Electronic Journal of Differential Equations, Vol. 2012 (2012), No. 163, pp. 1–9.
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu
ftp ejde.math.txstate.edu
INFINITELY MANY SOLUTIONS FOR CLASS OF NAVIER
BOUNDARY (p, q)-BIHARMONIC SYSTEMS
MOHAMMED MASSAR, EL MILOUD HSSINI, NAJIB TSOULI
Abstract. This article shows the existence and multiplicity of weak solutions
for the (p, q)-biharmonic type system
∆(|∆u|p−2 ∆u) = λFu (x, u, v)
∆(|∆v|
q−2
∆v) = λFv (x, u, v)
u = v = ∆u = ∆v = 0
in Ω,
in Ω,
on ∂Ω.
Under certain conditions on F , we show the existence of infinitely many weak
solutions. Our technical approach is based on Bonanno and Molica Bisci’s
general critical point theorem.
1. Introduction
In this paper we are concerned with the existence and multiplicity of weak solutions for the (p, q)-biharmonic type system
∆(|∆u|p−2 ∆u) = λFu (x, u, v)
q−2
∆(|∆v|
∆v) = λFv (x, u, v)
u = v = ∆u = ∆v = 0
in Ω,
in Ω,
(1.1)
on ∂Ω,
N
where Ω is an open bounded subset of R (N ≥ 1), with smooth boundary, λ ∈
(0, ∞), p > max{1, N2 }, q > max{1, N2 }. F : Ω × R2 → R is a function such
that F (., s, t) is continuous in Ω, for all (s, t) ∈ R2 and F (x, ., .) is C 1 in R2 for
every x ∈ Ω, and Fu , Fv denote the partial derivatives of F , with respect to u, v
respectively.
The investigation of existence and multiplicity of solutions for problems involving
biharmonic and p-biharmonic operators has drawn the attention of many authors,
see [5, 9, 12, 15] and references therein. Candito and Livrea [5] considered the
nonlinear elliptic Navier boundary-value problem
∆(|∆u|p−2 ∆u) = λf (x, u)
u = ∆u = 0
i nΩ,
on ∂Ω.
There the authors established the existence of infinitely many solutions.
2000 Mathematics Subject Classification. 35J40, 35J60.
Key words and phrases. Navier value problem; infinitely many solutions;
Ricceri’s variational principle.
c
2012
Texas State University - San Marcos.
Submitted June 4, 2012. Published September 21, 2012.
1
(1.2)
2
M. MASSAR, E. M. HSSINI, N. TSOULI
EJDE-2012/163
In the present paper, we look for the existence of infinitely many solutions of
system (1.1). More precisely, we will prove the existence of well precise intervals
of parameters such that problem (1.1) admits either an unbounded sequence of
solutions provided that F (x, u, v) has a suitable behaviour at infinity or a sequence
of nontrivial solutions converging to zero if a similar behaviour occurs at zero. Our
main tool is a general critical points theorem due to Bonanno and Molica Bisci [2]
that is a generalization of a previous result of Ricceri [11].
In the sequel, X will denote the space W 2,p (Ω) ∩ W01,p (Ω) × W 2,q (Ω) ∩
W01,q (Ω) , which is a reflexive Banach space endowed with the norm
k(u, v)k = kukp + kvkq ,
where
kukp =
Z
p
|∆u| dx
1/p
and kvkq =
Z
Ω
|∆v|q dx
1/q
.
Ω
Let
K := max
n
max|u(x)|p
x∈Ω
sup
u∈W 2,p (Ω)∩W01,p (Ω)\{0}
,
kukpp
max|v(x)|q o
x∈Ω
.
1,q
kvkqq
2,q
v∈W
(Ω)∩W
(Ω)\{0}
sup
0
(1.3)
Since p > max{1, N2 } and q > max{1, N2 }, the Rellich Kondrachov theorem assures
that W 2,p (Ω) ∩ W01,p (Ω) ,→ C(Ω) and W 2,q (Ω) ∩ W01,q (Ω) ,→ C(Ω) are compact,
and hence K < ∞.
Definition 1.1. We say that (u, v) ∈ X is a weak solution of problem (1.1) if
Z
Z
p−2
|∆u| ∆u∆ϕ dx +
|∆v|q−2 ∆v∆ψ dx
Ω
Ω
Z
Z
−λ
Fu (x, u, v)ϕ dx − λ
Fv (x, u, v)ψ dx = 0,
Ω
Ω
for all (ϕ, ψ) ∈ X.
Define the functional Iλ : X → R, given by
Iλ (u, v) = Φ(u, v) − λΨ(u, v),
for all (u, v) ∈ X, where
1
1
Φ(u, v) = kukpp + kvkqq
p
q
Z
and Ψ(u, v) =
F (x, u, v)dx.
Ω
Since X is compactly embedded in C 0 (Ω) × C 0 (Ω), it is well known that Φ and
Ψ are well defined Gâteaux differentiable functionals whose Gâteaux derivatives at
(u, v) ∈ X are given by
Z
Z
hΦ0 (u, v), (ϕ, ψ)i =
|∆u|p−2 ∆u∆ϕdx +
|∆v|q−2 ∆v∆ψdx,
Ω
Z
ZΩ
0
hΨ (u, v), (ϕ, ψ)i =
Fu (x, u, v)ϕdx +
Fv (x, u, v)ψdx,
Ω
Ω
for all (ϕ, ψ) ∈ X. Moreover, by the weakly lower semicontinuity of norm, we
see that Φ is sequentially weakly lower semi continuous. Since Ψ has compact
derivative, it follows that Ψ is sequentially weakly continuous.
EJDE-2012/163
INFINITELY MANY SOLUTIONS
3
In view of (1.3), for every (u, v) ∈ X, we have
sup |u(x)|p ≤ Kkukpp
and
x∈Ω
thus
sup |v(x)|q ≤ Kkvkqq ,
x∈Ω
1
1
1
|u(x)|p + |v(x)|q ≤ K kukpp + kvkqq .
(1.4)
q
p
q
x∈Ω p
Hence, for every r > 0
Φ−1 (] − ∞, r[) : = (u, v) ∈ X : Φ(u, v) < r
1
1
= (u, v) ∈ X : kukpp + kvkqq < r
(1.5)
p
q
1
1
⊆ (u, v) ∈ X : |u(x)|p + |v(x)|q < Kr, ∀ x ∈ Ω .
p
q
Let us recall for the reader’s convenience a smooth version of a previous result
of Ricceri [11].
sup
1
Theorem 1.2. Let X be a reflexive real Banach space, let Φ, Ψ : X → R be two
Gâteaux differentiable functionals such that Φ is sequentially weakly lower semicontinuous and coercive and Ψ is sequentially weakly upper semicontinuous. For every
r > inf X Φ, let us put
supv∈Φ−1 (]−∞,r[) Ψ(v) − Ψ(u)
ϕ(r) :=
inf
r − Φ(u)
u∈Φ−1 (]−∞,r[)
and
γ := lim inf ϕ(r),
r→+∞
δ :=
lim inf
r→(inf X Φ)+
ϕ(r).
Then, one has
1
(a) for every r > inf X Φ and every λ ∈]0, ϕ(r)
[, the restriction of the functional
−1
Iλ = Φ − λΨ to Φ (] − ∞, r[) admits a global minimum, which is a critical point
(local minimum) of Iλ in X.
(b) If γ < +∞ then, for each λ ∈]0, γ1 [, the following alternative holds: either
(b1) Iλ possesses a global minimum,or
(b2) there is a sequence (un ) of critical points (local minima) of Iλ such that
limn→+∞ Φ(un ) = +∞.
(c) If δ < +∞ then, for each λ ∈]0, 1δ [, the following alternative holds: either
(c1) there is a global minimum of Φ which is a local minimum of Iλ ,or
(c2) there is a sequence of pairwise distinct critical points (local minima) of Iλ
which weakly converges to global minimum of Φ.
2. Main results
Fix x0 ∈ Ω and pick R2 > R1 > 0 such that B(x0 , R2 ) ⊆ Ω. Set
R 2 − R 2 p
1
Γ(1 + N/2)
2
1
,
Lp :=
min(p,q)
N − RN
2N
R
1/p
1/q
N/2
2
1
(Kp) + (Kq)
π
R 2 − R 2 q
Γ(1 + N/2)
1
2
1
Lq :=
min(p,q)
N − RN
2N
R
1/p
1/q
N/2
2
1
(Kp) + (Kq)
π
(2.1)
where Γ denotes the Gamma function and K is given by (1.3). Now we are ready
to state our main results.
4
M. MASSAR, E. M. HSSINI, N. TSOULI
EJDE-2012/163
Theorem 2.1. Assume that
(i1) F (x, s, t) ≥ 0 for every (x, s, t) ∈ Ω × [0, +∞)2 ;
(i2) There exist x0 ∈ Ω, 0 < R1 < R2 as considered in (2.1) such that, if we
put
R
R
F (x, s, t)dx
sup|s|+|t|≤σ F (x, s, t)dx
B(x0 ,R1 )
Ω
, β := lim sup
α := lim inf
,
p
s
tq
min(p,q)
σ→+∞
σ
s,t→+∞
p + q
one has
α < Lβ,
(2.2)
where L := min{Lp , Lq }.
Then, for every
λ ∈ Λ :=
1 1
min(p,q) Lβ , α
(Kp)1/p + (Kq)1/q
1
problem (1.1) admits an unbounded sequence of weak solutions.
Theorem 2.2. Assume that (i1) holds and
(i3) F (x, 0, 0) = 0 for every x ∈ Ω.
(i4) There exist x0 ∈ Ω, 0 < R1 < R2 as considered in (2.1) such that, if we
put
R
R
F (x, s, t)dx
sup
F
(x,
s,
t)dx
|s|+|t|≤σ
B(x0 ,R1 )
0
,
α0 := lim inf Ω
,
β
:=
lim
sup
p
s
tq
σ→0+
σ min(p,q)
s,t→0+
p + q
one has
α0 < Lβ 0 .
(2.3)
where L := min{Lp , Lq }.
Then, for every
λ ∈ Λ :=
1
1
1 min(p,q) Lβ 0 , α0 ,
(Kp)1/p + (Kq)1/q
problem (1.1) admits a sequence (un ) of weak solutions such that un * 0.
3. Proofs of main results
Proof of Theorem 2.1. To apply Theorem 1.2, we set
sup(w,z)∈Φ−1 (]−∞,r[) Ψ(w, z) − Ψ(u, v)
ϕ(r) :=
inf
r − Φ(u, v)
(u,v)∈Φ−1 (]−∞,r[)
Note that Φ(0, 0) = 0, and by (i1), Ψ(0, 0) ≥ 0. Therefore, for every r > 0,
sup(w,z)∈Φ−1 (]−∞,r[) Ψ(w, z) − Ψ(u, v)
ϕ(r) =
inf
r − Φ(u, v)
(u,v)∈Φ−1 (]−∞,r[)
supΦ−1 (]−∞,r[) Ψ
≤
r R
supΦ(u,v)<r Ω F (x, u, v)dx
=
.
r
(3.1)
EJDE-2012/163
INFINITELY MANY SOLUTIONS
5
Hence, from (1.5), we have
1
ϕ(r) ≤
sup
r {(u,v)∈X: |u(x)|p + |v(x)|q <Kr, ∀ x∈Ω}
p
Z
F (x, u, v)dx
Ω
q
Let (σn ) a sequence of positive numbers such that σn → +∞ and
R
sup|s|+|t|≤σn F (x, s, t)dx
lim Ω
= α < +∞.
min(p,q)
n→+∞
σn
(3.2)
Put
rn :=
min(p,q)
σn
(Kp)1/p + (Kq)1/q
Let (u, v) ∈ Φ−1 (] − ∞, rn [), from (1.5) we have
|v(x)|q
|u(x)|p
+
< Krn ,
p
q
for all x ∈ Ω.
Thus
|u(x)| ≤ (Kprn )1/p
and |v(x)| ≤ (Kqrn )1/q ,
hence, for n large enough (rn > 1),
|u(x)| + |v(x)| ≤ (Kprn )1/p + (Kqrn )1/q
1
≤ (Kp)1/p + (Kq)1/q rnmin(p,q) = σn .
Therefore,
R
sup{(u,v)∈X:|u(x)|+|v(x)|<σn , ∀ x∈Ω} Ω F (x, u, v)dx
ϕ(rn ) ≤
min(p,q)
σ
n
(Kp)1/p +(Kq)1/q
≤ (Kp)1/p + (Kq)1/q
min(p,q)
(3.3)
R
Ω
sup|s|+|t|<σn F (x, s, t)dx
min(p,q)
.
σn
Let
γ := lim inf ϕ(r).
r→+∞
It follows from (3.2) and (3.3) that
γ ≤ lim inf ϕ(rn )
n→+∞
1/p
≤ (Kp)
1/q
+ (Kq)
min(p,q)
R
lim
Ω
n→+∞
sup|s|+|t|<σn F (x, s, t)
min(p,q)
σn
(3.4)
min(p,q)
= α (Kp)1/p + (Kq)1/q
< +∞.
From (3.4), it is clear that Λ ⊆]0, γ1 [.
For λ ∈ Λ, we claim that the functional Iλ is unbounded from below. Indeed,
min(p,q)
since λ1 < (Kp)1/p + (Kq)1/q
Lβ, we can consider a sequence (τn ) of positive numbers and η > 0 such that τn → +∞ and
R
min(p,q)
F (x, τn , τn )dx
1
B(x0 ,R1 )
< η < L (Kp)1/p + (Kq)1/q
,
(3.5)
q
p
τn
λ
+ τn
p
q
6
M. MASSAR, E. M. HSSINI, N. TSOULI
EJDE-2012/163
for n large enough. Define a sequence (un ) as follows

x ∈ Ω\B(x0 , R2 )

0,
PN
2
i
i 2
n
[R
−
(x
−
x
)
],
x ∈ B(x0 , R2 )\B(x0 , R1 )
un (x) = R2τ−R
2
2
0
i=1

 2 1
τn ,
x ∈ B(x0 , R1 )
(3.6)
Then (un , un ) ∈ X and
2N τ p
π N/2
n
(R2N − R1N ),
Γ(1 + N/2) R22 − R12
2N τ q
π N/2
n
kun kqq =
(R2N − R1N ).
Γ(1 + N/2) R22 − R12
kun kpp =
This and (2.1) imply that
τp
τnq n
min(p,q) pL + qL .
p
q
(Kp)1/p + (Kq)1/q
1
Φ(un , un ) =
(3.7)
By (i1), we have
Z
Z
F (x, un , un )dx ≥
Ψ(un , un ) =
F (x, τn , τn )dx.
(3.8)
B(x0 ,R1 )
Ω
Combining (3.5), (3.7) and (3.8), we obtain
Iλ (un , un )
= Φ(un , un ) − λΨ(un , un )
Z
τp
1
τnq n
≤
+
−
λ
F (x, τn , τn )dx
min(p,q) pL
qLq
p
B(x0 ,R1 )
(Kp)1/p + (Kq)1/q
Z
(3.9)
τp
τnq 1
n
+
−
λ
F
(x,
τ
,
τ
)dx
≤
n n
min(p,q) p
q
B(x0 ,R1 )
L (Kp)1/p + (Kq)1/q
τp
τnq 1 − λη
n
+
,
<
min(p,q) p
q
L (Kp)1/p + (Kq)1/q
for n large enough, so
lim Iλ (un , un ) = −∞,
n→+∞
and hence the claim follows.
The alternative of Theorem 1.2 case (b) assures the existence of unbounded
sequence (un ) of critical points of the functional Iλ and the proof of Theorem 2.1
is complete.
Proof of Theorem 2.2. First, note that
min Φ = Φ(0, 0) = 0.
X
Let (σn ) be a sequence of positive numbers such that σn → 0+ and
R
sup|s|+|t|≤σn F (x, s, t)dx
= α0 < +∞.
lim Ω
min(p,q)
n→+∞
σn
Put
min(p,q)
σn
rn =
, δ := lim inf
ϕ(r).
r→0+
(Kp)1/p + (Kq)1/q
(3.10)
(3.11)
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INFINITELY MANY SOLUTIONS
7
It follows from (3.1) and (3.11) that
δ ≤ lim inf ϕ(rn )
n→+∞
1/p
≤ (Kp)
1/q
+ (Kq)
min(p,q)
= α0 (Kp)1/p + (Kq)1/q
R
lim
Ω
sup|s|+|t|<σn F (x, s, t)
n→+∞
min(p,q)
min(p,q)
σn
(3.12)
< +∞.
By (3.12), we see that Λ ⊆]0, 1δ [.
Now, for λ ∈ Λ, we claim that Iλ has not a local minimum at zero. Indeed, since
min(p,q) 0
1
1/p
+ (Kq)1/q
Lβ , we can consider a sequence (τn ) of positive
λ < (Kp)
numbers and η > 0 such that τn → 0+ and
R
min(p,q)
F (x, τn , τn )dx
1
B(x0 ,R1 )
1/p
1/q
< η < L (Kp) + (Kq)
,
(3.13)
q
p
τn
τ
n
λ
p + q
for n large enough. Let (un ) be the sequence defined in (3.6). By combining (3.7),
(3.8) and (3.13), and taking into account (i3 ), we obtain
Iλ (un , un )
= Φ(un , un ) − λΨ(un , un )
Z
τp
τnq 1
n
+
−
λ
F (x, τn , τn )dx
≤
min(p,q) pL
qLq
p
B(x0 ,R1 )
(Kp)1/p + (Kq)1/q
Z
τp
1
τnq (3.14)
n
F (x, τn , τn )dx
≤
min(p,q) p + q − λ
B(x0 ,R1 )
L (Kp)1/p + (Kq)1/q
τp
1 − λη
τnq n
<
min(p,q) p + q
L (Kp)1/p + (Kq)1/q
< 0 = Iλ (0, 0)
for n large enough. This together with the fact that k(un , un )k → 0 shows that Iλ
has not a local minimum at zero, and the claim follows.
The alternative of Theorem 1.2 case (c) ensures the existence of sequence (un )
of pairwise distinct critical points (local minima) of Iλ which weakly converges to
0. This completes the proof of Theorem 2.2.
Example. It could be possible to consider the same example given in [14] for the
p-Laplacian system. Let Ω ⊂ R2 , p = 3, q = 4 and F : R2 → R be a function defined
by
(
1
−
(an+1 )5 e 1−[(s−an+1 )2 +(t−an+1 )2 ] (s, t) ∈ ∪n≥1 B((an+1 , an+1 ), 1)
F (s, t) =
0
otherwise,
(3.15)
for all x ∈ Ω, where
a1 := 2, an+1 := n!(an )5/4 + 2
and B((an+1 , an+1 ), 1) is an open unit ball of center (an+1 , an+1 ).
We see that F is non-negative and F ∈ C 1 (R2 ). For every n ∈ N, the restriction
of F on B((an+1 , an+1 ), 1) attains its maximum in (an+1 , an+1 ) and
F (an+1 , an+1 ) = (an+1 )5 e−1 ,
8
M. MASSAR, E. M. HSSINI, N. TSOULI
then
lim sup
F (an+1 , an+1 )
a3n+1
3
n→+∞
+
a4n+1
4
EJDE-2012/163
= +∞.
So,
R
F (s, t)dx
B(x0 ,R1 )
s3
t4
3 + 4
β : = lim sup
s,t→+∞
= |B(x0 , R1 )| lim sup
s,t→+∞
F (s, t)
4 = +∞.
+ t4
s3
3
On the other hand, for every n ∈ N, we have
sup
|s|+|t|≤an+1 −1
Then
lim
F (s, t) = a5n e−1
sup|s|+|t|≤an+1 −1 F (s, t)
(an+1 − 1)3
n→+∞
and hence
for all n ∈ N.
= 0,
sup|s|+|t|≤σ F (s, t)
= 0.
σ→+∞
σ3
lim
Finally
R
sup|s|+|t|≤σ F (s, t)dx
σ3
sup|s|+|t|≤σ F (s, t)
= |Ω| lim inf
σ→+∞
σ3
= 0 < Lβ
α : = lim inf
Ω
σ→+∞
So, applying Theorem 2.1, we have that for every λ ∈]0, +∞[ the system
∆(|∆u|∆u) = λFu (u, v)
2
∆(|∆v| ∆v) = λFv (u, v)
u = v = ∆u = ∆v = 0
in Ω,
in Ω,
(3.16)
on ∂Ω,
admits an unbounded sequence of weak solutions.
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Mohammed Massar
University Mohamed I, Faculty of Sciences, Department of Mathematics, Oujda, Morocco
E-mail address: massarmed@hotmail.com
El Miloud Hssini
University Mohamed I, Faculty of Sciences, Department of Mathematics, Oujda, Morocco
E-mail address: hssini1975@yahoo.fr
Najib Tsouli
University Mohamed I, Faculty of Sciences, Department of Mathematics, Oujda, Morocco
E-mail address: tsouli@hotmail.com