- Email: [email protected]
Solutions to Kirchhoff equations with critical exponent
A RAB J OURNAL OF M ATHEMATICAL S CIENCES
Arab J Math Sci 22 (2016) 138–149
Solutions to Kirchhoff equations with critical exponent E L M ILOUD H SSINI a,∗ , M OHAMMED M ASSAR b , NAJIB T SOULI a a University
Mohamed I, Faculty of Sciences, Department of Mathematics, Oujda, Morocco Mohamed I, Faculty of Sciences & Techniques, Al-Hoceima, Morocco
b University
Received 2 February 2015; received in revised form 30 March 2015; accepted 14 April 2015 Available online 28 April 2015 Abstract.
In this paper, we study the following problems
∆ u−M
u = ∆u = 0
2
∗ |∇u| dx ∆u = λf (x, u) + |u|2 −2 u in Ω 2
Ω
on ∂Ω ,
where 2∗ = N2N is the critical exponent. Under some conditions on M and f , we prove the −4 existence of nontrivial solutions by using variational methods. Keywords: Fourth-order elliptic equations; Nonlocal problem; Critical exponent; Lions principle 2010 Mathematics Subject Classification: 34B10; 35B33
1. I NTRODUCTION AND MAIN RESULTS In this paper we are concerned with the existence of nontrivial solutions for the following nonlocal elliptic problems ∗ 2 2 ∆ u−M |∇u| dx ∆u = λf (x, u) + |u|2 −2 u in Ω (1.1) Ω u = ∆u = 0 on ∂Ω ,
∗
Corresponding author. E-mail addresses: [email protected] (E.M. Hssini), [email protected] (M. Massar), [email protected] (N. Tsouli). Peer review under responsibility of King Saud University.
Production and hosting by Elsevier http://dx.doi.org/10.1016/j.ajmsc.2015.04.001 c 2015 The Authors. Production and Hosting by Elsevier B.V. on behalf of King Saud University. This is an open access 1319-5166 ⃝ article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/).
Solutions to Kirchhoff equations with critical exponent
139
where Ω is a bounded smooth domain of RN , N ≥ 5, 2∗ = N2N −4 is the critical Sobolev exponent, and M : R+ → R+ , f : Ω × R → R are continuous functions that satisfy some conditions which will be stated later on. Problem (1.1) is related to the stationary problem of a model introduced by Kirchhoff [22]. More precisely, the authors in [8,22] introduced a model given by the following equation L 2 2 ∂u ρ0 E ∂2u dx ∂ u = 0, (1.2) + ρ 2 − ∂t h 2L 0 ∂x ∂x2 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. Later (1.2) was developed to form 2 utt − M |∇u| dx ∆u = f (x, u) in Ω . (1.3) Ω
After that, many authors studied the following nonlocal elliptic boundary value problem 2 −M |∇u| dx ∆u = f (x, u) in Ω , u = 0 on ∂Ω . (1.4) Ω
Problems like (1.4) can be used for modeling several physical and biological systems where u describes a process which depends on the average of itself, such as the population density, see [4]. Many interesting results for problems of Kirchhoff type were obtained see, for example, [1,13,20,21]. The investigation of fourth order boundary value problems has drawn the attention of many authors, because the static form change of beam or the sport of rigid body can be described by a fourth order equation, and specially a model to study traveling waves in suspension bridges can be furnished by the fourth order equation of nonlinearity. Several results are known concerning the existence and multiplicity of solutions for fourth order boundary value problems, see [10,11,18] and the references therein. In [26], using the mountain pass theorem, Wang and An established the existence and multiplicity of solutions for the following fourthorder nonlocal elliptic problem 2 ∆ u−M |∇u|2 dx ∆u = λf (x, u) in Ω (1.5) Ω u = ∆u = 0 on ∂Ω . Also, in [24] employing a smooth version of Ricceri’s variational principle [25], the authors ensured the existence of infinitely many solutions for the problem p−1 ∆(|∆u|p−2 ∆u) − M |∇u|p dx ∆p u + ρ|u|p−2 u = λf (x, u) in Ω (1.6) Ω u = ∆u = 0 on ∂Ω . Much interest has grown on problems involving critical exponents, starting from the celebrated paper by Brezis and Nirenberg [12]. This pioneering work has stimulated a vast amount of research on this class of problems. We refer the reader to [2,7,3,5,9,14–17,19,28] and reference therein for the study of problems with critical exponent. Before stating our main result, we need the following hypotheses on the function M : R+ → R+ :
140
E.M. Hssini et al.
(m1 ) There is a positive constant m0 such that M (t) ≥ m0
for all t ≥ 0.
(m2 ) There exists σ > 2/2∗ such that (t) ≥ σM (t)t for all t ≥ 0, M (t) = t M (s)ds. where M 0 A typical example of a function satisfying the conditions (m1 )–(m2 ) is given by M (t) = m0 + bt with b ≥ 0 and for all t ≥ 0. The hypotheses on function f : Ω × R → R are the following: (f 1 ) f (x, t) = o(|t|) as t → 0 uniformly in x ∈ Ω . (f 2 ) There exists q ∈ (2, 2∗ ) such that lim
|t|→+∞
f (x, t) = 0 uniformly in x ∈ Ω . |t|q−2 t
(f 3 ) There exists θ ∈ (max(2, 2/σ), 2∗ ) such that t 0 < θF (x, t) = θ f (x, s)ds ≤ tf (x, t) for all x ∈ Ω and t ∈ R \ {0}, 0
where σ is given in (m2 ). A typical example of a function satisfying the conditions (f 1 )–(f 3 ) is given by f (x, t) =
k
ai (x)|t|qi −2 t,
i=1
where k ≥ 1, 2 < qi < 2∗ , ai ∈ C(Ω ) with ai ≥ 0 for all x ∈ Ω . Now, we formulate our main result as follows. Theorem 1.1. Suppose that (m1 ), (m2 ) and (f 1 )–(f 3 ) hold. Then, there exists λ∗ > 0, such that problem (1.1) has a nontrivial solution for all λ ≥ λ∗ . 2. P RELIMINARY RESULTS We denote by H = H 2 (Ω ) ∩ H01 (Ω ) the Hilbert space equipped with the inner product (u, v) = (∆u∆v + ∇u∇v)dx, Ω
and the deduced norm 2 ∥u∥ = (|∆u|2 + |∇u|2 )dx. Ω
Solutions to Kirchhoff equations with critical exponent
We consider the following energy functional Iλ : H → R, defined by 1 1 Iλ (u) = |∆u|2 dx + M |∇u|2 dx 2 Ω 2 Ω ∗ 1 −λ F (x, u)dx − ∗ |u|2 dx. 2 Ω Ω
141
(2.1)
It is well known that a critical point of Iλ is a weak solution of problem (1.1). In the sequel, we show that the functional Iλ has the mountain pass geometry. Lemma 2.1. Suppose that (m1 ), (m2 ) and (f 1 )–(f 3 ) hold, then we have (i) There exist r, ρ > 0 such that inf ∥u∥=r Iλ (u) ≥ ρ > 0. (ii) There exists a nonnegative function e ∈ H such that ∥e∥ > r and Iλ (e) < 0. Proof. (i) It follows from (f 1 ) and (f 2 ) that for any ε > 0, there exists C(ε) > 0 such that F (x, t) ≤
1 2 εt + C(ε)|t|q . 2
(2.2)
Together with (m1 ) and Sobolev’s inequalities, we have ∗ m0 1 |∆u|2 dx + |∇u|2 dx − λC1 ε∥u∥2 − λC2 (ε)∥u∥q − C3 ∥u∥2 Iλ (u) ≥ 2 Ω 2 Ω ∗ min(1, m0 ) − λC1 ε ∥u∥2 − λC2 (ε)∥u∥q − C3 ∥u∥2 . (2.3) ≥ 2 0) We take ε < min(1,m , since 2 < q < 2∗ , choosing ∥u∥ = r small enough, we can obtain a 2λC1 positive constant ρ such that Iλ (u) ≥ ρ as ∥u∥ = r. (ii) Choose a nonnegative function φ1 ∈ C0∞ (Ω ) with ∥φ1 ∥ = 1. By integrating (m2 ), we get
0) 1 1 (t) ≤ M (t M t σ = C0 t σ for all t ≥ t0 > 0. 1 σ t0 Moreover, from (f 3 ), one has Ω F (x, tφ1 ) > 0. Hence for t ≥ t0 , we obtain t2 Iλ (tφ1 ) ≤ 2 ≤
t2 2
≤
t2 2
∗ ∗ 1 2 t2 2 |∆φ1 | dx + M t |∇φ1 | dx − ∗ φ21 dx 2 2 Ω Ω Ω σ1 ∗ 2 ∗ t2 C0 t σ 2 2 |∆φ1 | dx + |∇φ1 | dx − ∗ φ21 dx 2 2 Ω Ω Ω ∗ 2 2 ∗ C0 t σ t + − ∗ φ2 dx. 2 2 Ω 1
(2.4)
2
(2.5)
The fact that max(2, 2/σ) < 2∗ , the assertion (ii) is proved by choosing e = t∗ φ1 with t∗ > 0 large enough.
142
E.M. Hssini et al.
From Lemma 2.1, using a version of the Mountain Pass theorem due to Ambrosetti and Rabinowitz [6], without (PS) condition (see [27]), there exists a sequence (un ) ⊂ H such that Iλ (un ) → c∗
Iλ′ (un ) → 0,
and
where c∗ = inf max Iλ (γ(t)) > 0,
(2.6)
γ∈Γ t∈[0,1]
with Γ = {γ ∈ C([0, 1], H) : γ(0) = 0, Iλ (γ(1)) < 0} . ∗
Let S∗ be the best positive constant of the Sobolev embedding H ↩→ L2 (Ω ) given by 2 2∗ S∗ = inf ∥u∥ : u ∈ H, |u| dx = 1 . (2.7) Ω
Lemma 2.2. that (m1 ), (m2 ) and (f 1 )–(f 3 ) hold. Then there exists λ∗ > 0 such Suppose N4 1 1 that c∗ ∈ 0, θ − 2∗ S∗ for all λ ≥ λ∗ . Proof. For the nonnegative function e given in (ii) of Lemma 2.1, we have limt→+∞ Iλ (te) = −∞, then there exists tλ > 0 such that Iλ (tλ e) = max Iλ (te). t≥0
Therefore 2 2 2 tλ |∆e| dx + M tλ |∇e| dx tλ |∇e|2 dx Ω Ω Ω ∗ 2∗ −1 =λ f (x, tλ e)edx + tλ |e|2 dx. Ω
Ω
By (m2 ) and (f 3 ) it follows that ∗ ∗ ∗ ∗ t2λ e2 dx ≤ λtλ f (x, tλ e)edx + t2λ e2 dx Ω Ω Ω = t2λ |∆e|2 dx + M t2λ |∇e|2 dx t2λ |∇e|2 dx Ω
≤ t2λ
Ω
|∆e|2 +
Ω
1 M σ
Ω
t2λ |∇e|2 dx .
(2.8)
Ω
Hence, from (2.4), we obtain ∗ t2λ
2∗
e dx ≤ Ω
t2λ
C0 2/σ |∆e| + t σ λ Ω 2
2
|∇e| dx Ω
1/σ ,
with tλ > t0 .
Solutions to Kirchhoff equations with critical exponent
143
Since 2∗ > max(2, 2/σ), (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 t2λn
|∆e|2 +
Ω
C0 2/σ t σ λn
1/σ ≤C |∇e|2 dx
for all n,
Ω
that is, λn tλn Ω
∗
f (x, tλn e)edx + t2λn
∗
e2 dx ≤ C
for all n.
Ω
If s0 > 0, the last inequality implies that ∗ ∗ λn tλn f (x, tλn e)e dx + t2λn e2 dx → +∞ ≤ C, Ω
as n → ∞,
Ω
which is impossible, and consequently, s0 = 0. Let γ∗ (t) = te for t ∈ [0, 1]. Clearly γ∗ ∈ Γ , thus 1 2 t2λ 2 2 |∆e| + M tλ |∇e| dx . 0 < c∗ ≤ max Iλ (γ∗ (t)) = Iλ (tλ e) ≤ t≥0 2 Ω 2 Ω N Since tλn → 0 and θ1 − 21∗ (m0 S∗ ) 4 > 0, for λ > 0 sufficiently large, we have N t2λ 1 2 1 1 |∆e|2 + M tλ |∇e|2 dx < − ∗ S∗4 , 2 Ω 2 θ 2 Ω and hence 0 < c∗ <
1 1 − ∗ θ 2
N
S∗4 .
Proof of Theorem 1.1. By Lemmas 2.1 and 2.2, there exists a sequence (un ) ⊂ H such that Iλ (un ) → c∗ and Iλ′ (un ) → 0, (2.9) N with c∗ ∈ 0, θ1 − 21∗ S∗4 for λ ≥ λ∗ . Then, there exists C > 0 such that |Iλ (un )| ≤ C and by (f 3 ) for n large enough, it follows from (m1 ) and (m2 ) that 1 C + ∥un ∥ ≥ Iλ (un ) − ⟨Iλ′ (un ), un ⟩ θ 1 1 σ 1 ≥ − |∆un |2 dx + − m0 |∇un |2 dx 2 θ 2 θ Ω Ω 1 1 σ 1 ≥ min − , − m0 ∥un ∥2 . 2 θ 2 θ
(2.10)
Since θ > max(2, 2/σ), (un ) is bounded. Hence, up to a subsequence, we may assume that un ⇀ u weakly in H 2 (Ω ) ∩ H01 (Ω ),
144
E.M. Hssini et al.
un → u a.e. in Ω , un → u in Ls (Ω ), 1 ≤ s < 2∗ ,
(2.11)
∗
2
|∆un | ⇀ µ (weak —sense of measures) ∗
|un |2 ⇀ ν (weak∗ —sense of measures), where µ and ν are nonnegative bounded measures on Ω . Then, by concentration-compactness principle due to Lions [23], there exists some at most countable index set J such that ∗ ν = |u|2 + νj δxj , j∈J µ ≥ |∆u|2 + µj δxj , j∈J 2/2∗ S∗ νj ≤ µj ,
νj > 0, (2.12)
µj > 0,
where δxj is the Dirac measure mass at xj ∈ Ω . For ε > 0 and j ∈ J, define a function ψεj (x) ∈ C0∞ such that 0 ≤ ψεj ≤ 1, ψεj (x)
=
1 0
if if
|x − xj | < ε |x − xj | ≥ 2ε,
(2.13)
|∇ψεj |∞ ≤ 2/ε and |∆ψεj |∞ ≤ 2/ε2 . Since Iλ′ (un ) → 0 and (ψεj un ) is bounded, ⟨Iλ′ (un ), ψεj un ⟩ → 0 as n → ∞, that is 2 j |∆un | ψε dx + ∆un 2∇un ∇ψεj dx + un ∆ψεj dx Ω Ω 2 j j 2 +M |∇un | dx un ∇un ∇ψε dx + ψε |∇un | dx Ω
=λ
Ω
f (x, un )un ψεj dx +
Ω
Ω ∗
|un |2 ψεj dx + on (1).
(2.14)
Ω
Note that ∥∇(un − u)∥2L2 (Ω) = −
(un − u)∆(un − u)dx ≤ ∥un − u∥.∥un − u∥L2 (Ω) , Ω
then, (2.11) implies ∇un → ∇u in L2 (Ω ).
(2.15)
Now, by Vitali’s theorem we get lim |un ∇ψεj |2 dx = |u∇ψεj |2 dx and n→∞ Ω Ω j 2 lim |un ∆ψε | dx = |u∆ψεj |2 dx. n→∞
Ω
Ω
Solutions to Kirchhoff equations with critical exponent
145
In what follows, the letter C will be indiscriminately used to denote various constants. By H¨older’s inequality, we obtain 12 12 |∇un |2 lim sup un ∇un ∇ψεj dx ≤ lim sup |un |2 |∇ψεj |2 dx n→∞
n→∞
Ω
Ω
Ω
|u|2 |∇ψεj |2 dx
≤C
12
Ω
21∗
N1 |∇ψεj |N dx
≤C
∗
|u|2 dx
B(xj ,ε)
B(xj ,ε)
21∗
2∗
≤C
→ 0,
|u| dx
ε→0
B(xj ,ε)
and similarly, we have 12 21 j 2 2 j 2 lim sup ∆un ∇un ∇ψε dx ≤ lim sup |∆un | |∇un | |∇ψε | dx n→∞ n→∞ Ω Ω Ω 21 2 j 2 ≤C |∇u| |∇ψε | dx Ω
N1
21∗ 2∗
|∇ψεj |N dx
≤C
|∇u| dx
B(xj ,ε)
B(xj ,ε)
21∗
∗
|∇u|2 dx
≤C B(xj ,ε)
→ 0,
ε→0
and 12 12 2 j 2 j 2 |un | |∆ψε | dx lim sup un ∆un ∆ψε dx ≤ lim sup |∆un | n→∞ n→∞ Ω Ω Ω 12 2 j 2 ≤C |u| |∆ψε | dx Ω
≤C
N |∆ψεj | 2
N2 dx
B(xj ,ε)
≤C B(xj ,ε)
21∗ 2∗
|u| dx
B(xj ,ε)
21∗ ∗
|u|2 dx
→ 0.
ε→0
It follows that lim lim ∆un (2∇un ∇ψεj + un ∆ψεj ) + un ∇un ∇ψεj + ψεj |∇un |2 dx ε→0 n→∞
= 0.
Ω
(2.16)
146
E.M. Hssini et al.
On the other hand, from (2.11) we have f (x, un )un → f (x, u)u a.e. in Ω , un → u strongly in L2 (Ω ) and in Lq (Ω ). By (m1 ) and (m2 ), for any ε > 0 there exists Cε > 0 such that |f (x, t)| ≤ ε|t| + Cε |t|q−1
for all (x, t) ∈ Ω × R,
(2.17)
thus |f (x, un )un | ≤ ε|un |2 + Cε |un |q . This is what we need to apply Vitali’s theorem, which yields f (x, un )un dx = f (x, u)udx. lim n→∞
Ω
Ω
Since ψεj has compact support, from (2.11), (2.14) and (2.16) we deduce
ψεj dµ
21∗
2∗
≤C
|u| dx
Ω
B(xj ,ε) 2
|∇u| dx + λ
+ B(xj ,ε)
|∇u| dx
+C
B(xj ,ε)
21∗ 2∗
f (x, u)udx +
B(xj ,ε)
ψεj dν,
Ω
letting ε → 0, we get µj ≤ νj . It follows from (2.12) that N
S∗4 ≤ νj .
(2.18)
Now, we shall prove that the above expression cannot occur, and therefore the set J is N
empty. Indeed, arguing by contradiction, let us suppose that S∗4 ≤ νj0 for some j0 ∈ J. Then, from the fact that 1 c∗ = Iλ (un ) − ⟨Iλ′ (un ), un ⟩ + on (1), θ we obtain
∗ 1 1 − ∗ |un |2 dx + on (1) θ 2 Ω ∗ 1 1 ≥ − ∗ ψεj |un |2 dx + on (1). θ 2 Ω
c∗ ≥
Solutions to Kirchhoff equations with critical exponent
147
Letting n → ∞, we obtain c∗ ≥ =
1 1 − ∗ θ 2
1 1 − θ 2∗
ψεj (xj )νj
j∈J
νj ≥
j∈J
1 1 − θ 2∗
N
S∗4 ∗
which contradicts Lemma 2.2. This implies that J = ∅ and it follows that un → u in L2 (Ω ). The relation (2.17) implies that
|f (x, un )(un − u)|dx ≤ Ω
ε|un | + Cε |un |q−1 |un − u|dx
Ω
12 12 |un |2 dx |un − u|2 dx
≤ε Ω
Ω
+ Cε
q−1 q1 q q , |un | dx |un − u|q dx
Ω
Ω
and using again (2.11), we get f (x, un )(un − u)dx = 0.
lim
n→∞
(2.19)
Ω ∗
Since un → u in L2 (Ω ), we see that lim
n→∞
∗
|un |2
−2
un (un − u)dx = 0.
(2.20)
Ω
From ⟨Iλ′ (un ), un − u⟩ = on (1), we deduce that ⟨Iλ′ (un ), un − u⟩ =
∆un ∆(un − u)dx + M |∇un |2 dx ∇un ∇(un − u) Ω Ω Ω ∗ −λ f (x, un )(un − u)dx − |un |2 −2 un (un − u)dx
Ω
= on (1). By continuity of M , (2.15), (2.19) and (2.20) we have ∆un ∆(un − u)dx = 0.
lim
n→∞
Ω
In the same way, we obtain ∆u∆(un − u)dx = 0.
lim
n→∞
Ω
Ω
148
E.M. Hssini et al.
Taking into account (2.15), we conclude that ∥un ∥ → ∥u∥. By the uniform convexity of H, it follows that un → u strongly in H, and hence Iλ′ (u) = 0 and
Iλ (u) = c∗ ̸= 0.
The proof of Theorem 1.1 is complete.
ACKNOWLEDGMENT The authors would like to thank the anonymous referee for his/her valuable suggestions and comments, which greatly improve the manuscript. R EFERENCES [1] M. Al-Gwaiz, V. Benci, F. Gazzola, Bending and stretching energies in a rectangular plate modeling suspension bridges, Nonlinear Anal. TMA 106 (2014) 18–34. [2] C.O. Alves, Existence of positive solutions for a problem with lack of compactness involving the p-Laplacian, Nonlinear Anal. 51 (2002) 1187–1206. [3] C.O. Alves, F.J.S.A. Corrˆea, G.M. Figueiredo, On a class of nonlocal elliptic problem with critical growth, Differ. Equ. Appl. 2 (3) (2010) 409–417. [4] C.O. Alves, F.J.S.A. Corrˆea, T.F. MA, Positive solutions for a quasilinear elliptic equation of Kirchhoff type, Comput. Math. Appl. 49 (2005) 85–93. ´ O.H. Migayaki, On a class of singular biharmonic problems involving critical [5] C.O. Alves, F.M. Do O, exponents, J. Math. Anal. Appl. 277 (2003) 12–26. [6] A. Ambrosetti, P.H. Rabinowitz, Dual variational methods in critical point theory and applications, J. Funct. Anal. 14 (1973) 349–381. [7] J.G. Azorero, I.P. Alonso, Multiplicity of solutions for elliptic problems with critical exponent or with a non symmetric term, Trans. Amer. Math. Soc. 323 (2) (1991) 877–895. [8] H.M. Berger, A new approach to the analysis of large deflections of plates, J. Appl. Mech. 22 (1955) 465–472. [9] F. Bernis, J. Garc´ıa Azorero, I. Peral Alonso, Existence and multiplicity of nontrivial solutions in semilinear critical problems of fourth order, Adv. Differential Equations 1 (1996) 219–240. [10] G. Bonanno, B. Di Bella, A Boundary value problem for fourth-order elastic beam equations, J. Math. Anal. Appl. 343 (2008) 1166–1176. [11] G. Bonanno, B. Di Bella, D. O’Regan, Non-trivial solutions for nonlinear fourth-order elastic beam equations, Comput. Math. Appl. 62 (2011) 1862–1869. [12] H. Brezis, L. Nirenberg, Positive solutions of nonlinear elliptic equations involving critical Sobolev exponents, Comm. Pure Appl. Math. 36 (1983) 437–477. [13] B. Cheng, X. Wu, J. Liu, Multiplicity of solutions for nonlocal elliptic system of (p, q)-Kirchhoff type, Abstr. Appl. Anal. 2011 (526026) (2015) 13. http://dx.doi.org/10.1155/2011/526026. [14] D.E. Edmunds, D. Fortunato, E. Jannelli, Critical exponents, critical dimensions and the biharmonic operator, Arch. Ration. Mech. Anal. 112 (1990) 269–289. [15] G.M. Figueiredo, Existence of a positive solution for a Kirchhoff problem type with critical growth via truncation argument, J. Math. Anal. Appl. 401 (2013) 706–713. [16] G.M. Figueiredo, M.F. Furtado, Positive solutions for some quasilinear equations with critical and supercritical growth, Nonlinear Anal. 66 (2007) 1600–1616. [17] F. Gazzola, Critical growth problems for polyharmonic operators, Proc. Roy. Soc. Edinburgh Sect. A 128 (1998) 251–263. [18] F. Gazzola, H.-Ch. Grunau, G. Sweers, Polyharmonic Boundary Value Problems, Springer, 2010. [19] F. Gazzola, B. Ruf, Lower-order perturbations of critical growth nonlinearities in semilinear elliptic equations, Adv. Differential Equations 2 (1997) 555–572. [20] A. Hamydy, M. Massar, N. Tsouli, Existence of solutions for p-Kirchhoff type problems with critical exponent, Electron. J. Differential Equations 2011 (105) (2011) 1–8. [21] X. He, W. Zou, Infinitely many positive solutions for Kirchhoff-type problems, Nonlinear Anal. TMA 70 (3) (2009) 1407–1414.
Solutions to Kirchhoff equations with critical exponent
149
[22] G. Kirchhoff, Mechanik, Teubner, Leipzig, Germany, 1883. [23] P.L. Lions, The concentration-compactness principle in the calculus of variations. The limit case, part 1, Rev. Mat. Iberoam. 1 (1985) 145–201. [24] M. Massar, El.M. Hssini, N. Tsouli, M. Talbi, Infinitely many solutions for a fourth-order Kirchhoff type elliptic problem, J. Math. Comput. Sci. 8 (2014) 33–51. [25] B. Ricceri, A general variational principle and some of its applications, J. Comput. Appl. Math. 113 (2000) 401–410. [26] F. Wang, Y. An, Existence and multiplicity of solutions for a fourth-order elliptic equation, Bound. Value Probl. 2012 (2012) 6. [27] M. Willem, Minimax Theorems, Birkhauser, 1996. [28] Y.X. Yao, R.X. Wang, Y.T. Shen Yaotian, Nontrivial solution for a class of semilinear biharmonic equation involving critical exponents, Acta Math. Sci. 27B (3) (2007) 509–514.
Arab J Math Sci 22 (2016) 138–149
Solutions to Kirchhoff equations with critical exponent E L M ILOUD H SSINI a,∗ , M OHAMMED M ASSAR b , NAJIB T SOULI a a University
Mohamed I, Faculty of Sciences, Department of Mathematics, Oujda, Morocco Mohamed I, Faculty of Sciences & Techniques, Al-Hoceima, Morocco
b University
Received 2 February 2015; received in revised form 30 March 2015; accepted 14 April 2015 Available online 28 April 2015 Abstract.
In this paper, we study the following problems
∆ u−M
u = ∆u = 0
2
∗ |∇u| dx ∆u = λf (x, u) + |u|2 −2 u in Ω 2
Ω
on ∂Ω ,
where 2∗ = N2N is the critical exponent. Under some conditions on M and f , we prove the −4 existence of nontrivial solutions by using variational methods. Keywords: Fourth-order elliptic equations; Nonlocal problem; Critical exponent; Lions principle 2010 Mathematics Subject Classification: 34B10; 35B33
1. I NTRODUCTION AND MAIN RESULTS In this paper we are concerned with the existence of nontrivial solutions for the following nonlocal elliptic problems ∗ 2 2 ∆ u−M |∇u| dx ∆u = λf (x, u) + |u|2 −2 u in Ω (1.1) Ω u = ∆u = 0 on ∂Ω ,
∗
Corresponding author. E-mail addresses: [email protected] (E.M. Hssini), [email protected] (M. Massar), [email protected] (N. Tsouli). Peer review under responsibility of King Saud University.
Production and hosting by Elsevier http://dx.doi.org/10.1016/j.ajmsc.2015.04.001 c 2015 The Authors. Production and Hosting by Elsevier B.V. on behalf of King Saud University. This is an open access 1319-5166 ⃝ article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/).
Solutions to Kirchhoff equations with critical exponent
139
where Ω is a bounded smooth domain of RN , N ≥ 5, 2∗ = N2N −4 is the critical Sobolev exponent, and M : R+ → R+ , f : Ω × R → R are continuous functions that satisfy some conditions which will be stated later on. Problem (1.1) is related to the stationary problem of a model introduced by Kirchhoff [22]. More precisely, the authors in [8,22] introduced a model given by the following equation L 2 2 ∂u ρ0 E ∂2u dx ∂ u = 0, (1.2) + ρ 2 − ∂t h 2L 0 ∂x ∂x2 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. Later (1.2) was developed to form 2 utt − M |∇u| dx ∆u = f (x, u) in Ω . (1.3) Ω
After that, many authors studied the following nonlocal elliptic boundary value problem 2 −M |∇u| dx ∆u = f (x, u) in Ω , u = 0 on ∂Ω . (1.4) Ω
Problems like (1.4) can be used for modeling several physical and biological systems where u describes a process which depends on the average of itself, such as the population density, see [4]. Many interesting results for problems of Kirchhoff type were obtained see, for example, [1,13,20,21]. The investigation of fourth order boundary value problems has drawn the attention of many authors, because the static form change of beam or the sport of rigid body can be described by a fourth order equation, and specially a model to study traveling waves in suspension bridges can be furnished by the fourth order equation of nonlinearity. Several results are known concerning the existence and multiplicity of solutions for fourth order boundary value problems, see [10,11,18] and the references therein. In [26], using the mountain pass theorem, Wang and An established the existence and multiplicity of solutions for the following fourthorder nonlocal elliptic problem 2 ∆ u−M |∇u|2 dx ∆u = λf (x, u) in Ω (1.5) Ω u = ∆u = 0 on ∂Ω . Also, in [24] employing a smooth version of Ricceri’s variational principle [25], the authors ensured the existence of infinitely many solutions for the problem p−1 ∆(|∆u|p−2 ∆u) − M |∇u|p dx ∆p u + ρ|u|p−2 u = λf (x, u) in Ω (1.6) Ω u = ∆u = 0 on ∂Ω . Much interest has grown on problems involving critical exponents, starting from the celebrated paper by Brezis and Nirenberg [12]. This pioneering work has stimulated a vast amount of research on this class of problems. We refer the reader to [2,7,3,5,9,14–17,19,28] and reference therein for the study of problems with critical exponent. Before stating our main result, we need the following hypotheses on the function M : R+ → R+ :
140
E.M. Hssini et al.
(m1 ) There is a positive constant m0 such that M (t) ≥ m0
for all t ≥ 0.
(m2 ) There exists σ > 2/2∗ such that (t) ≥ σM (t)t for all t ≥ 0, M (t) = t M (s)ds. where M 0 A typical example of a function satisfying the conditions (m1 )–(m2 ) is given by M (t) = m0 + bt with b ≥ 0 and for all t ≥ 0. The hypotheses on function f : Ω × R → R are the following: (f 1 ) f (x, t) = o(|t|) as t → 0 uniformly in x ∈ Ω . (f 2 ) There exists q ∈ (2, 2∗ ) such that lim
|t|→+∞
f (x, t) = 0 uniformly in x ∈ Ω . |t|q−2 t
(f 3 ) There exists θ ∈ (max(2, 2/σ), 2∗ ) such that t 0 < θF (x, t) = θ f (x, s)ds ≤ tf (x, t) for all x ∈ Ω and t ∈ R \ {0}, 0
where σ is given in (m2 ). A typical example of a function satisfying the conditions (f 1 )–(f 3 ) is given by f (x, t) =
k
ai (x)|t|qi −2 t,
i=1
where k ≥ 1, 2 < qi < 2∗ , ai ∈ C(Ω ) with ai ≥ 0 for all x ∈ Ω . Now, we formulate our main result as follows. Theorem 1.1. Suppose that (m1 ), (m2 ) and (f 1 )–(f 3 ) hold. Then, there exists λ∗ > 0, such that problem (1.1) has a nontrivial solution for all λ ≥ λ∗ . 2. P RELIMINARY RESULTS We denote by H = H 2 (Ω ) ∩ H01 (Ω ) the Hilbert space equipped with the inner product (u, v) = (∆u∆v + ∇u∇v)dx, Ω
and the deduced norm 2 ∥u∥ = (|∆u|2 + |∇u|2 )dx. Ω
Solutions to Kirchhoff equations with critical exponent
We consider the following energy functional Iλ : H → R, defined by 1 1 Iλ (u) = |∆u|2 dx + M |∇u|2 dx 2 Ω 2 Ω ∗ 1 −λ F (x, u)dx − ∗ |u|2 dx. 2 Ω Ω
141
(2.1)
It is well known that a critical point of Iλ is a weak solution of problem (1.1). In the sequel, we show that the functional Iλ has the mountain pass geometry. Lemma 2.1. Suppose that (m1 ), (m2 ) and (f 1 )–(f 3 ) hold, then we have (i) There exist r, ρ > 0 such that inf ∥u∥=r Iλ (u) ≥ ρ > 0. (ii) There exists a nonnegative function e ∈ H such that ∥e∥ > r and Iλ (e) < 0. Proof. (i) It follows from (f 1 ) and (f 2 ) that for any ε > 0, there exists C(ε) > 0 such that F (x, t) ≤
1 2 εt + C(ε)|t|q . 2
(2.2)
Together with (m1 ) and Sobolev’s inequalities, we have ∗ m0 1 |∆u|2 dx + |∇u|2 dx − λC1 ε∥u∥2 − λC2 (ε)∥u∥q − C3 ∥u∥2 Iλ (u) ≥ 2 Ω 2 Ω ∗ min(1, m0 ) − λC1 ε ∥u∥2 − λC2 (ε)∥u∥q − C3 ∥u∥2 . (2.3) ≥ 2 0) We take ε < min(1,m , since 2 < q < 2∗ , choosing ∥u∥ = r small enough, we can obtain a 2λC1 positive constant ρ such that Iλ (u) ≥ ρ as ∥u∥ = r. (ii) Choose a nonnegative function φ1 ∈ C0∞ (Ω ) with ∥φ1 ∥ = 1. By integrating (m2 ), we get
0) 1 1 (t) ≤ M (t M t σ = C0 t σ for all t ≥ t0 > 0. 1 σ t0 Moreover, from (f 3 ), one has Ω F (x, tφ1 ) > 0. Hence for t ≥ t0 , we obtain t2 Iλ (tφ1 ) ≤ 2 ≤
t2 2
≤
t2 2
∗ ∗ 1 2 t2 2 |∆φ1 | dx + M t |∇φ1 | dx − ∗ φ21 dx 2 2 Ω Ω Ω σ1 ∗ 2 ∗ t2 C0 t σ 2 2 |∆φ1 | dx + |∇φ1 | dx − ∗ φ21 dx 2 2 Ω Ω Ω ∗ 2 2 ∗ C0 t σ t + − ∗ φ2 dx. 2 2 Ω 1
(2.4)
2
(2.5)
The fact that max(2, 2/σ) < 2∗ , the assertion (ii) is proved by choosing e = t∗ φ1 with t∗ > 0 large enough.
142
E.M. Hssini et al.
From Lemma 2.1, using a version of the Mountain Pass theorem due to Ambrosetti and Rabinowitz [6], without (PS) condition (see [27]), there exists a sequence (un ) ⊂ H such that Iλ (un ) → c∗
Iλ′ (un ) → 0,
and
where c∗ = inf max Iλ (γ(t)) > 0,
(2.6)
γ∈Γ t∈[0,1]
with Γ = {γ ∈ C([0, 1], H) : γ(0) = 0, Iλ (γ(1)) < 0} . ∗
Let S∗ be the best positive constant of the Sobolev embedding H ↩→ L2 (Ω ) given by 2 2∗ S∗ = inf ∥u∥ : u ∈ H, |u| dx = 1 . (2.7) Ω
Lemma 2.2. that (m1 ), (m2 ) and (f 1 )–(f 3 ) hold. Then there exists λ∗ > 0 such Suppose N4 1 1 that c∗ ∈ 0, θ − 2∗ S∗ for all λ ≥ λ∗ . Proof. For the nonnegative function e given in (ii) of Lemma 2.1, we have limt→+∞ Iλ (te) = −∞, then there exists tλ > 0 such that Iλ (tλ e) = max Iλ (te). t≥0
Therefore 2 2 2 tλ |∆e| dx + M tλ |∇e| dx tλ |∇e|2 dx Ω Ω Ω ∗ 2∗ −1 =λ f (x, tλ e)edx + tλ |e|2 dx. Ω
Ω
By (m2 ) and (f 3 ) it follows that ∗ ∗ ∗ ∗ t2λ e2 dx ≤ λtλ f (x, tλ e)edx + t2λ e2 dx Ω Ω Ω = t2λ |∆e|2 dx + M t2λ |∇e|2 dx t2λ |∇e|2 dx Ω
≤ t2λ
Ω
|∆e|2 +
Ω
1 M σ
Ω
t2λ |∇e|2 dx .
(2.8)
Ω
Hence, from (2.4), we obtain ∗ t2λ
2∗
e dx ≤ Ω
t2λ
C0 2/σ |∆e| + t σ λ Ω 2
2
|∇e| dx Ω
1/σ ,
with tλ > t0 .
Solutions to Kirchhoff equations with critical exponent
143
Since 2∗ > max(2, 2/σ), (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 t2λn
|∆e|2 +
Ω
C0 2/σ t σ λn
1/σ ≤C |∇e|2 dx
for all n,
Ω
that is, λn tλn Ω
∗
f (x, tλn e)edx + t2λn
∗
e2 dx ≤ C
for all n.
Ω
If s0 > 0, the last inequality implies that ∗ ∗ λn tλn f (x, tλn e)e dx + t2λn e2 dx → +∞ ≤ C, Ω
as n → ∞,
Ω
which is impossible, and consequently, s0 = 0. Let γ∗ (t) = te for t ∈ [0, 1]. Clearly γ∗ ∈ Γ , thus 1 2 t2λ 2 2 |∆e| + M tλ |∇e| dx . 0 < c∗ ≤ max Iλ (γ∗ (t)) = Iλ (tλ e) ≤ t≥0 2 Ω 2 Ω N Since tλn → 0 and θ1 − 21∗ (m0 S∗ ) 4 > 0, for λ > 0 sufficiently large, we have N t2λ 1 2 1 1 |∆e|2 + M tλ |∇e|2 dx < − ∗ S∗4 , 2 Ω 2 θ 2 Ω and hence 0 < c∗ <
1 1 − ∗ θ 2
N
S∗4 .
Proof of Theorem 1.1. By Lemmas 2.1 and 2.2, there exists a sequence (un ) ⊂ H such that Iλ (un ) → c∗ and Iλ′ (un ) → 0, (2.9) N with c∗ ∈ 0, θ1 − 21∗ S∗4 for λ ≥ λ∗ . Then, there exists C > 0 such that |Iλ (un )| ≤ C and by (f 3 ) for n large enough, it follows from (m1 ) and (m2 ) that 1 C + ∥un ∥ ≥ Iλ (un ) − ⟨Iλ′ (un ), un ⟩ θ 1 1 σ 1 ≥ − |∆un |2 dx + − m0 |∇un |2 dx 2 θ 2 θ Ω Ω 1 1 σ 1 ≥ min − , − m0 ∥un ∥2 . 2 θ 2 θ
(2.10)
Since θ > max(2, 2/σ), (un ) is bounded. Hence, up to a subsequence, we may assume that un ⇀ u weakly in H 2 (Ω ) ∩ H01 (Ω ),
144
E.M. Hssini et al.
un → u a.e. in Ω , un → u in Ls (Ω ), 1 ≤ s < 2∗ ,
(2.11)
∗
2
|∆un | ⇀ µ (weak —sense of measures) ∗
|un |2 ⇀ ν (weak∗ —sense of measures), where µ and ν are nonnegative bounded measures on Ω . Then, by concentration-compactness principle due to Lions [23], there exists some at most countable index set J such that ∗ ν = |u|2 + νj δxj , j∈J µ ≥ |∆u|2 + µj δxj , j∈J 2/2∗ S∗ νj ≤ µj ,
νj > 0, (2.12)
µj > 0,
where δxj is the Dirac measure mass at xj ∈ Ω . For ε > 0 and j ∈ J, define a function ψεj (x) ∈ C0∞ such that 0 ≤ ψεj ≤ 1, ψεj (x)
=
1 0
if if
|x − xj | < ε |x − xj | ≥ 2ε,
(2.13)
|∇ψεj |∞ ≤ 2/ε and |∆ψεj |∞ ≤ 2/ε2 . Since Iλ′ (un ) → 0 and (ψεj un ) is bounded, ⟨Iλ′ (un ), ψεj un ⟩ → 0 as n → ∞, that is 2 j |∆un | ψε dx + ∆un 2∇un ∇ψεj dx + un ∆ψεj dx Ω Ω 2 j j 2 +M |∇un | dx un ∇un ∇ψε dx + ψε |∇un | dx Ω
=λ
Ω
f (x, un )un ψεj dx +
Ω
Ω ∗
|un |2 ψεj dx + on (1).
(2.14)
Ω
Note that ∥∇(un − u)∥2L2 (Ω) = −
(un − u)∆(un − u)dx ≤ ∥un − u∥.∥un − u∥L2 (Ω) , Ω
then, (2.11) implies ∇un → ∇u in L2 (Ω ).
(2.15)
Now, by Vitali’s theorem we get lim |un ∇ψεj |2 dx = |u∇ψεj |2 dx and n→∞ Ω Ω j 2 lim |un ∆ψε | dx = |u∆ψεj |2 dx. n→∞
Ω
Ω
Solutions to Kirchhoff equations with critical exponent
145
In what follows, the letter C will be indiscriminately used to denote various constants. By H¨older’s inequality, we obtain 12 12 |∇un |2 lim sup un ∇un ∇ψεj dx ≤ lim sup |un |2 |∇ψεj |2 dx n→∞
n→∞
Ω
Ω
Ω
|u|2 |∇ψεj |2 dx
≤C
12
Ω
21∗
N1 |∇ψεj |N dx
≤C
∗
|u|2 dx
B(xj ,ε)
B(xj ,ε)
21∗
2∗
≤C
→ 0,
|u| dx
ε→0
B(xj ,ε)
and similarly, we have 12 21 j 2 2 j 2 lim sup ∆un ∇un ∇ψε dx ≤ lim sup |∆un | |∇un | |∇ψε | dx n→∞ n→∞ Ω Ω Ω 21 2 j 2 ≤C |∇u| |∇ψε | dx Ω
N1
21∗ 2∗
|∇ψεj |N dx
≤C
|∇u| dx
B(xj ,ε)
B(xj ,ε)
21∗
∗
|∇u|2 dx
≤C B(xj ,ε)
→ 0,
ε→0
and 12 12 2 j 2 j 2 |un | |∆ψε | dx lim sup un ∆un ∆ψε dx ≤ lim sup |∆un | n→∞ n→∞ Ω Ω Ω 12 2 j 2 ≤C |u| |∆ψε | dx Ω
≤C
N |∆ψεj | 2
N2 dx
B(xj ,ε)
≤C B(xj ,ε)
21∗ 2∗
|u| dx
B(xj ,ε)
21∗ ∗
|u|2 dx
→ 0.
ε→0
It follows that lim lim ∆un (2∇un ∇ψεj + un ∆ψεj ) + un ∇un ∇ψεj + ψεj |∇un |2 dx ε→0 n→∞
= 0.
Ω
(2.16)
146
E.M. Hssini et al.
On the other hand, from (2.11) we have f (x, un )un → f (x, u)u a.e. in Ω , un → u strongly in L2 (Ω ) and in Lq (Ω ). By (m1 ) and (m2 ), for any ε > 0 there exists Cε > 0 such that |f (x, t)| ≤ ε|t| + Cε |t|q−1
for all (x, t) ∈ Ω × R,
(2.17)
thus |f (x, un )un | ≤ ε|un |2 + Cε |un |q . This is what we need to apply Vitali’s theorem, which yields f (x, un )un dx = f (x, u)udx. lim n→∞
Ω
Ω
Since ψεj has compact support, from (2.11), (2.14) and (2.16) we deduce
ψεj dµ
21∗
2∗
≤C
|u| dx
Ω
B(xj ,ε) 2
|∇u| dx + λ
+ B(xj ,ε)
|∇u| dx
+C
B(xj ,ε)
21∗ 2∗
f (x, u)udx +
B(xj ,ε)
ψεj dν,
Ω
letting ε → 0, we get µj ≤ νj . It follows from (2.12) that N
S∗4 ≤ νj .
(2.18)
Now, we shall prove that the above expression cannot occur, and therefore the set J is N
empty. Indeed, arguing by contradiction, let us suppose that S∗4 ≤ νj0 for some j0 ∈ J. Then, from the fact that 1 c∗ = Iλ (un ) − ⟨Iλ′ (un ), un ⟩ + on (1), θ we obtain
∗ 1 1 − ∗ |un |2 dx + on (1) θ 2 Ω ∗ 1 1 ≥ − ∗ ψεj |un |2 dx + on (1). θ 2 Ω
c∗ ≥
Solutions to Kirchhoff equations with critical exponent
147
Letting n → ∞, we obtain c∗ ≥ =
1 1 − ∗ θ 2
1 1 − θ 2∗
ψεj (xj )νj
j∈J
νj ≥
j∈J
1 1 − θ 2∗
N
S∗4 ∗
which contradicts Lemma 2.2. This implies that J = ∅ and it follows that un → u in L2 (Ω ). The relation (2.17) implies that
|f (x, un )(un − u)|dx ≤ Ω
ε|un | + Cε |un |q−1 |un − u|dx
Ω
12 12 |un |2 dx |un − u|2 dx
≤ε Ω
Ω
+ Cε
q−1 q1 q q , |un | dx |un − u|q dx
Ω
Ω
and using again (2.11), we get f (x, un )(un − u)dx = 0.
lim
n→∞
(2.19)
Ω ∗
Since un → u in L2 (Ω ), we see that lim
n→∞
∗
|un |2
−2
un (un − u)dx = 0.
(2.20)
Ω
From ⟨Iλ′ (un ), un − u⟩ = on (1), we deduce that ⟨Iλ′ (un ), un − u⟩ =
∆un ∆(un − u)dx + M |∇un |2 dx ∇un ∇(un − u) Ω Ω Ω ∗ −λ f (x, un )(un − u)dx − |un |2 −2 un (un − u)dx
Ω
= on (1). By continuity of M , (2.15), (2.19) and (2.20) we have ∆un ∆(un − u)dx = 0.
lim
n→∞
Ω
In the same way, we obtain ∆u∆(un − u)dx = 0.
lim
n→∞
Ω
Ω
148
E.M. Hssini et al.
Taking into account (2.15), we conclude that ∥un ∥ → ∥u∥. By the uniform convexity of H, it follows that un → u strongly in H, and hence Iλ′ (u) = 0 and
Iλ (u) = c∗ ̸= 0.
The proof of Theorem 1.1 is complete.
ACKNOWLEDGMENT The authors would like to thank the anonymous referee for his/her valuable suggestions and comments, which greatly improve the manuscript. R EFERENCES [1] M. Al-Gwaiz, V. Benci, F. Gazzola, Bending and stretching energies in a rectangular plate modeling suspension bridges, Nonlinear Anal. TMA 106 (2014) 18–34. [2] C.O. Alves, Existence of positive solutions for a problem with lack of compactness involving the p-Laplacian, Nonlinear Anal. 51 (2002) 1187–1206. [3] C.O. Alves, F.J.S.A. Corrˆea, G.M. Figueiredo, On a class of nonlocal elliptic problem with critical growth, Differ. Equ. Appl. 2 (3) (2010) 409–417. [4] C.O. Alves, F.J.S.A. Corrˆea, T.F. MA, Positive solutions for a quasilinear elliptic equation of Kirchhoff type, Comput. Math. Appl. 49 (2005) 85–93. ´ O.H. Migayaki, On a class of singular biharmonic problems involving critical [5] C.O. Alves, F.M. Do O, exponents, J. Math. Anal. Appl. 277 (2003) 12–26. [6] A. Ambrosetti, P.H. Rabinowitz, Dual variational methods in critical point theory and applications, J. Funct. Anal. 14 (1973) 349–381. [7] J.G. Azorero, I.P. Alonso, Multiplicity of solutions for elliptic problems with critical exponent or with a non symmetric term, Trans. Amer. Math. Soc. 323 (2) (1991) 877–895. [8] H.M. Berger, A new approach to the analysis of large deflections of plates, J. Appl. Mech. 22 (1955) 465–472. [9] F. Bernis, J. Garc´ıa Azorero, I. Peral Alonso, Existence and multiplicity of nontrivial solutions in semilinear critical problems of fourth order, Adv. Differential Equations 1 (1996) 219–240. [10] G. Bonanno, B. Di Bella, A Boundary value problem for fourth-order elastic beam equations, J. Math. Anal. Appl. 343 (2008) 1166–1176. [11] G. Bonanno, B. Di Bella, D. O’Regan, Non-trivial solutions for nonlinear fourth-order elastic beam equations, Comput. Math. Appl. 62 (2011) 1862–1869. [12] H. Brezis, L. Nirenberg, Positive solutions of nonlinear elliptic equations involving critical Sobolev exponents, Comm. Pure Appl. Math. 36 (1983) 437–477. [13] B. Cheng, X. Wu, J. Liu, Multiplicity of solutions for nonlocal elliptic system of (p, q)-Kirchhoff type, Abstr. Appl. Anal. 2011 (526026) (2015) 13. http://dx.doi.org/10.1155/2011/526026. [14] D.E. Edmunds, D. Fortunato, E. Jannelli, Critical exponents, critical dimensions and the biharmonic operator, Arch. Ration. Mech. Anal. 112 (1990) 269–289. [15] G.M. Figueiredo, Existence of a positive solution for a Kirchhoff problem type with critical growth via truncation argument, J. Math. Anal. Appl. 401 (2013) 706–713. [16] G.M. Figueiredo, M.F. Furtado, Positive solutions for some quasilinear equations with critical and supercritical growth, Nonlinear Anal. 66 (2007) 1600–1616. [17] F. Gazzola, Critical growth problems for polyharmonic operators, Proc. Roy. Soc. Edinburgh Sect. A 128 (1998) 251–263. [18] F. Gazzola, H.-Ch. Grunau, G. Sweers, Polyharmonic Boundary Value Problems, Springer, 2010. [19] F. Gazzola, B. Ruf, Lower-order perturbations of critical growth nonlinearities in semilinear elliptic equations, Adv. Differential Equations 2 (1997) 555–572. [20] A. Hamydy, M. Massar, N. Tsouli, Existence of solutions for p-Kirchhoff type problems with critical exponent, Electron. J. Differential Equations 2011 (105) (2011) 1–8. [21] X. He, W. Zou, Infinitely many positive solutions for Kirchhoff-type problems, Nonlinear Anal. TMA 70 (3) (2009) 1407–1414.
Solutions to Kirchhoff equations with critical exponent
149
[22] G. Kirchhoff, Mechanik, Teubner, Leipzig, Germany, 1883. [23] P.L. Lions, The concentration-compactness principle in the calculus of variations. The limit case, part 1, Rev. Mat. Iberoam. 1 (1985) 145–201. [24] M. Massar, El.M. Hssini, N. Tsouli, M. Talbi, Infinitely many solutions for a fourth-order Kirchhoff type elliptic problem, J. Math. Comput. Sci. 8 (2014) 33–51. [25] B. Ricceri, A general variational principle and some of its applications, J. Comput. Appl. Math. 113 (2000) 401–410. [26] F. Wang, Y. An, Existence and multiplicity of solutions for a fourth-order elliptic equation, Bound. Value Probl. 2012 (2012) 6. [27] M. Willem, Minimax Theorems, Birkhauser, 1996. [28] Y.X. Yao, R.X. Wang, Y.T. Shen Yaotian, Nontrivial solution for a class of semilinear biharmonic equation involving critical exponents, Acta Math. Sci. 27B (3) (2007) 509–514.