Electronic Journal of Differential Equations, Vol. 2011 (2011), No. 105, pp. 1–8.
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu
ftp ejde.math.txstate.edu
EXISTENCE OF SOLUTIONS FOR P-KIRCHHOFF TYPE
PROBLEMS WITH CRITICAL EXPONENT
AHMED HAMYDY, MOHAMMED MASSAR, NAJIB TSOULI
Abstract. We study the existence of solutions for the p-Kirchhoff type problem involving the critical Sobolev exponent,
”i
h “Z
?
|∇u|p dx ∆p u = λf (x, u) + |u|p −2 u in Ω,
− g
Ω
u=0
on ∂Ω,
where Ω is a bounded smooth domain of RN , 1 < p < N , p? = N p/(N − p) is
the critical Sobolev exponent, λ is a positive parameter, f and g are continuous
functions. The main results of this paper establish, via the variational method.
The concentration-compactness principle allows to prove that the Palais-Smale
condition is satisfied below a certain level.
1. Introduction and main results
We are concerned with the existence of solutions for the p-Kirchhoff type problem
i
h Z
?
|∇u|p dx ∆p u = λf (x, u) + |u|p −2 u in Ω
− g
(1.1)
Ω
u = 0 on ∂Ω,
where Ω is a bounded smooth domain of RN , 1 < p < N , p? = N p/(N − p) is
the critical Sobolev exponent, and f : Ω × R → R, g : R+ → R+ are continuous
functions that satisfy the following conditions:
(F1) f (x, t) = o(|t|p−1 ) as t → 0, uniformly for x ∈ Ω;
(F2) There exists q ∈ (p, p? ) such that
f (x, t)
= 0,
|t|→+∞ |t|q−2 t
uniformly forx ∈ Ω.
lim
(F3) There exists θ ∈ (p/σ, p? ) such that 0 < θF (x, t) ≤ tf (x, t) for all x ∈ Ω
Rt
and t 6= 0, where F (x, t) = 0 f (x, s)ds and σ is given by (G2) below.
(G1) There exists α0 > 0 such that g(t) ≥ α0 for all t ≥ 0;
(G2) There exists σ > p/p? such that G(t) ≥ σg(t)t for all t ≥ 0, where G(t) =
Rt
g(s)ds;
0
2000 Mathematics Subject Classification. 35A15, 35B33, 35J62.
Key words and phrases. p-Kirchhoff; critical exponent; parameter; Lions principle.
c
2011
Texas State University - San Marcos.
Submitted July 26, 2011. Published August 16, 2011.
1
2
A. HAMYDY, M. MASSAR, N. TSOULI
EJDE-2011/105
Much interest has grown on problems involving critical exponents, starting from
the celebrated paper by Brezis and Nirenberg [5], where the case p = 2 is considered.
We refer the reader to [1, 9, 10] and reference therein for the study of problems
with critical exponent.
Problem (1.1) is a general version of a model presented by Kirchhoff [11]. More
precisely, Kirchhoff introduced a model
Z L
∂2u
∂ 2 u ρ0
E
∂u
ρ 2 −
+
| |2 dx
= 0,
(1.2)
∂t
h
2L 0 ∂x
∂x2
where ρ, ρ0 , h, E, L are constants, which extends the classical D’Alembert’s wave
equation by considering the effects of the changes in the length of the strings during
the vibrations. The problem
Z
− a+b
|∇u|2 dx ∆u = f (x, u) in Ω
(1.3)
Ω
u = 0 on ∂Ω
received much attention, mainly after the article by Lions [12]. Problems like
(1.3) are also introduced as models for other physical phenomena as, for example,
biological systems where u describes a process which depends on the average of itself
(for example, population density). See [3] and its references therein. For a more
detailed reference on this subject we refer the interested reader to [4, 6, 7, 8, 14, 15].
Motivated by the ideas in [2], our approach for studying problem (1.1) is variational and uses minimax critical point theorems. The difficulty is due to the lack
?
of compactness of the imbedding W01,p (Ω) ,→ Lp (Ω) and the Palais-Smale condition for the corresponding energy functional could not be checked directly. So the
concentration-compact principle of Lions [13] is applied to deal with this difficulty.
The main result of this paper is the following theorem.
Theorem 1.1. Suppose that (G1)–(G2), (F1)–(F3) hold. Then, there exists λ∗ > 0,
such that (1.1) has a nontrivial solution for all λ ≥ λ∗ .
2. Preliminary results
We consider the energy functional I : W01,p (Ω) → R defined by
Z
Z
?
1
1
F (x, u)dx − ?
|u|p dx,
I(u) = G(kukp ) − λ
p
p Ω
Ω
(2.1)
R
where W01,p (Ω) is the Sobolev space endowed with the norm kukp = Ω |∇u|p dx. It
is well known that a critical point of I is a weak solution of problem (1.1).
To use variational methods, we give some results related to the Palais-Smale
compactness condition. Recall that a sequence (un ) is a Palais-Smale sequence of
I at the level c, if I(un ) → c and I 0 (un ) → 0.
In the sequel, we show that the functional I has the mountain pass geometry.
This purpose is proved in the next lemmas.
Lemma 2.1. Suppose that (F1), (F2), (G1) hold. Then, there exist r, ρ > 0 such
that inf kuk=r I(u) ≥ ρ > 0.
Proof. It follows from (F1) and (F2) that for any ε > 0, there exists C(ε) > 0.
1
F (x, t) ≤ ε|t|p + C(ε)|t|q for all t.
(2.2)
p
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EXISTENCE OF SOLUTIONS
3
By (G1) and the Sobolev embdding, we have
?
α0
kukp − λC1 εkukp − λC2 (ε)kukq − C3 kukp
I(u) ≥
p
i
h α
?
0
= kuk
− λC1 ε kukp−1 − λC2 (ε)kukq−1 − C3 kukp −1 .
p
(2.3)
Taking ε = α0 /(2pλC1 ) and setting
?
α0 p−1
t
− λC2 tq−1 − C3 tp −1 .
ξ(t) =
2p
Since p < q < p? , we see that there exist r > 0 such that maxξ(t) = ξ(r). Then,
t≥0
by (2.3), there exists ρ > 0 such that I(u) ≥ ρ for all kuk = r.
Lemma 2.2. Suppose that (G2), (F3) hold. Then for all λ > 0, there exists
a nonnegative function e ∈ W01,p (Ω) independent of λ, such that kek > r and
I(e) < 0.
Proof. Choose a nonnegative function φ0 ∈ C0∞ (Ω) with kφ0 k = 1. By integrating
(G2), we obtain
G(t) ≤
G(t0 )
1/σ
t0
By (F3),
R
Ω
t1/σ = C0 t1/σ
for all t ≥ t0 > 0.
(2.4)
F (x, tφ0 )dx ≥ 0. Hence
?
I(tφ0 ) ≤
C0 p/σ tp
t
− ?
p
p
Z
?
φp0 dx
for all t ≥ t0 .
Ω
Since p/σ < p? , the lemma is proved by choosing e = t∗ φ0 with t∗ > 0 large
enough.
In view of Lemmas 2.1 and 2.2, we may apply a version of the Mountain Pass
theorem without Palais-Smale condition to obtain a sequence (un ) ⊂ W01,p (Ω) such
that
I(un ) → c∗ and I 0 (un ) → 0,
where
c∗ = inf max I(γ(t)) > 0,
(2.5)
γ∈Γ t∈[0,1]
with
Γ = γ ∈ C([0, 1], W01,p (Ω)) : γ(0) = 0, I(γ(1)) < 0 .
Denoted by S∗ the best positive constant of the Sobolev embedding W01,p (Ω) ,→
?
Lp (Ω) given by
Z
Z
?
S∗ = inf
|∇u|p dx : u ∈ W01,p (Ω),
|u|p dx = 1 .
(2.6)
Ω
Ω
Lemma 2.3. Suppose that (G1)–(G2), (F1)–(F3) hold. Then there exists λ∗ > 0
N such that c∗ ∈ 0, ( θ1 − p1? )(α0 S∗ ) p for all λ ≥ λ∗ , where c∗ is given by (2.5).
Proof. For e given by Lemma 2.2, we have limt→+∞ I(te) = −∞, then there exists
tλ > 0 such that I(tλ e) = maxI(te). Therefore,
t≥0
Z
Z
?
p−1
p? −1
p
p
tλ g(ktλ ek )kek = λ
f (x, tλ e)e dx + tλ
ep dx;
Ω
Ω
4
A. HAMYDY, M. MASSAR, N. TSOULI
thus
p
Z
p
g(ktλ ek )ktλ ek = λtλ
f (x, tλ e)e dx +
Ω
By (2.4), it follows that
Z
?
?
C0
p/σ
kekp/σ tλ ≥ tpλ
ep dx,
σ
Ω
EJDE-2011/105
?
tpλ
Z
?
ep dx.
(2.7)
Ω
with t0 < tλ .
Since p/σ < p? , (tλ ) is bounded. So, there exists a sequence λn → +∞ and s0 ≥ 0
such that tλn → s0 as n → ∞. Hence, there exists C > 0 such that
g(ktλn ekp )ktλn ekp ≤ C
for all n;
that is,
Z
λn tλn
f (x, tλn e)e dx +
Ω
?
tpλn
Z
?
ep dx ≤ C
for all n.
Ω
If s0 > 0, the above inequality implies that
Z
Z
?
p?
λn tλn
f (x, tλn e)e dx + tλn
ep dx → +∞ ≤ C,
Ω
as n → ∞,
Ω
which is impossible, and consequently s0 = 0. Let γ∗ (t) = te. Clearly γ∗ ∈ Γ, thus
1
0 < c∗ ≤ max I(γ∗ (t)) = I(tλ e) ≤ G(ktλ ekp ).
t≥0
p
Since tλn → 0 and ( θ1 −
1
N/p
p? )(α0 S∗ )
> 0, for λ > 0 sufficiently large, we have
1
1
1
G(ktλ ekp ) <
− ? (α0 S∗ )N/p ,
p
θ p
and hence
0 < c∗ <
1
1
− ? (α0 S∗ )N/p .
θ p
This completes the proof.
Proof of Theorem 1.1. From Lemmas 2.1, 2.2 and 2.3, there exists a sequence
(un ) ⊂ W01,p (Ω) such that
I(un ) → c∗ and I 0 (un ) → 0,
(2.8)
1
1
N/p
with c∗ ∈ 0, ( θ − p? )(α0 S∗ )
for λ ≥ λ∗ . Then, there exists C > 0 such that
|I(un )| ≤ C, and by (F3) for n large enough, it follows from (G1) and (G2) that
1
C + kun k ≥ I(un ) − hI 0 (un ), un i
θ
1
1
p
≥ G(kuk ) − g(kun kp )kun kp
(2.9)
p
θ
σ 1
≥
− α0 kun kp .
p
θ
Since θ > p/σ, (un ) is bounded. Hence, up to a subsequence, we may assume that
un * u weakly in W01,p (Ω),
un → u a.e. in Ω,
un → u
in Ls (Ω), 1 ≤ s < p? ,
p
|∇un | * µ (weak*-sense of measures )
?
|un |p * ν
(weak*-sense of measures),
(2.10)
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EXISTENCE OF SOLUTIONS
5
where µ and ν are a nonnegative bounded measures on Ω. Then, by concentrationcompactness principle due to Lions [13], there exists some at most countable index
set J such that
X
?
ν = |u|p +
νj δxj , νj > 0,
j∈J
p
µ ≥ |∇u| +
X
µj δxj ,
(2.11)
µj > 0,
j∈J
p/p?
S∗ νj
≤ µj ,
where δxj is the Dirac measure mass at xj ∈ Ω.
Let ψ(x) ∈ C0∞ such that 0 ≤ ψ ≤ 1,
(
1 if |x| < 1
ψ(x) =
0 if |x| ≥ 2
(2.12)
and |∇ψ|∞ ≤ 2.
For ε > 0 and j ∈ J, denote ψεj (x) = ψ((x − xj )/ε). Since I 0 (un ) → 0 and
j
(ψε un ) is bounded, hI 0 (un ), ψεj un i → 0 as n → ∞; that is,
Z
g(kun kp )
|∇un |p ψεj dx
Ω
Z
p
= −g(kun k )
un |∇un |p−2 ∇un ∇ψεj dx
(2.13)
Ω
Z
Z
?
+λ
f (x, un )un ψεj dx +
|un |p ψεj dx + on (1).
Ω
Ω
By (2.10) and Vitali’s theorem, we see that
Z
Z
j p
lim
|un ∇ψε | dx =
|u∇ψεj |p dx
n→∞
Ω
Ω
Hence, by Hölder’s inequality we obtain
Z
lim sup un |∇un |p−2 ∇un ∇ψεj dx
n→∞
Ω
1/p
(p−1)/p Z
Z
|un ∇ψεj |p dx
|∇un |p dx
≤ lim sup
n→∞
Ω
Ω
Z
1/p
≤ C1
|u|p |∇ψεj |p dx
B(xj ,2ε)
≤ C1
Z
|∇ψεj |N dx
1/N Z
B(xj ,2ε)
≤ C2
Z
?
|u|p dx
(2.14)
1/p?
B(xj ,2ε)
?
|u|p dx
1/p?
→0
as ε → 0 .
B(xj ,2ε)
On the other hand, from (2.10) we have
f (x, un )un → f (x, u)u
a.e. in Ω,
and un → u strongly in Lp (Ω) and in Lq (Ω). By (F1)–(F3), for any ε > 0 there
exists Cε > 0 such that
|f (x, t)| ≤ ε|t|p−1 + Cε |t|q−1
for all (x, t) ∈ Ω × R;
(2.15)
6
A. HAMYDY, M. MASSAR, N. TSOULI
EJDE-2011/105
thus
|f (x, un )un | ≤ ε|un |p + Cε |un |q .
This is what we need to apply Vitali’s theorem, which yields
Z
Z
lim
f (x, un )un dx =
f (x, u)u dx.
n→∞
Ω
Ω
ψεj
Since
has compact support, letting n → ∞ in (2.13) we deduce from (2.10) and
(2.14) that
Z
Z
Z
Z
1/p?
j
p?
α0
ψε dµ ≤ C2
|u| dx
+λ
f (x, u)udx +
ψεj dν.
Ω
B(xj ,2ε)
B(xj ,2ε)
Ω
Letting ε → 0, we obtain α0 µj ≤ νj . Therefore,
(α0 S∗ )N/p ≤ νj .
(2.16)
N/p
We will prove that this inequality is not possible. Let us assume that (α0 S∗ )
νj0 for some j0 ∈ J. From (G2) we see that
1
1
G(kun kp ) − g(kun kp )kun kp ≥ 0
p
θ
≤
for all n.
Since
1
c∗ = I(un ) − hI 0 (un ), un i + on (1),
θ
it follows that
Z
?
1
1
|un |p dx + on (1)
− ?
θ p
ZΩ
?
1
1
≥
− ?
ψεj0 |un |p dx + on (1)
θ p
Ω
c∗ ≥
Letting n → ∞, we obtain
c∗ ≥
1
1 X j0
1
1
ψε (xj )νj ≥
− ?
− ? (α0 S∗ )N/p .
θ p
θ p
j∈J
?
This contradicts Lemma 2.3. Then J = ∅, and hence un → u in Lp (Ω). By (2.15)
we have
Z
Z
|f (x, un )(un − u)|dx ≤
ε|un |p−1 + Cε |un |q−1 |un − u|dx
Ω
Ω
Z
p−1)/p Z
1/p
≤ε
|un |p dx
|un − u|p dx
Ω
Ω
(q−1)/q Z
1/q
Z
q
|un − u|q dx
.
+ Cε
|un | dx
Ω
Then, using again (2.10), we obtain
Z
lim
f (x, un )(un − u)dx = 0.
n→∞
Ω
(2.17)
Ω
?
Since un → u in Lp (Ω), we see that
Z
?
lim
|un |p −2 un (un − u)dx = 0.
n→∞
Ω
(2.18)
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EXISTENCE OF SOLUTIONS
7
From hI 0 (un ), un − ui = on (1), we deduce that
Z
0
p
hI (un ), un − ui = g(kun k )
|∇un |p−2 ∇un ∇(un − u)dx
Ω
Z
Z
?
−λ
f (x, un )(un − u)dx −
|un |p −2 un (un − u)dx = on (1)
Ω
Ω
This, (2.17) and (2.18) imply
p
Z
lim g(kun k )
n→∞
|∇un |p−2 ∇un ∇(un − u)dx = 0.
Ω
Since un is bounded and g is continuous, up to subsequence, there is t0 ≥ 0 such
that
g(kun kp ) → g(tp0 ) ≥ α0 ,
as n → ∞,
and so
Z
|∇un |p−2 ∇un ∇(un − u)dx = 0.
lim
n→∞
Ω
Thus by the (S+ ) property, un → u strongly in W01,p (Ω), and hence I 0 (u) = 0. The
proof is complete.
3. A special case
We consider the problem
Z
?
− α+β
|∇u|p dx ∆p u = λf (x, u) + |u|p −2 u
in Ω
Ω
u=0
(3.1)
on ∂Ω,
where Ω is a bounded smooth domain of RN , 1 < p < N < 2p, α and β are a
positive constants.
Set g(t) = α + βt. Then, g(t) ≥ α and
Z
G(t) =
0
1
1
1
g(s)ds = αt + βt2 ≥ (α + βt)t = σg(t)t
2
2
where σ = 1/2. Hence the conditions (G1) and (G2) are satisfied.
For this case, a typical example of a function satisfying the conditions (F1)–(F3)
is given by
f (x, t) =
k
X
ai (x)|t|qi −2 t,
i=1
?
where k ≥ 1, 2p < qi < p and ai (x) ∈ C(Ω). In view of Theorem 1.1, we have the
following corollary.
Corollary 3.1. Suppose that (F1)–(F3) hold. Then, there exists λ∗ > 0, such that
problem (3.1) has a nontrivial solution for all λ ≥ λ∗ .
8
A. HAMYDY, M. MASSAR, N. TSOULI
EJDE-2011/105
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Ahmed Hamydy
University Mohamed I, Faculty of sciences, Department of Mathematics, Oujda, Morocco
E-mail address: a.hamydy@yahoo.fr
Mohammed Massar
University Mohamed I, Faculty of sciences, Department of Mathematics, Oujda, Morocco
E-mail address: massarmed@hotmail.com
Najib Tsouli
University Mohamed I, Faculty of sciences, Department of Mathematics, Oujda, Morocco
E-mail address: tsouli@hotmail.com