Electronic Journal of Differential Equations, Vol. 2011 (2011), No. 167, pp. 1–9.
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu
ftp ejde.math.txstate.edu
EXISTENCE OF SOLUTIONS FOR NON-UNIFORMLY
NONLINEAR ELLIPTIC SYSTEMS
GHASEM ALIZADEH AFROUZI, SOMAYEH MAHDAVI,
NIKOLAOS B. ZOGRAPHOPOULOS
Abstract. Using a variational approach, we prove the existence of solutions
for the degenerate quasilinear elliptic system
− div(ν1 (x)|∇u|p−2 ∇u) = λFu (x, u, v) + µGu (x, u, v),
− div(ν2 (x)|∇v|q−2 ∇v) = λFv (x, u, v) + µGv (x, u, v),
with Dirichlet boundary conditions.
1. Introduction
In this article, we study the degenerate quasilinear elliptic system
− div(ν1 (x)|∇u|p−2 ∇u) = λFu (x, u, v) + µGu (x, u, v),
q−2
− div(ν2 (x)|∇v|
∇v) = λFv (x, u, v) + µGv (x, u, v),
u = v = 0,
in Ω,
in Ω,
(1.1)
on ∂Ω.
where Ω is a bounded smooth domain in RN , N ≥ 2 and 1 < p, q < N . The
parameters λ and µ are nonnegative real numbers.
Throughout this work we assume that
(Fu , Fv ) = ∇F
and (Gu , Gv ) = ∇G
(1.2)
which stand for gradient of F and G, respectively, in the variables w = (u, v) ∈
R2 . Systems of form (1.1), where hypothesis (1.2) is satisfied, are called potential
systems. In recent years, more and more attention have been paid to the existence
and multiplicity of positive solutions for potential systems. For more details about
this kind of systems see [1, 2, 3, 4, 7, 8, 12, 13, 14, 17, 20] and references therein.
The degeneracy of this system is considered in the sense that the measurable,
non-negative diffusion coefficients ν1 , ν2 are allowed to vanish in Ω, (as well as
at the boundary ∂Ω) and/or to blow up in Ω̄. The point of departure for the
consideration of suitable assumptions on the diffusion coefficients is the work [9],
where the degenerate scalar equation was studied.
2000 Mathematics Subject Classification. 34B18, 35B40, 35J65.
Key words and phrases. Non-uniformly elliptic system; mountain pass theorem;
minimum principle.
c
2011
Texas State University - San Marcos.
Submitted November 12, 2011. Published December 14, 2011.
1
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G. A. AFROUZI, S. MAHDAVI, N. B. ZOGRAPHOPOULOS
EJDE-2011/167
We introduce the space (H)p consisting of functions ν : Ω ⊂ RN → R, such that
1
ν ∈ L1 (Ω), ν −1/(p−1) ∈ L1 (Ω) and ν −s ∈ L1 (Ω), for some p > 1, s > max{ Np , p−1
}
satisfying ps ≤ N (s + 1).
Then for the weight functions ν1 , ν2 we assume the hypothesis:
(H1) There exist µ1 in the space (H)p for some sp , and there exists µ2 in the
spaces (H)q for some sp , such that
µ2 (x)
≤ ν2 (x) ≤ c2 µ2 (x),
c2
µ1 (x)
≤ ν1 (x) ≤ c1 µ1 (x),
c1
(1.3)
a.e. in Ω, for some constants c1 > 1 and c2 > 1.
There exists a vast literature on non-uniformly nonlinear elliptic problems in
bounded or unbounded domains. Many authors studied the existence of solutions
for such problems (equations or systems); see for example [5, 6, 11, 15, 16, 18, 19].
Recently in [6], the authors considered the system
− div(h1 (x)∇u) = λFu (x, u, v),
in Ω,
− div(h2 (x)∇v) = λFv (x, u, v),
in Ω,
u = v = 0,
on ∂Ω.
They are concerned with the nonexistence and multiplicity of nonnegative, nontrivial solutions. In [19], the author studied the principal eigenvalue of the system
−∇(ν1 (x)|∇u|p−2 ∇u) = λa(x)|u|p−2 u + λb(x)|u|α |v|β v,
q−2
−∇(ν2 (x)|∇v|
q−2
∇v) = λd(x)|v|
u = v = 0,
α
β
v + λb(x)|u| |v| u,
in Ω,
in Ω,
on ∂Ω.
While in [11] the following system was considered
− div(|x|−ap |∇u|p−2 ∇u) = λg1 (x, u, v),
−bq
− div(|x|
q−2
|∇v|
∇v) = λg2 (x, u, v),
u = v = 0,
in Ω,
in Ω,
on ∂Ω,
where g1 , g2 : Ω × RtimesR are continuous and monotone functions.
The aim of this work is to extend or complete some of the above results for
system (1.1). Our assumptions are as follows: F (x, t, s) and G(x, t, s) are C 1 functions satisfying the hypotheses below:
(F1) There exist positive constants c1 , c2 > 0 such that
|Fu (x, t, s)| ≤ c1 |t|θ |s|δ+1 ,
|Fv (x, t, s)| ≤ c2 |t|θ+1 |s|δ
for all (t, s) ∈ R2 , a.e. x ∈ Ω and some θ, δ > 0 with
θ+1 δ+1
+
= 1.
p
q
(F2)
1
1
Fu (x, s, t) + Fv (x, s, t) − F (x, s, t) = ∞
q
|(s,t)|→∞ p
lim
(G1) There exist positive constants c01 , c02
Gu (x, t, s) ≤ c01 |t|α |s|γ+1 ,
Gv (x, t, s) ≤ c02 |t|α+1 |s|γ ;
(1.4)
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EXISTENCE OF SOLUTIONS
3
for all (t, s) ∈ R2 , a.e. x ∈ Ω and for some α, γ > 0. We will distinguish
the following cases:
α+1 γ+1
+
< 1;
p
q
α+1 γ+1
α+1 γ+1
+
> 1 and
+
< 1;
p
q
p∗
q∗
(1.5)
(1.6)
(G2)
1
1
Gu (x, s, t) + Gv (x, s, t) − G(x, s, t) = ∞
p
q
The main results of this paper are the following two theorems.
lim
|(s,t)|→∞
Theorem 1.1. In addition to (F1), (G1) and (1.5), assume that there exist p1 ∈
γ+1
(1, p) and q1 ∈ (1, q), such that α+1
p1 + q1 = 1. Then there exists λ0 > 0, such that
(1.1) possesses a weak solution for all µ > 0 and 0 ≤ λ < λ0 .
Theorem 1.2. In addition to (F1), (G1), (F2) or (G2) and (1.6), assume that
γ+1
there exist p2 ∈ (p, p∗ ) and q2 ∈ (q, q ∗ ), such that α+1
p2 + q2 = 1. Then there
exists λ0 > 0 such that system (1.1) possesses a weak solution for all µ > 0 and
0 ≤ λ < λ0 .
The quantities p∗ and q ∗ are defined in the next section.
2. Preliminaries
Let ν(x) be a nonnegative weight function in Ω which satisfies condition Hp . We
consider the weighted Sobolev space D01,p (Ω, ν) defined as the closure of C0∞ (Ω)
with respect to the norm
Z
1/p
.
kukD1,p (Ω,ν) :=
ν(x)|∇u|p
0
Ω
D01,p (Ω, ν)
The space
is a reflexive Banach space. For a discussion about the space
setting we refer the reader to [9] and the references therein. Let
p∗s :=
N ps
.
N (s + 1) − ps
(2.1)
Lemma 2.1. Assume that Ω is a bounded domain in RN and the weight ν satisfies
(N )p . Then the following embeddings hold:
∗
(i) D01,p (Ω, ν) ,→ Lps (Ω) continuously for 1 < p∗s < N ,
(ii) D01,p (Ω, ν) ,→ Lr (Ω) compactly for any r ∈ [1, p∗s ).
In the sequel we denote by p∗ and q ∗ the quantities p∗sp and p∗sq , respectively,
where sp and sq are induced by condition (H), recall that ν1 , ν2 satisfy (H).
The space setting for our problem is the product space H := D01,p (Ω, ν1 ) ×
D01,q (Ω, ν2 ) equipped with the norm
khkH := kukD1,p (Ω,ν1 ) + kvkD1,q (Ω,ν2 ) ,
0
0
h = (u, v) ∈ H.
Observe that (1.3) in condition (H) implies that the spaces D01,p (Ω, ν1 )×D01,q (Ω, ν2 )
and D01,p (Ω, µ1 ) × D01,q (Ω, µ2 ) are equivalent. Next, we introduce the functionals
4
G. A. AFROUZI, S. MAHDAVI, N. B. ZOGRAPHOPOULOS
EJDE-2011/167
I, J, J˜ : H → R as follows:
1
I(u, v) :=
p
Z
Z
1
ν1 (x)|∇u| dx +
ν2 (x)|∇v|q dx,
q Ω
Ω
Z
J(u, v) :=
F (x, u, v) dx,
ZΩ
˜ v) :=
J(u,
G(x, u, v) dx.
p
Ω
It is a standard procedure (see [10, 13]) to prove the following properties of these
functionals.
Lemma 2.2. The functionals I, J, J˜ are well defined. Moreover, I is continuous
and J, J˜ are compact.
We say that (u, v) is a weak solution of problem (1.1) if (u, v) is a critical point
˜ v); i.e.,
of the functional Φ(u, v) := I(u, v) − λJ(u, v) − µJ(u,
Z
Z
Z
ν1 (x)|∇u|p−2 ∇u · ∇φ dx = λ
Fu (x, u, v)φ dx + µ
Gu (x, u, v)φ dx, (2.2)
ZΩ
ZΩ
ZΩ
ν2 (x)|∇v|q−2 ∇v · ∇ψ dx = λ
Fv (x, u, v)ψ dx + µ
Gv (x, u, v)ψ dx, (2.3)
Ω
Ω
Ω
for any (φ, ψ) ∈ H.
Also, we mention some results concerning the associated eigenvalue problem. Let
λ1 be the first eigenvalue of the Dirichlet problem
− div(ν1 (x)|∇u|p−2 ∇u) = λ|u|θ−1 |v|δ+1 u,
q−2
− div(ν2 (x)|∇v|
θ+1
∇v) = λ|u|
u = v = 0,
δ−1
|v|
v,
in Ω,
in Ω,
(2.4)
on ∂Ω.
where the functions ν1 (x) and ν2 (x) satisfy (H1), and the exponents θ, δ satisfy
(1.4). Then, we have that λ1 is a positive number, which is characterized variationally by
R θ+1
q
( p ν1 (x)|∇u|p + δ+1
q ν2 (x)|∇v| ) dx
R
λ1 =
inf
.
(u,v)∈H−{(0,0)}
|u|θ+1 |v|δ+1 dx
Moreover, λ1 is isolated, the associated eigenfunction (ϕ1 , ϕ2 ) is componentwise
nonnegative and λ1 is the only eigenvalue of (2.4) to which corresponds a componentwise nonnegative eigenfunction. In addition, the set of all eigenfunctions corresponding to the principal eigenvalue λ1 forms a one-dimensional manifold E1 ⊂ H,
which is defined by
p/q
E1 = {(t1 ϕ1 , t1 ϕ2 ); t1 ∈ R}.
In the rest of this article, the following assumption is required.
λ1 ≤ lim inf
|(t,s)|→∞
λF (x, t, s) + µG(x, t, s)
|t|θ+1 |s|δ+1
(2.5)
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EXISTENCE OF SOLUTIONS
5
3. Proof of main theorems
To prove Theorem 1.1 we need following two Lemmas.
Lemma 3.1. Let {wm } be a sequence weakly converging to w in H. Then we have
(i) Φ(w) ≤ lim inf m→∞ Φ(wm )
(ii) limm→∞ J(wm ) = J(w)
˜ m ) = J(w)
˜
(iii) limm→∞ J(w
Proof. (i) Let {wm } = {(um , vm )} be a sequence that converges weakly to w =
(u, v) ∈ H. By the weak lower semicontinuity of the norm in the space D01,p (Ω, ν1 )
and D01,q (Ω, ν2 ), we deduce that
Z
Z
Z
Z
lim inf
ν1 (x)|∇um |p +
ν2 (x)|∇vm |q ≥
ν1 (x)|∇u|p +
ν2 (x)|∇v|q .
m→∞
Ω
Ω
Ω
Ω
˜ by Lemma(2.2), imply the conclusion.
The compactness of operators J and J,
Lemma 3.2. The functional φ is coercive and bounded from below.
Proof. By (F1) and (G1), there exists c3 , c03 , such that for all (t, s) ∈ R2 and a. e.
x ∈ Ω, we deduce that
F (x, t, s) ≤ c3 |t|θ+1 |s|δ+1 ,
G(x, t, s) ≤ c03 |t|α+1 |s|γ+1 .
γ+1
By taking p1 ∈ (1, p, q1 ∈ (1, q) such that α+1
p1 + q1 = 1 and applying Young’s
inequality, we obtain
Z
Z
F (x, u, v)dx ≤ c3 |u|θ+1 |v|δ+1 dx
Z
Z
θ+1
δ+1
≤ c3 (
|u|p dx +
|v|q dx)
p
q
Z
Z
(3.1)
δ+1
θ+1
p
q
s1 ν1 (x)|∇u| dx +
s2 ν2 (x)|∇v| dx)
≤ c3 (
p
q
θ+1
δ+1
≤ c(
kukpD1,p (Ω,ν ) +
kvkqD1,q (Ω,ν ) )
1
2
p
q
0
0
where s1 , s2 are the embedding constants of D01,p (Ω, ν1 ) ,→ Lp (Ω), D01,q (Ω, ν2 ) ,→
Lq (Ω) and c = max{c3 s1 , c3 s2 }, while
Z
Z
G(x, u, v)dx ≤ c03 |u|α+1 |v|γ+1 dx
Z
Z
γ+1
α+1
(3.2)
|u|p1 dx + c03
|v|q1 dx
≤ c03
p1
q1
γ+1
α+1
≤ c0 (
kukpD11,p (Ω,ν ) +
kvkqD11,q (Ω,ν ) )
1
2
p1
q1
0
0
Consequently, using (3.1), (3.2), we obtain the estimate
θ+1
δ+1
1
1
)kukpD1,p (Ω,ν ) + ( − λc
)kvkqD1,q (Ω,ν )
Φ(u, v) ≥ ( − λc
1
2
p
p
q
q
0
0
α
+
1
γ
+
1
− µc0
kukpD11,p (Ω,ν ) − µc0
kvkqD11,q (Ω,ν ) .
1
2
p1
q1
0
0
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G. A. AFROUZI, S. MAHDAVI, N. B. ZOGRAPHOPOULOS
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Taking λ0 > 0 such that min{1 − λ(θ + 1)c, 1 − λ(δ + 1)c} > 0 for all 0 ≤ λ < λ0 ,
it follows that for µ > 0 and 0 ≤ λ < λ0 , φ is coercive, indeed φ(u, v) → ∞ as
k(u, v)kH → ∞.
Proof of Theorem 1.1. The coerciveness of Φ and the weak sequential lower semicontinuity are enough in order to prove that Φ attains its infimum, so the system
(1.1) has at least one weak solution.
Proof of theorem 1.2. To prove the existence of a weak solution we apply a version
of the Mountain Pass Theorem due to Ambrosetti and Rabinowitz [1]. For this
purpose we verify that Φ satisfies:
(i) the mountain pass type geometry,
(ii) the (P S)c condition.
γ+1
(i) By choosing p2 ∈ (p, p∗ ) and q2 ∈ (q, q ∗ ) such that α+1
p2 + q2 = 1 and
applying the Young’s inequality, we obtain
Z
Z
G(x, u, v)dx ≤ c03 |u|α+1 |v|γ+1 dx
Z
Z
γ+1
p2
0 α+1
|u| dx +
|v|q1 dx)
≤ c3 (
p2
q2
α+1
γ+1
q2
kukpD21,p (Ω,ν ) +
kvkD
≤ c(
),
1,q
1
p2
q2
0
0 (Ω,ν2 )
which implies
1
θ+1
1
δ+1
Φ(u, v) ≥ ( − λc
)kukpD1,p (Ω,ν ) + ( − λc
)kvkqD1,q (Ω,ν )
1
2
p
p
q
q
0
0
γ
+
1
α
+
1
kukpD21,p (Ω,ν ) − µc0
kvkqD21,q (Ω,ν ) .
− µc0
1
2
p2
q2
0
0
Hence, there exists r > 0, small enough, such that
inf
k(u,v)k=r
Φ(u, v) > 0 = Φ(0, 0).
On the other hand by using (2.5) we have
Φ(t1/p ϕ1 , t1/q ϕ2 )
Z
Z
Z
1
t
t
≤
ν1 |∇ϕ1 |p dx +
ν2 |∇ϕ2 |q dx − (λ1 + ε) (|t1/p ϕ1 |θ+1 |t q ϕ2 |δ+1 )dx
p
q
Z
= −tε (|ϕ1 |θ+1 |ϕ2 |δ+1 )dx.
Thus, we conclude that there exists t > 0, large enough, such that for e =
(t1/p ϕ1 , t1/q ϕ2 ), we have kek > r and Φ(e) < 0.
(ii) Let {wn }∞
n=1 ∈ H be such that there exists c > 0, with
|Φ(wn )| ≤ c,
∀n ∈ N,
and there exists a strictly decreasing sequence
|hΦ0 (wn ), zi| ≤ εn kzkH ,
{εn }∞
n=1 , limn→∞ εn
∀n ∈ N, z ∈ H.
(3.3)
= 0, such that
(3.4)
We will prove that {wn } contains a subsequence which converges strongly in H.
Let us begin by proving that {wn } is bounded in H. Suppose, by contradiction,
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EXISTENCE OF SOLUTIONS
7
that kwn kH → ∞. We have
|hΦ0 (un , vn ), (un , vn )i|
Z
Z
Z
p
q
= | ν1 (x)|∇un | dx + ν2 (x)|∇vn | dx − λ Fu (x, un , vn )un dx
Z
Z
Z
− λ Fv (x, un , vn )vn dx − µ Gu (x, un , vn )un dx − µ Gv (x, un , vn )vn dx|
≤ εn k(un , vn )kH .
On the other hand
|Φ(un , vn )| = |
1
p
Z
ν1 (x)|∇un |p dx +
1
q
Z
Z
−λ
ν2 (x)|∇vn |q dx
Z
F (x, un , vn )dx − µ
G(x, un , vn )dx| ≤ c.
Thus one has
c + εn k(un , vn )kH
un vn
≥ Φ(un , vn ) − hΦ0 (un , vn ), ( , )i
p q
Z 1
1
Fu (x, un , vn )un + Fv (x, un , vn )vn − F (x, un , vn ) dx
=λ
p
q
Z 1
1
µ
Gu (x, un , vn )un + Gv (x, un , vn )vn − G(x, un , vn ) dx,
p
q
which contradicts both (F2) and (G2). So {wn } is bounded. This imply that there
exists (u, v) ∈ H such that at least its subsequence, wn converges and strongly in
Lp (Ω) × Lq (Ω). Choosing z = (un − u, 0) in (3.4), we obtain
Z
Z
p−2
ν1 (x)|∇un | ∇un ∇(un − u)dx − λ Fu (x, un , vn )(un − u)dx
Z
− µ Gu (x, un , vn )(un − u)dx
≤ εn kun − ukD1,p (Ω,ν1 ) ,
0
Z
Z
Fu (x, un , vn )(un − u)dx ≤ |Fu (x, un , vn )k(un − u)|dx
Z
≤ |un |θ |vn |γ+1 |un − u|dx
γ+1
≤ kun kθLp kvn kL
q kun − ukLp ,
and
Z
Z
G
(x,
u
,
v
)(u
−
u)dx
≤ |Gu (x, un , vn )k(un − u)|dx
u
n n
n
Z
≤ |un |α |vn |δ+1 |un − u|dx
δ+1
≤ kun kα
Lp kvn kLq kun − ukLp .
Thus, we obtain
Z
ν1 (x)|∇un |p−2 ∇un (∇un − ∇u)dx → 0,
8
G. A. AFROUZI, S. MAHDAVI, N. B. ZOGRAPHOPOULOS
EJDE-2011/167
as n → ∞. In the same way we obtain
Z
ν1 (x)|∇u|p−2 ∇u(∇un − ∇u)dx,
as n → ∞. Finally, we conclude that
Z
lim
ν1 (x)(|∇un |p−2 ∇un − |∇u|p−2 ∇u)(∇un − ∇u)dx = 0.
n→∞
(3.5)
Observe now that for all ξ, η ∈ RN , there exists constant c3 > 0, such that
(|ξ|p−2 ξ − |η|p−2 η, ξ − η) ≥ c(|ξ| + |η|)p−2 |ξ − η|2
p−2
(|ξ|
p−2
ξ − |η|
p
η, ξ − η) ≥ c|ξ − η|
if 1 < p < 2
if p ≥ 2.
(3.6)
where (·, ·) denotes the usual product in RN .
1/p
1/p
So, for 1 < p < 2, by Hölder’s inequality and substituting zn = ν1 un , z = ν1 u
∗
in (3.6), there exists c > 0, such that
Z
0 ≤ |∇zn − ∇z|p dx
Z
= |∇zn − ∇z|p (|∇zn | + |∇z|)p(p−2)/2 (|∇zn | + |∇z|)p(2−p)/2 dx
Z
(2−p)/2
p/2 Z
(|∇zn | + |∇z|)p dx
≤
|∇zn − ∇z|2 (|∇zn | + |∇z|)p−2 dx
Z
p/2
1
(|∇zn |p−2 ∇zn − |∇z|p−2 ∇z, (∇zn − ∇z)dx
≤ ∗
c
Z
(2−p)/2
×
(|∇zn | + |∇z|)p dx
Z
p/2
c
(|∇zn |p−2 ∇zn − |∇z|p−2 ∇z, (∇zn − ∇z)dx
,
≤ ∗
c
which implies kun − ukD1,p (Ω,ν1 ) → 0, by (3.5), as n → ∞. While, for p ≥ 2, by
0
(3.6), one has
Z
1
(|∇zn |p−2 ∇zn − |∇z|p−2 ∇z, (∇zn − ∇z)dx ,
0 ≤ kun − ukD1,p (Ω,ν1 ) ≤ ∗
0
c
so we have kun − ukD1,p (Ω,ν1 ) → 0, by (3.5), as n → ∞.
0
ukD1,p (Ω,ν1 ) → 0 for p > 1, as n → ∞, that is, un → u in
0
Therefore, kun −
D01,p (Ω, ν1 )
as n → ∞.
D01,q (Ω, ν2 )
Similarly , we obtain vn → v in
as n → ∞. Consequently, Φ satisfies
the (P S)c condition and the proof of is completed.
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Ghasem Alizadeh Afrouzi
Department of Mathematics, Faculty of Mathematical Sciences, University of Mazandaran, Babolsar, Iran
E-mail address: afrouzi@umz.ac.ir
Somayeh Mahdavi
Department of Mathematics, Faculty of Mathematical Sciences, University of Mazandaran, Babolsar, Iran
E-mail address: smahdavi@umz.ac.ir
Nikolaos B. Zographopoulos
University of Military Education, Hellenic Army Academy, Department of Mathematics
& Engineering Sciences, Vari - 16673, Athens, Greece
E-mail address: nzograp@gmail.com, zographopoulosn@sse.gr