1037-1045.
[15] E. Zeidler, Nonlinear Functional Analysis and its Applications, Vol III Variational Methods
and Optimization. Springer Verlag, Berlin 1990.
Ali Djellit
University of Annaba, Faculty of Sciences, Department of Mathematics, B.P. 12, 23000,
Annaba, Algeria
E-mail address: a djellit@hotmail.com
Saadia Tas
University of Annaba, Faculty of Sciences, Department of Mathematics, B.P. 12, 23000,
Annaba, Algeria
E-mail address: tas saadia@yahoo.fr