On nonlocal p⃗(x) -Laplacian equations

Nonlinear Analysis 73 (2010) 3364–3375

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On nonlocal p (x)-Laplacian equationsI Xianling Fan ∗ Department of Mathematics, Lanzhou University, Lanzhou, 730000, PR China

article

info

abstract − →

This paper deals with the nonlocal anisotropic p (x)-Laplacian Dirichlet problems with non-variational form

Article history: Received 3 April 2010 Accepted 7 July 2010

−

MSC: 35J70 35J92 58E30 47H05

N X

Ai (u)Di |Di u|pi (x)−2 Di u = B (u)f (x, u(x)) in Ω ; u|∂ Ω = 0,




i=1

and with variational form

!
|Di u|pi (x) dx Di |Di u|pi (x)−2 Di u − ai p ( x ) i Ω i=1 Z  =b F (x, u)dx f (x, u(x)) in Ω ; u|∂ Ω = 0, N X

Keywords: Nonlocal differential equation Variable exponent Anisotropic Variational method

Z

Ω

Rt

where F (x, t ) = 0 f (x, s)ds, and ai is allowed to be singular at zero. Using (S+ ) mapping theory and the variational method, some results on existence and multiplicity for the problems are obtained. © 2010 Elsevier Ltd. All rights reserved.

1. Introduction Nonlocal differential equations and differential equations with variable exponents have been two very active fields of investigation in recent years. Both of them have important and extensive application backgrounds. The study of nonlocal differential equations with variable exponents is worthy of attention. Nonlocal differential equations are also called Kirchhoff-type equations because Kirchhoff [1] has investigated an equation of the form

Z L 2 ! 2 ∂u dx ∂ u = 0. + h 2L 0 ∂ x ∂ x2

∂ 2u ρ 2 − ∂t

P0

E

A distinguishing feature of Eq. (1.1) is that the equation contains a nonlocal coefficient (

| ∂ u |2 dx of the kinetic energy 21 | ∂∂ux |2 0 ∂x

RL

(1.1) P0 h

+

E 2L

RL 0

| ∂∂ux |2 dx) which depends

on the average on [0, L], and hence the equation is no longer a pointwise identity. Lions [2] has proposed an abstract framework for the Kirchhoff-type equations. After the work of Lions [2], various equations of Kirchhoff-type have been studied extensively, see e.g. [3–8]. Recently, many authors (see e.g. [9–19]) have studied the 1 2L

I Research supported by National Natural Science Foundation of China (10971087).

∗

Tel.: +86 931 8911173; fax: +86 931 8912481. E-mail addresses: [email protected], [email protected].

0362-546X/$ – see front matter © 2010 Elsevier Ltd. All rights reserved. doi:10.1016/j.na.2010.07.018

X. Fan / Nonlinear Analysis 73 (2010) 3364–3375

3365

p-Kirchhoff Dirichlet problem of the form



−a(I (u))1p u = f (x, u(x)) in Ω u = 0 on ∂ Ω ,

(1.2)

where Ω ⊂ RN is a bounded domain, p ∈ (1, +∞) and I (u) is an integral functional. In particular, in the case that I (u) = R |∇ u|p dx, problem (1.2) possesses a variational structure. Ω p For the advances of the study of differential equations with variable exponents see the overview paper [20]. From [20] it can be seen that various differential equations of p(x)-Laplace type have been studied extensively. More recently, several authors (see e.g. [21–26]) have studied the anisotropic quasilinear elliptic equations with variable exponents, i.e. the − → quasilinear elliptic equations involving the p (x)-Laplacian → −1− p (x) u := −

N X

Di (|Di (u)|pi (x)−2 Di u),

i=1

− →

where p (x) = (p1 (x), p2 (x), . . . , pN (x)) and Di = ∂∂x . i The nonlocal p(x)-Laplacian equations have already been studied by several authors (see [27–30]). In [30], the nonlocal p(x)-Laplacian Dirichlet problems of the form



−A(u)1p(x) u = B (u)f (x, u(x)) in Ω u = 0 on ∂ Ω

(1.3)

and of the form

 Z  |∇ u|p(x) −a dx 1p(x) u = b F (x, u)dx f (x, u(x)) in Ω (1.4) p(x) Ω Ω  u = 0 on ∂ Ω Rt have been studied, where F (x, t ) = 0 f (x, s)ds. (1.4) is a special case of (1.3) having the variational structure. − → The purpose of the present paper is to study the nonlocal anisotropic p (x)-Laplacian Dirichlet problems of the form  N 
 X − Ai (u)Di |Di u|pi (x)−2 Di u = B (u)f (x, u(x)) in Ω (1.5)  u =i=01 on ∂ Ω  

Z

and of the form

 Z  Z  N 
|Di u|pi (x)  X − ai dx Di |Di u|pi (x)−2 Di u = b F (x, u)dx f (x, u(x)) in Ω pi (x) Ω Ω  u =i=01 on ∂ Ω ,

(1.6)

Rt

where F (x, t ) = 0 f (x, s)ds. (1.5) and (1.6) are an anisotropic variant of (1.3) and (1.4) respectively. The anisotropic problems are more complicated than the isotropic problems. The underlying ideas of the present paper are essentially the same as those of [30], but the study of (1.5) and (1.6) needs − →

− →

embedding theorem for

1, p (·) W0

− →

1, p (·)

the theory of the anisotropic variable exponent Sobolev spaces W 1, p (·) (Ω ) and W0

(Ω ), (in particular, the compact

(Ω )), established in [26], as a theoretic basis. It is well known that (see e.g. [26]), in the − → − → 1, p (·) 1, p (·) − →

→ (Ω ) → (W0 (Ω ))∗ is a strictly case where pi,− > 1 for i = 1, . . . , N, the p (x)-Laplacian operator −1− p (x) : W0 monotone homeomorphism. However, in general, the operator T defined by the left side of the equation in (1.5) or (1.6) is not monotone and is not a homeomorphism. An important result obtained in the present paper is that T is a mapping of type (S+ ) (see Propositions 3.1 and 4.2 below), which plays an important role in the study of existence and multiplicity for (1.5) and (1.6). The paper is organized as follows. In Section 2 we present some preliminary results on the anisotropic variable exponent − →

− →

1, p (·)

Sobolev spaces W 1, p (·) (Ω ) and W0 (Ω ). In Section 3 we consider problem (1.5) and give an existence result for (1.5) by using the (S+ ) mapping theory. In Section 4 we consider problem (1.6) and obtain some results on existence and multiplicity for (1.6) by using the variational method. The hypotheses used in the present paper for (1.6) are weaker than those used in some existing works on (1.2). For example, in some existing works (see e.g. [14–19]) it is assumed that a ∈ C 0 ([0, +∞)) and a(t ) ≥ m0 > 0

for all t ≥ 0 (or for large t > 0),

but in the present paper condition (1.7) is not necessary for ai , and ai is allowed to be singular at 0. The results obtained in the present paper are also new for the case that each pi is a constant.

(1.7)

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X. Fan / Nonlinear Analysis 73 (2010) 3364–3375

2. Preliminaries Let Ω ⊂ RN be a bounded domain. Define S (Ω ) = {u|u : Ω → R is measurable}. For q ∈ S (Ω ) and E ⊂ Ω , define q− (E ) = ess inf q(x), q+ (E ) = ess sup q(x). x∈E

x∈E

For simplicity we write q− and q+ instead of q− (Ω ) and q+ (Ω ) respectively. ∞ ∞ q(·) Define L∞ (Ω ) is defined by + (Ω ) = {q ∈ L (Ω ) : q− ≥ 1}. Let q ∈ L+ (Ω ). The variable exponent Lebesgue space L L

q(·)



(Ω ) = u ∈ S (Ω ) :

Z

q(x)

| u| Ω

dx < ∞



with the norm

  Z q(x) u |u|Lq(·) (Ω ) = |u|q(·) = inf σ > 0 | dx ≤ 1 . Ω σ For the variable exponent Lebesgue and Sobolev spaces see [31,32]. Let pi ∈ L∞ + (Ω ) for i = 1, 2, . . . , N. Write

− → p (x) = (p1 (x), . . . , pN (x)) and p∨ (x) = max{p1 (x), . . . , pN (x)} for x ∈ Ω . − →

Define the anisotropic variable exponent Sobolev space W 1, p (·) (Ω ) by − →

W 1, p (·) (Ω ) =

n

∨ u ∈ Lp (·) (Ω ) : Di u ∈ Lpi (·) (Ω ) for i = 1, . . . , N

o

 = u ∈ S (Ω ) : u ∈ Lpi (·) (Ω ), Di u ∈ Lpi (·) (Ω ) for i = 1, . . . , N with the norm

kukW 1,−→p (·) (Ω ) = |u|p∨ (·) +

N X

|Di u|pi (·) .

i =1

− →

− →

1, p (·)

− →

− →

− →

− →

1, p (·)

Proposition 2.1 ([26]). W 1, p (·) (Ω ) and W0 i = 1, . . . , N.


N

N N P i=1

(Ω ),

(Ω ) are separable Banach spaces, and they are reflexive if pi,− > 1 for

For p (·) = (p1 (·), p2 (·), . . . , pN (·)) ∈ L∞ + (Ω ) p(x) =

− →

1, p (·)

We denote by W0 (Ω ) the closure of C0∞ (Ω ) in W 1, p (·) (Ω ). For the basic properties of W 1, p (·) (Ω ) and W0 in particular, for the following Propositions 2.1–2.5, see [26].

and x ∈ Ω , we define

,

1 pi (x)

  Np(x) ∗ p (x) = N − p(x) +∞

if p(x) < N if p(x) ≥ N .

For the Banach spaces X and Y , we denote by X ,→ Y that X is continuously embedded into Y , and by X ,→,→ Y that X is compactly embedded into Y . In what follows, for brevity, a rectangular domain (or a cube) with edges parallel to the coordinate axes in RN is simply said to be a rectangular domain (or a cube). For x ∈ RN and ε > 0, we denote by K (x, ε) the N-dimensional open cube with center x and edge length ε . We say that Ω ⊂ RN is a rectangular-like domain if Ω is a union of finitely many rectangular domains. Define 0 C+ (Ω ) = {u ∈ C 0 (Ω ) : u(x) ≥ 1 for all x ∈ Ω }.

− →

0 Proposition 2.2 ([26]). 1. Let Ω ⊂ RN be a rectangular-like domain and p (·) ∈ (C+ (Ω ))N . 0 (1) If q ∈ C+ (Ω ) and

q(x) < max{p (x), p∨ (x)} for all x ∈ Ω , ∗

− → 1, p (·)

then W (Ω ) ,→,→ Lq(·) (Ω ). − → − → (2) If p(x) > N for all x ∈ Ω , then there exists β ∈ (0, 1) such that W 1, p (·) (Ω ) ,→ C 0,β (Ω ), and consequently, W 1, p (·) (Ω ) ,→,→ C 0 (Ω ). − →

1, p (·)

2. The same statements as in part 1 are true for any bounded domain Ω ⊂ RN if use W0

− →

(Ω ) instead of W 1, p (·) (Ω ).

X. Fan / Nonlinear Analysis 73 (2010) 3364–3375

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− →

0 Proposition 2.3 ([26]). Let Ω ⊂ RN be a bounded domain and p (·) ∈ (C+ (Ω ))N . Suppose that

p∨ (x) < p (x) for all x ∈ Ω . ∗

(2.1)

Then the following Poincaré-type inequality holds:

|u|Lp∨ (·) (Ω ) ≤ C

N X

− →

|Di u|Lpi (·) (Ω ) for all u ∈ W01, p (·) (Ω ),

i=1

− →

1, p (·)

(Ω ). Thus

where C is a positive constant independent of u ∈ W0

PN

i=1

− →

|Di u|Lpi (·) (Ω ) is an equivalent norm on W01, p (·) (Ω ).

Remark 2.1. Under the assumptions of Proposition 2.3, max{|Di u|Lpi (·) (Ω ) : i = 1, . . . , N } is also an equivalent norm on − →

1, p (·)

W0

(Ω ).

This equivalent norm is used in Sections 3 and 4.

− →

0 Proposition 2.4 ([26]). Let Ω ⊂ RN be a rectangular-like domain and p (·) ∈ (C+ (Ω ))N . Suppose that (2.1) holds. Then

− →

W 1, p (·) (Ω ) = u ∈ L1 (Ω ) : Di u ∈ Lpi (·) (Ω ) for i = 1, . . . , N .





Let us recall the definition of the mapping of type (S+ ). Let X be a Banach space and D ⊂ X . The notations ‘‘un * u’’ and ‘‘un → u’’ denote respectively the weak convergence and the strong convergence in X . A mapping A : D → X ∗ is said to be of type (S+ ) if for any sequence {un } ⊂ D for which un * u in X and limn→∞ A(un )(un − u) ≤ 0, un must converge strongly to u in X . For the (S+ ) mapping theory, including the degree theory and the surjection theorem, see [33–35]. ∞ Proposition 2.5 ([26]). Let q ∈ L∞ + (Ω ) with q− > 1, b ∈ L (Ω ) with b− > 0, and ρ(u) = Then the following assertions hold.

R

Ω

b(x)|u|q(x) dx for u ∈ Lq(·) (Ω ).

(1) If {un } ⊂ Lq(·) (Ω ), un * u in Lq(·) (Ω ) and ρ(un ) → ρ(u) as n → ∞, then un → u in Lq(·) (Ω ). (2) ρ ∈ C 1 (Lq(·) (Ω ), R), and the mapping ρ 0 : Lq(·) (Ω ) → (Lq(·) (Ω ))∗ is a bounded, strictly monotone homeomorphism, and is of type (S+ ). − →

1, p (·)

− →

N Proposition 2.6. Let p (·) ∈ (L∞ + (Ω )) and u ∈ W0

(Ω ) \ {0}. Then |Di u|pi (·) 6= 0 for all i = 1, 2, . . . , N.

− → 1, p (·)

Proof. Let u ∈ W0 (Ω ) and |Di0 u|pi0 (·) = 0 for some i0 ∈ {1, 2, . . . , N }. Then |Di0 u|pi0 ,− = 0. By a result of [36] (see the proof of Theorem 1 in [36]), |u|pi ,− ≤ c |Di0 u|pi ,− = 0. This shows that u = 0.  0

0

3. The non-variational case In this section we consider problem (1.5).

− →

1, p (·)

Let Ω ⊂ RN be a bounded domain. In what follows, for brevity, we shall write X instead of W0 assumptions are used. → 0 (Ω ))N , pi,− > 1 for i = 1, . . . , N, and (2.1) holds. (P0 ) − p (·) = (p1 (·), p2 (·), . . . , pN (·)) ∈ (C+ (f0 ) f : Ω × R → R is a Carathéodory function, and

(Ω ). The following

|f (x, t )| ≤ c1 |t |q(x)−1 + h(x) for x ∈ Ω and t ∈ R, r ( x)

0 where c1 is a positive constant, q ∈ C+ (Ω ) with q(x) < p (x) for x ∈ Ω , h is a nonnegative function, h ∈ L r (x)−1 (Ω ) for some ∗ 0 r ∈ C+ (Ω ) with r (x) < p (x) for x ∈ Ω . (A0 ) For each i = 1, . . . , N, Ai : X → [0, +∞) is continuous and bounded on any bounded subset of X , Ai (u) > 0 for all u ∈ X \ {0}, and for any bounded sequence {un } ⊂ X for which Ai (un ) → 0, Di un must converge strongly to 0 in Lpi (·) (Ω ). (B0 ) The functional B : X → R is continuous and bounded on any bounded subset of X . For u ∈ X , we write kuk = max{|Di u|Lpi (·) (Ω ) : i = 1, . . . , N }. By Remark 2.1, in the case when (P0 ) holds, k · k is an equivalent norm on X . u ∈ X is called a weak solution of (1.5) if

∗

N X i =1

Ai (u)

Z Ω

|Di u(x)|pi (x)−2 Di u(x)Di v(x)dx = B (u)

Z Ω

f (x, u(x))v(x)dx,

∀v ∈ X .

(3.1)

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X. Fan / Nonlinear Analysis 73 (2010) 3364–3375

Define the mappings T , G and Nf : X → X ∗ respectively by T (u)v =

N X

Ai (u)

i =1

G(u)v = B (u) Nf (u)v =

Z Ω

Z Ω

Z Ω

|Di u(x)|pi (x)−2 Di u(x)Di v(x)dx,

f (x, u(x))v(x)dx,

f (x, u(x))v(x)dx,

∀ u, v ∈ X ,

∀u, v ∈ X ,

∀u, v ∈ X .

Then G(u) = B (u)Nf (u) for u ∈ X . It is clear that, u ∈ X is a weak solution of (1.5) if and only if T (u) − G(u) = 0. Proposition 3.1. Let (P0 ) and (A0 ) hold. Then the mapping T : X → X ∗ is continuous and bounded, and is of type (S+ ). Proof. It is obvious that the mapping T : X → X ∗ is continuous and bounded. To prove that T is of type (S+ ), assuming that {un } ⊂ X , un * u in X and limn→∞ T (un )(un − u) ≤ 0, it is sufficient to show that any subsequence of {un } has a (strongly) convergent subsequence. Let {unk } be a subsequence of {un }. Then there exists a subsequence of {unk }, denoted still by {unk }, such that Ai (unk ) → ci ≥ 0 for each i = 1, . . . , N. Since un * u in X , we have that, for each i = 1, . . . , N, Di un * Di u in Lpi (·) (Ω ) and

Z limn→∞

|Di un |pi (x)−2 Di un (Di un − Di u) dx ≥ 0,

Ω

and consequently

Z ci lim

nk →∞

Ω


Di un pi (x)−2 Di un Di un − Di u dx = 0. k k k

(3.2)

In the case where ci = 0, it follows immediately from assumption (A0 ) that Di unk → 0 = Di u in Lpi (·) (Ω ). In the case where ci > 0, it follows from (3.2) that

Z lim

nk →∞

Ω


|Di unk |pi (x)−2 Di unk Di unk − Di u dx = 0,

and thus, by Proposition 2.5, Di unk → Di u in Lpi (·) (Ω ). This shows that unk → u in X as nk → ∞. The proof is complete.



Proposition 3.2. Let (P0 ), (f0 ) and (B0 ) hold. Then the mapping G : X → X ∗ is completely continuous, that is, G is continuous and G(D) is relatively compact for any bounded set D ⊂ X . Proof. By Proposition 2.2, there is a compact embedding X ,→,→ Lq(x) (Ω ). From (f0 ) we can see that the mapping Nf : X → X ∗ is sequentially weakly–strongly continuous. The continuity of G is obvious. Let {un } ⊂ X be bounded. Then there exists a subsequence {unk } of {un } such that {Nf (unk )} and {B (unk )} are strongly convergent, and consequently, {G(unk )} is strongly convergent. This shows that G : X → X ∗ is completely continuous.  Noting that the sum of an (S+ ) type mapping and a completely continuous mapping is of type (S+ ), then from Propositions 3.1 and 3.2 we have the following: Corollary 3.1. Let (P0 ), (A0 ), (f0 ) and (B0 ) hold. Then the mapping T − G : X → X ∗ is continuous and bounded, and is of type (S+ ). Theorem 3.1. Let (P0 ), (A0 ), (f0 ) and (B0 ) hold. Suppose that the following conditions are satisfied:

(A1 ) There are constants αi ∈ R, M ≥ 1 and C0 > 0 such that for each i = 1, . . . , N,
α Ai (u) ≥ C0 |Di u|pi (·) i for u ∈ X with |Di u|pi (·) ≥ M . (B1 ) There are constants β ∈ R, M ≥ 1 and c2 > 0 such that |B (u)| ≤ c 2 kukβ for u ∈ X with kuk ≥ M . (H0 ) δ := min{αi + pi,− : i = 1, . . . , N } > β + q+ and δ > 0. Then problem (1.5) has at least one solution. If, in addition, δ > 1, then the mapping T − G : X → X ∗ is surjective, and consequently, for any ψ ∈ X ∗ the operator equation T (u) − G(u) = ψ has at least one solution.

X. Fan / Nonlinear Analysis 73 (2010) 3364–3375

3369

∗ Proof. Under the hypotheses of Theorem 3.1, by Corollary 2.1, the mapping T − G : X → X is continuous and bounded, and is of type (S+ ). Let u ∈ X with kuk ≥ M, and let kuk = Di0 u p (·) . Then, by (A0 ) and (A1 ), i0

T (u)u =

N X

Ai (u)

Z Ω

i=1

 ≥ C0 Di0 u p

|Di u|pi (x) dx ≥ Ai0 (u)

i0 (·)

αi  0 Di u 0

Z Ω

pi

0 ,−

Di u pi0 (x) dx 0

= C0 kukαi0 +pi0 ,− ≥ C0 kukδ ,

pi (·) 0

and by (f0 ) and (B1 ), G(u)u = B (u)

Z Ω

≤ c2 kukβ ≤ c2 kuk

β

f (x, u(x))u(x)dx

Z

c1 |u|q(x) + h(x)|u| dx




Ω

c3 kukq+ + 2 |h|

 r (·)

r (·)−1

≤ c2 kukβ c3 kukq+ + c4 kuk

|u|r (·)




≤ c5 kukβ+q+ . Thus, for sufficiently large kuk, noting that δ > β + q+ and δ > 0, we have that

(T (u) − G(u))u ≥ C0 kukδ − c5 kukβ+q+ ≥

C0 2

kukδ > 0.

By the degree theory for (S+ ) type mappings (see [34]), for R > 0 large enough, we have deg(T − G, B(0, R), 0) = 1, and consequently, there exists u ∈ B(0, R) such that T (u) − G(u) = 0, that is, problem (1.5) has at least one solution u ∈ B(0, R). If in addition, δ > 1, then lim

kuk→∞

(T (u) − G(u))u C0 ≥ lim kukδ−1 = +∞, kuk→∞ 2 k uk

that is, the mapping T − G is coercive, and consequently, by the surjection theorem for the pseudomonotone mappings (see [35, Theorem 27.A]), the mapping T − G is surjective.  In Theorem 3.1, taking specially f (x, t ) = h(x) and B (u) ≡ 1, we obtain the following corollary. s(x)

0 Corollary 3.2. Let (P0 ), (A0 ) and (A1 ) hold, and let g ∈ L s(x)−1 (Ω ) for some s ∈ C+ (Ω ) with s(x) < p (x) for x ∈ Ω . Suppose that δ := min{αi + pi,− : i = 1, . . . , N } > 1. Then the following problem

∗

 N 
 X − Ai (u)Di |Di u|pi (x)−2 Di u = g (x) in Ω  u =i=01 on ∂ Ω

(3.3)

has at least one solution. Remark 3.1. In Theorem 3.1, αi and β are allowed to be negative. It is well known that, if in Corollary 3.2 Ai (u) ≡ 1 for i = 1, . . . , N, then the solution of (3.3) is unique. However, for general Ai , the solution of (3.3) is not necessarily unique. 4. The variational case In this section we consider problem (1.6). We assume that (P0 ), (f0 ) and the following conditions are satisfied. (a0 ) For each i = 1, . . . , N, ai : (0, +∞) → (0, +∞) is continuous and ai ∈ L1 (0, t ) for any t > 0. (b0 ) b : R → R is continuous. Note that the function ai satisfying (a0 ) may be singular at t = 0. u ∈ X \ {0} is called a weak solution of (1.6) if N X i =1

Z ai

Ω

|Di u|pi (x) dx pi (x)

Z

pi (x)−2

Ω

|Di u|

Di uDi v dx = b

Z

Z

Ω

F (x, u)dx

Ω

f (x, u)v dx,

∀v ∈ X .

(4.1)

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X. Fan / Nonlinear Analysis 73 (2010) 3364–3375

Define for i = 1, . . . , N,

b ai ( t ) =

t

Z

ai (s)ds,

∀t ≥ 0;

b b(t ) =

t

Z

b(s)ds,

∀t ∈ R,

0

0

|Di u|pi (x) dx, ϕ(u) = F (x, u)dx, ∀u ∈ X , pi (x) Ω Ω Z  |Di u|pi (x) dx , ∀u ∈ X , J i ( u) = b ai (Ii (u)) = b ai pi (x) Ω N X J (u) = Ji (u), ∀u ∈ X , I i ( u) =

Z

Z

i=1

Φ ( u) = b b(ϕ(u)) = b b E (u) = J (u) − Φ (u),



Z Ω

F (x, u)dx ,

∀u ∈ X ,

∀u ∈ X .

Proposition 4.1. Let (P0 ), (f0 ), (a0 ) and (b0 ) hold. Then the following statements hold: (1) b ai ∈ C 0 ([0, ∞)) ∩ C 1 ((0, ∞)), b ai (0) = 0, b ai 0 (t ) = ai (t ) > 0 for t > 0, b ai is strictly increasing on [0, ∞); b b ∈ C 1 (R), 0 b b b(0) = 0 and b (t ) = b(t ) for t ∈ R. (2) Ji , Φ , E ∈ C 0 (X ), Ji (0) = Φ (0) = E (0) = 0. Ji ∈ C 1 (X \ {0}), Φ ∈ C 1 (X ), E ∈ C 1 (X \ {0}). u ∈ X \ {0} is a weak solution of (1.6) if and only if u is a nontrivial critical point of E. (3) The functional Ji : X → R is sequentially weakly lower semi-continuous, Φ : X → R is sequentially weakly continuous, and thus E is sequentially weakly lower semi-continuous. (4) The mapping Φ 0 : X → X ∗ is sequentially weakly–strongly continuous. Proof. The proof of statements (1) and (2) is immediate. Since the function b ai (t ) is increasing and the convex functional Ii is sequentially weakly lower semi-continuous, we can see that the functional Ji : X → R is sequentially weakly lower semi-continuous. From (f0 ), (b0 ) and the compact embedding X ,→,→ Lq(x) (Ω ), we can see that Φ : X → R is sequentially weakly continuous and Φ 0 : X → X ∗ is sequentially weakly–strongly continuous. Statements (3) and (4) are proved.  Proposition 4.2. Let (P0 ), (f0 ), (a0 ) and (b0 ) hold. Then the mappings J 0 and E 0 : X \ {0} → X ∗ are of type (S+ ). Proof. Let {un } ⊂ X \ {0}, un * u in X and limn→∞ J 0 (un )(un − u) ≤ 0. We will prove that un → u in X in two cases respectively: (i) u = 0, (ii) u 6= 0. In case (i), u = 0, then limn→∞ J 0 (un )un ≤ 0, that is, limn→∞

N X

Z ai

Ω

i=1

!Z |Di un |pi (x) |Di un |pi (x) dx ≤ 0. dx p i ( x) Ω

(4.2)

Noting that {kun k} is bounded, from (4.2) and (a0 ) we obtain that

Z Ω

|Di un |pi (x) dx → 0 as n → ∞ for i = 1, . . . , N ,

and consequently un → 0 in X . In case (ii), u 6= 0, then by Proposition 2.6, |D u |pi (x) limn→∞ Ω i p n(x) dx i

R

Z Ω

≥

|Di u|pi (x) dx, Ω pi (x)

R

R

Ω

|Di u|pi (x) dx pi (x)

> 0 for all i = 1, . . . , N. Note that

we can see that there exists a positive constant c such that for sufficiently large n,

|Di un |pi (x) dx ≥ c for i = 1, . . . , N . pi (x)

(4.3)

Since {kun k} is bounded, there exists a positive constant C such that

Z Ω

|Di un |pi (x) dx ≤ C for i = 1, . . . , N . pi (x)

(4.4)

In order to prove un → u in X , it is sufficient to show that any subsequence of {un } has a strongly convergent subsequence. Let {unk } is a subsequence of {un }. By (4.3) and (4.4), there exists a subsequence of {unk }, denoted still by {unk }, such that

pi (x) Z D i un k

Ω

p i ( x)

dx → di ∈ [c , C ],

∀i = 1, . . . , N .

X. Fan / Nonlinear Analysis 73 (2010) 3364–3375

 Thus ai

R |Di unk |pi (x)

dx

pi (x)

Ω

limn→∞



N X

ai (di )

Z

i=1

Ω

3371

→ ai (di ) ∈ (0, +∞) and


Di un pi (x)−2 Di un Di un − Di u dx ≤ 0. k k k

(4.5)

Because

 
Di un (x) pi (x)−2 Di un (x) − |Di u(x)|pi (x)−2 Di u(x) Di un (x) − Di u(x) ≥ 0, k k k

∀x ∈ Ω ,

and

Z


|Di u|pi (x)−2 Di u Di unk − Di u dx = 0,

lim

n→∞

Ω

it follows from (4.5) that

Z lim

n→∞

Ω


Di un pi (x)−2 Di un Di un − Di u dx = 0. k k k

(4.6)

By Proposition 2.5, it follows from (4.6) that Di unk → Di u in Lpi (·) (Ω ) for i = 1, . . . , N, and hence unk → u in X . This shows that the mappings J 0 : X \ {0} → X ∗ is of type (S+ ). Since Φ 0 : X → X ∗ is sequentially weakly–strongly continuous, the mappings E 0 = J 0 − Φ 0 : X \ {0} → X ∗ is also of type (S+ ).  Corollary 4.1. Let (P0 ), (f0 ), (a0 ) and (b0 ) hold. Suppose that {un } ⊂ X \ {0} is bounded and E 0 (un ) → 0 as n → ∞. Then {un } has a strongly convergent subsequence {unk } : unk → u in X . If, in addition, E (un ) → c 6= 0, then E (u) = c, u 6= 0 and E 0 (u) = 0, and consequently, u is a non-zero solution of (1.6). Proof. Let {un } ⊂ X \ {0} be bounded and E 0 (un ) → 0 as n → ∞. Then there exists a subsequence {unk } of {un } such that unk * u in X . Since E 0 (un ) → 0, limn→∞ E 0 (unk )(unk − u) = 0. Since E 0 : X \ {0} → X ∗ is of type (S+ ), we have unk → u in X . If, in addition, E (un ) → c 6= 0, then, by the continuity of E at u, E (u) = c 6= 0 = E (0). Thus u 6= 0, and by the continuity of E 0 at u, E 0 (u) = limnk →∞ E 0 (unk ) = 0.  Remark 4.1. Under assumption (a0 ), the function ai may be singular at 0 and in this case the energy functional E may be non-differentiable at 0. It is obvious that, under assumptions (P0 ), (f0 ), (a0 ) and (b0 ), if in addition, for each i = 1, . . . , N, ai is continuous at 0, then E ∈ C 1 (X ) and E : X → X ∗ is of type (S+ ). In the sequel, we denote by C , Cj or cj a positive constant. Theorems 4.1–4.4 below deal with the case where the functional E is coercive, that is, limkuk→∞ E (u) = +∞. Theorem 4.1. Let (P0 ), (f0 ), (a0 ), (b0 ) and the following conditions hold:

(a1 ) For each i = 1, . . . , N, there are positive constants γi , M and C such that b ai (t ) ≥ Ct γi for t ≥ M. (b1 ) There are positive constants β1 and C1 such that |b b(t )| ≤ C1 + C1 |t |β1 for t ∈ R. (H1 ) β1 q+ < γi pi,− for i = 1, . . . , N. Then the functional E is coercive, that is, E (u) → +∞ as kuk → ∞, and E attains its infimum in X at some u0 ∈ X . Therefore, u0 is a solution of (1.6) if E is differentiable at u0 , and in particular, if u0 6= 0. Proof. Set ε = min{γi pi,− − β1 q+ : i = 1, . . . , N }. Then by (H1 ), ε > 0. By (P0 ), (a0 ), (a1 ) and (H1 ), for each i = 1, . . . , N, we have that, for sufficiently large |Di u|pi (·) ,

!  
p |Di u|pi (x) 1 Ji (u) = b ai dx ≥ b ai |Di u|pi (·) i,− pi (x) pi,+ Ω
γi pi,−
β q +ε ≥ C |Di u|pi (·) ≥ C |Di u|pi (·) 1 + , Z

and hence, for sufficiently large kuk, J (u) =

N X

Ji (u) ≥

i =1

N X

C |Di u|pi (·)


β1 q+ +ε

≥ C kukβ1 q+ +ε .

i =1

By (f0 ), (b0 ) and (b1 ),

Z Z β1  |Φ (u)| = b b F (x, u)dx ≤ C1 + C1 F (x, u)dx Ω Ω
q+ β1 ≤ C1 + C1 C2 + C3 kuk ≤ C4 + C5 kukβ1 q+ .

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X. Fan / Nonlinear Analysis 73 (2010) 3364–3375

Thus, when kuk → ∞, E (u) = J (u) − Φ (u) ≥ C kukβ1 q+ +ε − C4 − C5 kukβ1 q+ → +∞, that is, E is coercive. Since E is sequentially weakly lower semi-continuous and X is reflexive, E attains its infimum in X at some u0 ∈ X . In the case where E is differentiable at u0 , u0 is a solution of (1.6).  Theorem 4.2. Let (P0 ), (f0 ), (a0 ), (a1 ), (b0 ), (b1 ), (H1 ) and the following conditions hold:

(a2 ) For each i = 1, . . . , N, there exists δi > 0 such that limt →0+ abti (δti ) < +∞. (b2 ) There exists β2 > 0 such that limt →0 |bt |(βt )2 > 0. b

(f1 ) There exist an open subset Ω0 of Ω and r > 0 such that limt →0 F |(tx|,rt ) > 0 uniformly in x ∈ Ω0 . (H2 ) β2 r < δi pi,− for i = 1, . . . , N. Then (1.6) has at least one nontrivial solution which is a global minimizer of the energy functional E. Proof. Setting ε1 = min{δi pi,− − β2 r : i = 1, . . . , N }, then by (H2 ), ε1 > 0. From Theorem 4.1 we know that E has a global minimizer u0 . It is clear that E (0) = 0. Take w ∈ C0∞ (Ω0 ) \ {0}. Then, by (a2 ), for sufficiently small λ ∈ (0, 1), we have that, for each i = 1, . . . , N,

! !δ i Z λpi (x) |Di w|pi (x) λpi (x) |Di w|pi (x) Ji (λw) = b ai dx ≤ C2 dx pi (x) pi (x) Ω Ω !δ i Z pi (x) | D w| i ≤ C3 λδi pi,− ≤ C3 λβ2 r +ε1 . ≤ C2 λδi pi,− dx pi (x) Ω R Note that when λ → 0, λw(x) → 0 uniformly in x ∈ Ω , and Ω F (x, λw)dx → 0. Then by (b2 ) and (f1 ), for sufficiently small λ ∈ (0, 1), Z  Z β2 Φ (λw) = b b F (x, λw)dx ≥ c2 F (x, λw)dx Z

Ω

Z ≥ c2

Ω

Ω

c3 (λw)r dx

β2

≥ c4 λβ2 r .

Thus for sufficiently small λ ∈ (0, 1), E (λw) = J (λw) − Φ (λw) ≤ NC3 λβ2 r +ε1 − c4 λβ2 r < 0. Hence E (u0 ) < 0 which shows u 6= 0.



Let c ∈ R. A sequence {un } ⊂ X is called a (PS)c sequence for E is E (un ) → c and E 0 (un ) → 0 as n → ∞. We say that E satisfies condition (PS)c if any (PS)c sequence has a (strongly) convergent subsequence.



Theorem 4.3. Let all the hypotheses of Theorem 4.2 hold, and let, in addition, f satisfy the following condition

(f2 ) f (x, −t ) = −f (x, t ) for x ∈ Ω and t ∈ R. Then (1.6) has a sequence of solutions {±un } such that E (±un ) < 0 for n = 1, 2, . . . . Proof. Condition (f2 ) implies that F (x, t ) is even in t, and therefore the functional E is even. Since E is coercive, by Corollary 4.1, E satisfies condition (PS)c for any c 6= 0. We already know that E (0) = 0 and infu∈X E (u) > −∞. By the Ljusternik–Schnirelman category theorem (see [37]), to prove Theorem 4.3, it is sufficient to prove that for every positive integer n, there exists a symmetric closed set An ⊂ X such that γ (An ) ≥ n and supu∈An E (u) < 0, where γ (·) is the Krasnoselskii genus. Now let any n be given. Noting that C0∞ (Ω0 ) is an infinite-dimensional subspace of X , we can take an n-dimensional subspace Yn ⊂ C0∞ (Ω0 ) ⊂ X . Set Sn−1 = {u ∈ Yn |kuk = 1}. From the proof of Theorem 4.2 we know that, for every w ∈ Sn−1 , there exists λw > 0 such that E (λw w) < 0. Since Sn−1 is compact, we can find a positive constant λ such that E (λw) < 0 for all w ∈ Sn−1 . Set An = λSn−1 . Then γ (An ) = n and supu∈An E (u) < 0. The proof is complete.  Theorem 4.4. Let (P0 ), (f0 ), (a0 ), (a1 ), (a2 ), (b0 ), (b1 ), (b2 ), (H1 ), (H2 ) and the following conditions hold:

(b+ ) b(t ) ≥ 0 for t ≥ 0. (f+ ) f (x, t ) ≥ 0 for x ∈ Ω and t ≥ 0. (f1 )+ There exist an open subset Ω0 of Ω and r > 0 such that limt →0+ F (txr,t ) > 0 uniformly in x ∈ Ω0 . Then (1.6) has at least one nontrivial nonnegative solution with negative energy.

X. Fan / Nonlinear Analysis 73 (2010) 3364–3375

3373

Proof. Define

  f (x, t ) if t ≥ 0, b(t ) if t ≥ 0, e e f ( x, t ) = b(t ) = f (x, 0) if t < 0, b(0) if t < 0, Z t Z t b e e e e b(s)ds, f (x, s)ds, ∀x ∈ Ω , t ∈ R, b(t ) = F (x, t ) = 0 0 Z  b e e E (u) = J (u) − e b F (x, u)dx , ∀u ∈ X .

∀t ∈ R,

Ω

Then, like in the proof of Theorem 4.2 (in the proof take w ∈ C0∞ (Ω0 ) \ {0} with w ≥ 0), we can prove that e E has a nontrivial global minimizer u0 with e E (u0 ) < 0, and thus u0 6= 0 is a solution of the following problem

 Z  Z  N 
|Di u|pi (x)  X e dx Di |Di u|pi (x)−2 Di u = e b F (x, u)dx e f (x, u(x)) in Ω − ai pi (x) Ω Ω i =1  u = 0 on ∂ Ω .

(4.7)

R |Di u0 |pi (x) − + Define u+ dx), i = 1, . . . , N, and µ0 = 0 (x) = max{u0 (x), 0} and u0 (x) = u0 (x) − u0 (x). Put λi = ai ( Ω p (x)

i R e F (x, u0 )dx). Then λi > 0 and µ0 ≥ 0. Taking u− b( Ω e 0 as a test function of (4.7) and using the condition (f+ ), we obtain

that

−

N X i =1

λi

Z Ω

− pi (x) Di u dx = µ0 0

Z Ω

e f (x, u0 )u− 0 dx ≥ 0,

which implies that u0 = 0. Thus u0 ≥ 0, e f (x, u0 (x)) = f (x, u0 (x)), e F (x, u0 (x)) = F (x, u0 (x)), e b( R b( Ω F (x, u0 )dx), and consequently, u0 is a nontrivial nonnegative solution of (1.6).  −

R

Ω

e F (x, u0 )dx) =

Example 4.1. Let ai (t ) = t γi −1 for t > 0 and i = 1, . . . , N, where γi > 0; b(t ) = |t |β1 −2 t for t ∈ R, where β1 ≥ 1; 0 f (x, t ) = |t |q(x)−2 t for t ∈ R, where q ∈ C+ (Ω ) with q(x) < p∗ (x) for x ∈ Ω . Suppose that β1 q+ < γi pi,− for i = 1, . . . , N. Then all hypotheses of Theorems 4.1–4.4 are satisfied. Note that when γi ∈ (0, 1), ai is singular at t = 0. Remark 4.2. Above we use assumptions (a0 ) and (b0 ). Thus ai is allowed to be singular at 0, but b is continuous at 0. In fact, as was mentioned in [30], we can also consider the case that b is also singular at 0. We now turn to the case where the functional E satisfies the mountain pass geometry. Proposition 4.3. Let (P0 ), (a0 ), (b0 ) and the following conditions be satisfied:

(f0 )0 (f0 ) holds and h ∈ L∞ (Ω ). (a1 )0 (a1 ) holds and γi pi,− > 1 for i = 1, . . . , N. (a3 ) For each i = 1, . . . , N, there exist λi > 0 and M > 0 such that λib ai (t ) ≥ ai (t )t for t ≥ M. (b3 ) There exist ν > 0 and M > 0 such that 0 < νb b(t ) ≤ b(t )t for t ≥ M. (f3 ) There exist µ > 0 and M > 0 such that 0 < µF (x, t ) ≤ f (x, t )t for |t | ≥ M and x ∈ Ω . (H3 ) λi pi,+ < νµ for i = 1, . . . , N. Then E satisfies condition (PS)c for any c 6= 0. Proof. By (a3 ), for each i = 1, . . . , N, and for sufficiently large |Di u|pi (·) ,

 |Di u|pi (x) dx pi (x) Ω Z Z |Di u|pi (x) |Di u|pi (x) ≥ pi,+ ai dx dx pi (x) p i ( x) Ω Ω Z Z |Di u|pi (x) ≥ ai dx |Di u|pi (x) dx = Ji0 (u)u. pi (x) Ω Ω

λi pi,+ Ji (u) = λi pi,+b ai

Z

(4.8)

In [30] it was proved that, (b3 ), (f0 )0 and (f3 ) imply that, given any ε ∈ (0, µ), there exists Cε > 0 such that

Φ 0 (u)u − ν(µ − ε)Φ (u) ≥ −Cε for u ∈ X ,

(4.9)

(see the proof of Proposition 4.1 in [30]). Now let {un } ⊂ X \ {0}, E 0 (un ) → 0 and E (un ) → c with c 6= 0. By (H3 ), there exists ε > 0 small enough that λi pi,+ < ν(µ − ε) for i = 1, . . . , N. Setting d = min{γi pi,− : i = 1, 2, . . . , N } and

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X. Fan / Nonlinear Analysis 73 (2010) 3364–3375

e = ν(µ − ε) − λi pi,+ , then d > 1 and e > 0. Since {un } is a (PS)c sequence, by (4.8), (4.9) and (a1 )0 , for sufficiently large n, we have

ν(µ − ε)c + 1 + kun k ≥ ν(µ − ε)E (un ) − E 0 (un )un N N X X = ν(µ − ε) Ji (un ) − Ji0 (un )un + Φ 0 (un )un − ν(µ − ε)Φ (un ) i =1

≥

N X

i=1


ν(µ − ε) − λi pi,+ Ji (un ) − C2 − Cε

i=1

≥ eJ (un ) − C2 − Cε ≥ C3 kun kd − C4 . This shows that {kun k} is bounded because d > 1. By Corollary 4.1, E satisfies condition (PS)c for any c 6= 0.



Proposition 4.4. Under the hypotheses of Proposition 4.3, for any w ∈ X \ {0}, E (sw) → −∞ as s → +∞. Proof. Setting τ = min{νµ−λi pi,+ : i = 1, 2, . . . , N }, then by (H3 ), τ > 0. Let w ∈ X \{0} be given. Then, by Proposition 2.6, |Di w|pi (·) 6= 0 for all i = 1, 2, . . . , N. (a3 ) implies that b ai (t ) ≤ Ci t λi for i = 1, 2, . . . , N and sufficiently large t > 0. λi pi,+ νµ−τ From this it follows that Ji (sw) ≤ d1 s ≤ d1 s for s large enough, where d1 is a positive constant depending on w . Thus J (sw) ≤ Nd1 sνµ−τ for s large enough. It follows from (f0 )0 and (f3 ) that that F (x, t ) ≥ C2 |t |µ − C1 for x ∈ Ω and R t ∈ R, and consequently, Ω F (x, sw)dx ≥ d2 sµ for s large enough, where d2 is a positive constant depending on w . (b3 ) implies that b b(t ) ≥ C3 t ν for t large enough, and thus Φ (sw) = b b Ω F (x, sw)dx ≥ d3 sνµ for s large enough, where d3 is a positive constant depending on w . Hence for s large enough, E (sw) ≤ d1 sνµ−τ − d3 sνµ , and consequently, E (sw) → −∞ as s → +∞. 

R




Proposition 4.5. Let (P0 ), (f0 )0 , (a0 ), (b0 ) and the following conditions be satisfied:

(a4 ) For each i = 1, 2, . . . , N, there exists σi > 0 such that limt →0+ abti σ(ti ) > 0. b (b4 ) There exists β3 > 0 such that limt →0 |bt |(βt )3 < +∞.

(f4 ) There exists r ∈ C+0 (Ω ) such that r (x) < p∗ (x) for x ∈ Ω and limt →0 |F|t(|xr (,xt ))| < +∞ uniformly in x ∈ Ω . (H4 ) σi pi,+ < β3 r− . Then there exist positive constants ρ and δ such that E (u) ≥ δ for kuk = ρ . Proof. Setting ε = min{β3 r− −σi pi,+ : i = 1, 2, . . . , N }, then by (H4 ), ε > 0. It follows from (a4 ) that for each i = 1, 2, . . . , N and sufficiently small |Di u|pi (·) , Ji (u) ≥ C2 (|Di u|pi (·) )σi pi,+ ≥ C2 (|Di u|pi (·) )β3 r− −ε , and consequently J (u) ≥ C2 kukβ3 r− −ε for sufficiently small kuk. It follows from (f4 ) and (f0 )0 that

|F (x, t )| ≤ c2 |t |r (x) + c3 |t |q(x)

for x ∈ Ω and t ∈ R.

Without loss of generality, we may assume that q(x) ≥ r (x) for x ∈ Ω . Then for sufficiently small kuk, we have that R | Ω F (x, u)dx| ≤ c4 kukr− . It follows from (b4 ) that, for sufficiently small kuk,

Z Z β3  b |Φ (u)| = b F (x, u)dx ≤ c5 F (x, u)dx ≤ c6 kukβ3 r− . Ω

Ω

Thus, for sufficiently small kuk, E (u) ≥ C2 kukβ3 r− −ε − c6 kukβ3 r− . From this we can see that the assertion of Proposition 4.5 is true.  By the famous Mountain Pass lemma (see e.g. [37]), from Propositions 4.3–4.5 we have the following: Theorem 4.5. Let all hypotheses of Propositions 4.3–4.5 hold. Then (1.6) has a nontrivial solution with positive energy. By the symmetric Mountain Pass lemma (see [37]), we have the following: Theorem 4.6. Under the hypotheses of Theorem 4.5, if, in addition, f satisfies (f2 ), then (1.6) has a sequence of solutions {±un } such that E (±un ) → +∞ as n → ∞. Like in the proof of Theorem 4.4, we can prove the following: Theorem 4.7. Under the hypotheses of Theorem 4.5, if, in addition, (b+ ) and (f+ ) hold, then (1.6) has at least one nontrivial nonnegative solution with positive energy. Example 4.2. Let ai (t ) = t γi −1 for t > 0 and i = 1, . . . , N, where γi > 0 and γi pi,− > 1; b(t ) = |t |β1 −2 t for t ∈ R, where β1 ≥ 1; f (x, t ) = |t |q(x)−2 t for t ∈ R, where q ∈ C+0 (Ω ) with q(x) < p∗ (x) for x ∈ Ω . Suppose that γi pi,+ < β1 q+ for i = 1, . . . , N. Then all hypotheses of Theorems 4.5–4.7 are satisfied.

X. Fan / Nonlinear Analysis 73 (2010) 3364–3375

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References [1] G. Kirchhoff, Mechanik, Teubner, Leipzig, 1883. [2] J.L. Lions, On some questions in boundary value problems of mathematical physics, in: Proceedings of International Symposium on Continuum Mechanics and Partial Differential Equations, Rio de Janeiro, 1977, in: de la Penha, Medeiros (Eds.), Math. Stud., vol. 30, North-Holland, 1978, pp. 284–346. [3] P. D’Ancona, S. Spagnolo, Global solvability for the degenerate Kirchhoff equation with real analytic date, Invent. Math. 108 (1992) 447–462. [4] A. Arosio, S. Panizzi, On the well-posedness of the Kirchhoff string, Trans. Amer. Math. Soc. 348 (1996) 305–330. [5] M. Chipot, B. Lovat, Some remarks on non local elliptic and parabolic problems, Nonlinear Anal. 30 (1997) 4619–4627. [6] M.M. Cavalcanti, V.N. Domingos Cavalcanti, J.A. Soriano, Global existence and uniform decay rates for the Kirchhoff–Carrier equation with nonlinear dissipation, Adv. Differential Equations 6 (2001) 701–730. [7] M. Dreher, The Kirchhoff equation for the p-Laplacian, Rend. Semin. Mat. Univ. Politec. Torino 64 (2006) 217–238. [8] M. Dreher, The ware equation for the p-Laplacian, Hokkaido Math. J. 36 (2007) 21–52. [9] D. Andrade, T.F. Ma, An operator equation suggested by a class of stationary problems, Commun. Appl. Nonlinear Anal. 4 (1997) 65–71. [10] C.O. Alves, F.J.S.A. Corrêa, On existence of solutions for a class of problem involving a nonlinear operator, Commun. Appl. Nonlinear Anal. 8 (2001) 43–56. [11] M. Chipot, V. Valente, G. Vergara Caffarelli, Remarks on a nonlocal problem involving the Dirichlet energy, Rendiconti Sem. Mat. Padova 110 (2003) 199–220. [12] F.J.S.A. Corrêa, On positive solutions of nonlocal and nonvariational elliptic problems, Nonlinear Anal. 59 (2004) 1147–1155. [13] F.J.S.A. Corrêa, S.D.B. Menezes, J. Ferreira, On a class of problems involving a nonlocal operator, Appl. Math. Comput. 147 (2004) 475–489. [14] T.F. Ma, Remarks on an elliptic equation of Kirchhoff type, Nonlinear Anal. 63 (2005) e1967–e1977. [15] C.O. Alves, F.J.S.A. Corrêa, T.F. Ma, Positive solutions for a quasilinear elliptic equation of Kirchhoff type, Comput. Math. Appl. 49 (2005) 85–93. [16] K. Perera, Z.T. Zhang, Nontrivial solutions of Kirchhoff-type problems via the Yang index, J. Differential Equations 221 (2006) 246–255. [17] F.J.S.A. Corrêa, G.M. Figueiredo, On an elliptic equation of p-Kirchhoff type via variational methods, Bull. Austral. Math. Soc. 74 (2006) 263–277. [18] X.M. He, W.M. Zou, Infinitely many positive solutions for Kirchhoff-type problems, Nonlinear Anal. 70 (2009) 1407–1414. [19] B. Ricceri, On an elliptic Kirchhoff-type problem depending on two parameters, J. Global Optim. 46 (2010) 543–549. [20] P. Harjulehto, P. Hästö, U.V. Lê, M. Nuortio, Overview of differential equations with non-standard growth, Nonlinear Anal. 72 (2010) 4551–4574. [21] S. Antontsev, S. Shmarev, Elliptic equations with anisotropic nonlinearity and nonstandard conditions, in: M. Chipot, P. Quittner (Eds.), Handbook of Differential Equations, Stationary Partial Differential Equations, vol. 3, Elsevier B.V., North Holland, Amsterdam, 2006, pp. 1–100. [22] M. Mihăilescu, P. Pucci, V. Rădulescu, Nonhomogeneous boundary value problems in anisotropic Sobolev spaces, C. R. Acad. Sci. Paris, Ser. I 345 (2007) 561–566. [23] M. Mihăilescu, P. Pucci, V. Rădulescu, Eigenvalue problems for anisotropic quasilinear elliptic equations with variable exponent, J. Math. Anal. Appl. 340 (2008) 687–698. [24] M. Mihăilescu, G. Moroşanu, Existence and multiplicity of solutions for an anisotropic elliptic problem involving variable exponent growth conditions, Appl. Anal. 89 (2010) 257–271. [25] C. Ji, An eigenvalue of an anisotropic quasilinear elliptic equation with variable exponent and Neumann boundary condition, Nonlinear Anal. 71 (2009) 4507–4514. − → [26] X.L. Fan, Anisotropic variable exponent Sobolev spaces and p (x)-Laplace equations, Complex Var. Elliptic Equ., in press (doi:10.1080/17476931003728412). First Published on 11 June 2010. [27] G. Autuori, P. Pucci, M.C. Salvatori, Asymptotic stability for anisotropic Kirchhoff systems, J. Math. Anal. Appl. 352 (2009) 149–165. [28] G.W. Dai, R.F. Hao, Existence of solutions for a p(x)-Kirchhoff-type equation, J. Math. Anal. Appl. 359 (2009) 275–284. [29] G.W. Dai, D.C. Liu, Infinitely many positive solutions for a p(x)-Kirchhoff-type equation, J. Math. Anal. Appl. 359 (2009) 704–710. [30] X.L. Fan, On nonlocal p(x)-Laplacian Dirichlet problems, Nonlinear Anal. 72 (2010) 3314–3323. [31] O. Ková˘cik, J. Rákosnik, On spaces Lp(x) and W k,p(x) , Czechoslovak Math. J. 41 (116) (1991) 592–618. [32] P. Harjulehto, P. Hästö, An overview of variable exponent Lebesgue and Sobolev spaces, in: D. Herron (Ed.), Future Trends in Geometric Function Theory, RNC Workshop, Jyväskylä, 2003, pp. 85–93. [33] F.E. Browder, Nonlinear Operators and Nonlinear Equations of Evolution in Banach Spaces, in: Proceedings of Symposia in Pure Mathematics, vol. 18, Part 2, American Mathematical Society, Providence, Rhode Island, 1976. [34] I.V. Skrypnik, Methods for Analysis of Nonlinear Elliptic Boundary Value Problems, ‘‘Nauka’’, Moscow, 1990, in: English transl. Translation of Mathematical Monographs, vol. 139, American Mathematical Society, Providence, Rhode Island, 1994 (in Russian). [35] E. Zeidler, Nonlinear Functional Analysis and its Applications, II/B: Nonlinear Monotone Operators, Springer-Verlag, New York, 1990. [36] I. Fragalà, F. Gazzola, B. Kawohl, Existence and nonexistence results for anisotropic quasilinear elliptic equations, Ann. Inst. H. Poincaré Anal. Non Linéaire 21 (2004) 715–734. [37] M. Struwe, Variational Mathods, Applications to Nonlinear Partial Differential Equations and Hamiltonian Systems, second ed., Springer-Verlag, Berlin, 1996.