Nonlinear Analysis 72 (2010) 302–308

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Nonlinear Analysis

journal homepage: www.elsevier.com/locate/na

On a p-Kirchhoff equation via Fountain Theorem and DualFountain TheoremI

Duchao LiuSchool of Mathematics and Statistics, Lanzhou University, Lanzhou 730000, PR China

a r t i c l e i n f o

Article history:Received 4 December 2008Accepted 12 June 2009

MSC:35B3835D0535J20

Keywords:Critical pointsp-Kirchhoff problemFountain TheoremDual Fountain Theorem

a b s t r a c t

In this paper, we show the existence of infinite solutions to the Kirchhoff type quasilinearelliptic equation[

M(∫

Ω

(| ∇u |p+λ(x) | u |p)dx)]p−1

(−∆pu+ λ(x) | u |p−2 u) = f (x, u)

in a smooth bounded domainΩ ⊂ RN with nonlinear boundary condition | ∇u |p−2 ∂u∂ν=

η | u |p−2 on ∂Ω . The method we used here is based on ‘‘Fountain Theorem’’ and ‘‘DualFountain Theorem’’.

© 2009 Elsevier Ltd. All rights reserved.

1. Introduction

In this paper we deal with the nonlocal elliptic problem (P) of the p-Kirchhoff type given by[M(∫

Ω

(|∇u|p + λ(x)|u|p)dx)]p−1

(−∆pu+ λ(x)|u|p−2u) = f (x, u), x ∈ Ω, (1)

|∇u|p−2∂u∂ν= η|u|p−2, x ∈ ∂Ω. (2)

whereΩ ⊂ RN is a bounded domain, ∂∂νis the outer unit normal derivative,∆pu = div(|∇u|p−2∇u) is the p-Laplacian with

1 < p < N .

λ(x) ∈ L∞(Ω) satisfying essinfx∈Ωλ(x) > 0. (3)

The functionM : R+ → R+ is a continuous nondecreasing function and there is a constantm0 > 0 such thatM(t) ≥ m0,for all t ≥ 0.The perturbation f (x, t) is a Caratheodory function with subcritical growth with respect to t , that is, |f (x, t)| ≤ C(1 +

|t|q−1) holds true for all x ∈ Ω and t ∈ Rwith 1 ≤ q < p∗ = NpN−p , where p

∗ is a critical exponent according to the SobolevembeddingW 1,p(Ω) → Lp

∗

(Ω), η is a real parameter.Problem (P) is called nonlocal because of the presence of the termM , which implies that the equation in (P) is no longer

pointwise identities. This provokes some mathematical difficulties which makes the study of such a problem particulary

I Research supported by the National Natural Science Foundation of China (10671084).E-mail address: liuduchao@yahoo.cn.

0362-546X/$ – see front matter© 2009 Elsevier Ltd. All rights reserved.doi:10.1016/j.na.2009.06.052

D. Liu / Nonlinear Analysis 72 (2010) 302–308 303

interesting. This problem has a physical motivation. Similar operatorM(∫Ω|∇u|2dx)∆u appears in the Kirchhoff equation,

which arises in nonlinear vibrations, namelyutt −M(‖u‖)∆u = f (x, u), inΩ × (0, T ),u = 0 on ∂Ω × (0, T ),u(x, 0) = u0(x), ut(x, 0) = u1(x).

p-Kirchhoff problem began to attract the attention of several researchers mainly after the work of Lions [1] where afunctional analysis approach was proposed to attack it.The reader may consult [2–4,6–11] and the references therein, for similar problem in several cases.Throughout this paper, byweak solutions of (P)weunderstand critical points of the associated energy functionalϕ acting

on the Sobolev spaceW 1,p(Ω):

ϕ(u) =1pM(∫

Ω

(|∇u|p + λ(x)|u|p)dx)−

∫Ω

F(x, u)dx−η

p

∫∂Ω

|u|pdS, (4)

where M(t) =∫ t0 [M(s)]

p−1ds, F(x, t) =∫ t0 f (x, s)ds, and dS is the surface measure on the boundary. Obviously ϕ(u) ∈

C1(W 1,p(Ω),R) and

〈ϕ′(u), v〉 =[M(∫

Ω

(|∇u|p + λ(x)|u|p)dx)]p−1 ∫

Ω

[|∇u|p−2∇u∇v

+ λ(x)|u|p−2uv]dx−∫Ω

f (x, u)v − η∫∂Ω

|u|p−2uvdS

for all u, v ∈ W 1,p(Ω), where

W 1,p(Ω) =u ∈ Lp(Ω) :

∫Ω

|∇u|pdx <∞

is a Banach space with the norm

‖u‖1,p :=(∫

Ω

[|∇u|p + λ(x)|u|p]dx) 1p

for u ∈ W 1,p(Ω).

Our main results are as following.

Theorem 1.1. Suppose that M : R+ → R+ is a continuous nondecreasing function satisfies the following conditions

(m0) there exists a constant m0 > 0 such that M(t) ≥ m0 for all t ≥ 0;(m1) there exists a constant m1 > 0 such that M(t) ≤ m1 for all t > 0 and p∗ > p(

m1m0)p−1.

Caratheodory function f satisfies

(f1) For some p < q < p∗, there exists a constant C > 0 such that

|f (x, t)| ≤ C(1+ |t|q−1) for all x ∈ Ω, t ∈ R;

(f2) There exist p∗ > α > p(m1m0 )p−1 and R > 0 such that

|t| ≥ R⇒ 0 < αF(x, t) ≤ tf (x, t) for all x ∈ Ω;

(f3) f (x, t) is odd with respect to t, that is,

f (x,−t) = −f (x, t) for all x ∈ Ω, t ∈ R.

Then there exists a constant Λ > 0 such that for any η < Λ, the problem (P) has a sequence of solutions uk ∈ W 1,p(Ω)such that ϕ(uk)→∞, as k→∞.

For a special f , we obtain a sequence of weak solutions with negative energy by dual fountain theorem.

Theorem 1.2. M satisfies (m0) (m1) in Theorem 1.1, let f (x, t) = µ|t|r−2t+λ|t|s−2t, where 1 < r < p < s < p∗,. Then thereexists a constant Λ > 0 such that for any η < Λ,

(1) For every λ > 0, µ ∈ R, problem (P) has a sequence of solutions uk such that ϕ(uk)→∞ as k→∞;(2) For every µ > 0, λ ∈ R, problem (P) has a sequence of solutions vk such that ϕ(vk) < 0, ϕ(vk)→ 0 as k→∞.

304 D. Liu / Nonlinear Analysis 72 (2010) 302–308

2. Preliminaries

2.1. Definitions

In this paper we use the following notations: X denotes Banach space with the norm ‖ · ‖X ; X∗ denotes the conjugatespace for X; Lp(Ω) denotes Lebesgue space with the norm | · |p; W 1,p(Ω) denotes Sobolec space with the norm ‖ · ‖1,p;〈·, ·〉 is the dual pairing of the space X∗ and X; by→ (resp.) we mean strong (resp. weak) convergence. |Ω| denotes theLebesgue measure of the setΩ ⊂ RN ; C1, C2, . . . denote (possibly different) positive constants.First, let us recall some definitions, Fountain Theorem and Dual Fountain Theorem.

Definition 2.1 (See [5]). The action of a topological group G on a normed space X is a continuous map G×X → X : [g, t] →gt such that 1 · t = t , (gh)t = g(ht), t 7→ gt is linear. The action is isometric if ‖gt‖X = ‖t‖X . A set A ⊂ X is invariant ifgA = A for every g ∈ G. A function ϕ : X → R is invariant if ϕ g = ϕ for every g ∈ G. A map f : X → X is equivariant ifg f = f g for every g ∈ G.

Definition 2.2 (See [5]). Assume that the compact group G acts diagonally on V k:

g(v1, v2, . . . , vk) := (gv1, gv2, . . . , gvk),

where V is a finite dimensional space. The action of G is admissible if every continuous equivariant map ∂U → V k−1, whereU is an open bounded invariant neighborhood of 0 in V k, k ≥ 2, has a zero.

Remark 2.1. The Borsuk–Ulam theorem says that the antipodal action of G := Z2 on V := R is admissible.

We say that (X,G) satisfies (A1) if the compact group G acts isometrically on the Banach space X =⊕j∈N Xj, the spaces

Xj are invariant under G and there exists a finite dimensional space V such that, for every j ∈ N , Xj ' V and the action of Gon V is admissible.Next we introduce the definitions of the (PS)c and (PS)∗c condition which play important roles in our paper.

Definition 2.3. Let ϕ ∈ C1(X,R) and c ∈ R. The function ϕ satisfies the (PS)c condition if any sequence un ∈ X such that

ϕ(un)→ c, ϕ′(un)→ 0 in X∗ as n→∞

has a convergent subsequence.

Let X be a reflexive and separable Banach space, then there are ej ∈ X and e∗j ∈ X∗ such that

X = spanej|j = 1, 2, . . ., X∗ = spane∗j |j = 1, 2, . . .,

and

〈e∗i , ej〉 =1, i = j,0, i 6= j.

For convenience, we write Xj := spanej, Yk :=⊕kj=1 Xj, Zk =

⊕∞

j=k Xj. And let Bk := u ∈ Yk|‖u‖X ≤ ρk, Nk := u ∈Zk|‖u‖X = γk, where ρk > γk > 0.

Definition 2.4. Let ϕ ∈ C1(X, R) and c ∈ R. The function ϕ satisfies (PS)∗c condition (with respect to (Yn)) if any sequenceunj ⊂ X such that

unj ∈ Ynj , ϕ(unj)→ c, ϕ |′Ynj(unj)→ 0 in X∗ as nj →∞

has a convergent subsequence.

2.2. Preliminary theorems

Theorem 2.1 (Fountain Theorem, Bartsch, 1993 [5]). Let (X,G) satisfies (A1) and ϕ ∈ C1(X,R) be an invariant functional.Suppose that, for every k ∈ N, there exists ρk > γk > 0 such that(A2) ak := maxu∈Yk,‖u‖X=ρk ϕ(u) ≤ 0;(A3) bk := infu∈Zk,‖u‖X=γk ϕ(u)→∞;(A4) ϕ satisfies the (PS)c condition for every c > 0.

Then ϕ has an unbounded sequence of critical values which have the form

ck := infr∈Γkmaxu∈Bk

ϕ(r(u)),

where Γk := r ∈ C(Bk, X)|r equivariant and r|∂Bk = id.

D. Liu / Nonlinear Analysis 72 (2010) 302–308 305

Theorem 2.2 (Dual Fountain Theorem, Bartsch andWillem, 1995 [5]). Let (X,G) satisfies (A1) and ϕ ∈ C1(X,R) be an invariantfunctional. Suppose that, for every k ≥ k0, there exists ρk > γk > 0 such that

(B1) ak := infu∈Zk,‖u‖X=ρk ϕ(u) ≥ 0;(B2) bk := maxu∈Yk,‖u‖X=γk ϕ(u) < 0;(B3) dk := infu∈Zk,‖u‖X≤ρk ϕ(u)→ 0 as k→∞;(B4) ϕ satisfies the (PS)∗c condition for every c ∈ [dk0 , 0).

Then ϕ has a sequence of negative critical values converging to 0.

3. The proof of Theorems 1.1 and 1.2

First, we give several lemmas.

Lemma 3.1. Under the hypotheses of Theorem 1.1, ϕ satisfies the (PS)c condition.

Proof. Suppose that un ⊂ W 1,p(Ω), for every c > 0,

ϕ(un)→ c, ϕ′(un)→ 0 in (W 1,p(Ω))∗ as n→∞.

After integrating, we obtain from the assumption (f2) that there exists C1 such that

C1(|u|α − 1) ≤ F(x, u) for all x ∈ Ω, u ∈ R.

Then for n sufficiently large, let β ∈ ( 1α, 1p (

m0m1)p−1); we have

c + 1+ ‖un‖1,p ≥ ϕ(un)− β〈ϕ′(un), un〉

=1pM(‖un‖

p1,p)− β(M(‖un‖

p1,p))

p−1‖un‖

p1,p +

∫Ω

(βf (x, un)un − F(x, un))dx

− η

(1p− β

)∫∂Ω

|un|pdS

≥

(mp−10p− βmp−11

)(M(‖un‖

p1,p))

p−1‖un‖

p1,p + (αβ − 1)

∫Ω

F(x, un)dx

− ηK(1p− β

)‖un‖

p1,p

≥

[(1p

(m0m1

)p−1− β

)mp−11 − ηK

(1p− β

)]‖un‖

p1,p + C1(αβ − 1)|un|

αα − C3.

ChoseΛ > 0 such that if η < Λwe can have ( 1p (m0m1)p−1 − β)mp−11 − ηK(

1p − β) > 0; noticing that αβ − 1 > 0, we obtain

the boundedness of un inW 1,p(Ω).Since un is bounded, by the self-reflexive ofW 1,p(Ω), there exists a subsequence of un (whichwe also denote by un)

and u ∈ W 1,p(Ω), such that un u inW 1,p(Ω) and ‖un‖p → t0 as n→∞.From (f1), Hölder inequality and compact Sobolev embedding, we see that∫

Ω

f (x, un)(un − u)→ 0,∫∂Ω

|un|p−2un(un − u)→ 0.

Let us now consider the sequence

Pn = ϕ′(un)un +∫Ω

f (x, un)undx+ η∫∂Ω

|un|pdS − ϕ′(un)u−∫Ω

f (x, un)udx− η∫∂Ω

|un|p−2unudS.

Then we have Pn → 0 since

|ϕ′(un)(un − u)| ≤ ‖ϕ′(un)‖(W1,p(Ω))∗‖un − u‖1,p → 0.

We can see that

Pn = [M(‖un‖p)]p−1‖un‖p − [M(‖un‖p)]p−1(∫

Ω

|∇un|p−2∇un∇udx+ λ(x)∫Ω

|un|p−2unudx). (5)

306 D. Liu / Nonlinear Analysis 72 (2010) 302–308

Moreover, from ‖un‖p → t0 and the weak convergence we have

on(1) = [M(‖un‖p)]p−1‖u‖p − [M(‖un‖p)]p−1(∫

Ω

|∇u|p−2∇u∇undx+ λ(x)∫Ω

|u|p−2uundx). (6)

Combine (5) and (6) we have

on(1)+ Pn = [M(‖un‖p)]p−1∫Ω

〈|∇un|p−2∇un − |∇u|p−2∇u,∇un −∇u〉n

+ λ(x)〈|un|p−2un − |u|p−2u, un − u〉1dx,

where 〈·, ·〉n denotes the scalar product in Rn. Using the standard inequality in Rn given by

〈|x|p−2x− |y|p−2y, x− y〉n ≥

Cp|x− y|p, p ≥ 2,

Cp|x− y|2

(|x| + |y|)2−p, p < 2,

we have

on(1)+ Pn ≥ mp−10 C‖un − u‖1,p

Thus un → u inW 1,p(Ω) as n→∞.

In order to prove Theorem 1.1 we also need the following lemma.

Lemma 3.2. If 1 ≤ q < p∗ = NpN−p , then we have

βk := supu∈Zk,‖u‖1,p=1

|u|q → 0 as k→∞.

Proof. Obviously 0 < βk+1 ≤ βk, so there exists β ≥ 0 such that βk → β as k→∞. Wewill prove β = 0. By the definitionof βk, for every k ≥ 0 there exists uk ∈ Zk such that ‖uk‖1,p = 1, 0 ≤ β − |uk|q < 1

k . Then there exists a subsequence of uk(which still denote by uk) such that uk u inW 1,p(Ω), and

〈e∗j , u〉 = limk→∞〈e∗j , uk〉 = 0, j = 1, 2, . . .

which implies that u = 0, and uk 0 in W 1,p(Ω). Since the Sobolev embedding W 1,p(Ω) → Lq(Ω) is compact thenuk → 0 in Lq(Ω). Thus we have proved that β = 0.

We now give the proof of Theorem 1.1.

Proof of Theorem 1.1. Assumption (f1) of Theorem 1.1 and Lemma 3.1 imply that ϕ is continuously differentiable onW 1,p(Ω) and satisfies the (PS)c condition for every c > 0. We obtain from the assumption (f2) and (m1) of Theorem 1.1that

ϕ(u) =1pM(‖u‖p1,p)−

∫Ω

F(x, u)dx−η

p|u|pLp(∂Ω)

≤mp−11p‖u‖p1,p − C1|u|

αα − C1|Ω| −

η

p|u|pLp(∂Ω).

Since on the finite dimensional space Yk all norms are equivalent, α > p(m1m0)p−1 > p implies that (A2) is satisfied for ρk > 0

large enough.After integrating, we obtain from the assumption of (f1) the existence of C2 > 0 such that

F(x, u) ≤ C2(1+ |u|q) for all x ∈ Ω, u ∈ R.

Let us define

βk := supu∈Zk,‖u‖1,p=1

|u|q.

ChooseΛ > 0 and let η < Λ; then from the trace embedding inequality we have

η

p|u|pLp(∂Ω) ≤

q− p2pq

mp−10 ‖u‖p1,p.

D. Liu / Nonlinear Analysis 72 (2010) 302–308 307

Hence on Zk,

ϕ(u) ≥M(‖u‖p1,p)

p− C2|u|qq − C2|Ω| −

η

p|u|pLp(∂Ω)

≥mp−10 ‖u‖

p1,p

p− C2β

qk‖u‖

q1,p − C2|Ω| −m

p−10q− p2pq‖u‖p1,p

= mp−10p+ q2pq‖u‖p1,p − C2β

qk‖u‖

q1,p − C2|Ω|.

Choosing γk := (m1−p0 C2qβ

qk )

1p−q , we obtain, if u ∈ Zk and ‖u‖1,p = γk, then

ϕ(u) ≥q− p2pq

mq(p−1)q−p0 (C2qβ

qk )

pp−q − C2|Ω|.

Since βk → 0 as k → ∞ from Lemma 3.2, (A3) is proved. It suffices then to use the fountain theorem with the antipodalaction of Z2.

We now give the proof of Theorem 1.2.

Proof of Theorem 1.2. The first conclusion of Theorem 1.2 is just a corollary of Theorem 1.1; we shall prove Theorem 1.2by using Theorem 2.2, so we need to verify the conditions (B1)–(B4). Now we assume that µ > 0.In order to verify (B1), we define

βk := supu∈Zk,‖u‖1,p=1

|u|r .

Observe that

|u|r ≤ βk‖u‖1,p (7)

for any u ∈ Zk. Note that s < p∗, there exists a constant C > 0 such that

|u|s ≤ C‖u‖1,p. (8)

On the other hand, we have the trace embedding inequality,

|u|pLp(∂Ω) ≤ K‖u‖p1,p.

Then we may takeΛ∗ = 14K such that for all η < Λ∗,

η

p|u|pLp(∂Ω) ≤

14p‖u‖p1,p. (9)

By (6)–(8) we obtain

ϕ(u) =1pM(‖u‖p1,p)−

µ

r|u|rr −

λ

s|u|pLp(∂Ω)

≥3mp−104p‖u‖p1,p −

µ

rβrk‖u‖

r1,p −

|λ|

sC s‖u‖s1,p.

Note that p < s; we may choose R > 0 sufficiently small that

mp−104p‖u‖p1,p −

|λ|

sC s‖u‖s1,p ≥ 0 (10)

holds true for any u ∈ W 1,p(Ω)with ‖u‖1,p ≤ R. So we have

ϕ(u) ≥mp−102p‖u‖p1,p −

µ

rβrk‖u‖

r1,p for any u ∈ Zk, ‖u‖1,p ≤ R. (11)

We choose ρk := (2m1−p0 pµβrk

r )1p−r ; by Lemma 3.2, for βk → 0, k → ∞, it follows that ρk → 0, k → ∞, so there exists k0

such that ρk ≤ Rwhen k ≥ k0. Thus, for k ≥ k0, u ∈ Zk and ‖u‖1,p = ρk, we have ϕ(u) ≥ 0 and relation (B1) is proved.Since on the finite dimensional space Yk all norms are equivalent, then when µ > 0 relation (B2) is satisfied for every

rk > 0 small enough since r < p < s.We obtain from (10), for k ≥ k0, u ∈ Zk, ‖u‖1,p ≤ ρk, ϕ(u) ≥ −

µ

r βrkρrk ; since βk → 0, and ρk → 0, k→∞, relation (B3)

is also satisfied.

308 D. Liu / Nonlinear Analysis 72 (2010) 302–308

Finally we prove the (PS)∗c condition. Consider a sequence unj ⊂ W1,p(Ω) such that nj → ∞, unj ∈ Ynj , ϕ(unj) → c ,

ϕ |′Ynj→ 0 in (W 1,p(Ω))∗. For nj big enough, let β < 1

p (m0m1)p−1 ≤ 1

p we have

c + 1+ ‖unj‖1,p ≥ ϕ(unj)− β〈ϕ′(unj), unj〉

=1pM(‖unj‖

p1,p)− β(M(‖unj‖

p1,p))

p−1‖unj‖

p1,p − µ

(1r− β

)|unj |

rr − η

(1p− β

)|unj |

pLp(∂Ω)

≥

(mp−10p− βmp−11

)‖unj‖

p1,p − µτ

rnj

(1r− β

)‖unj‖

r1,p − ηK

(1p− β

)‖unj‖

p1,p

≥

(mp−10p− βmp−11 − ηK

(1p− β

))(1p−1s

)‖unj‖

p1,p − µτ

rnj

(1r−1s

)‖unj‖

r1,p

where τk := supu∈Yk,‖u‖1,p=1 |u|r . Choose 0 < Λ < Λ∗ and if η < Λ such that mp−10p − βm

p−11 − ηK( 1p − β) > 0, we can

obtain the boundedness of unj inW1,p(Ω). Going if necessary to a subsequence, we can assume that unj u inW 1,p(Ω).

It is easy to conclude as in Lemma 3.1 that unj → u inW 1,p(Ω) and ϕ′(u) = 0. This ends the proof.

Remark. If we substitute the condition (m1)′: for all t ≥ 0 the following inequality holds: M(t) ≥ (M(t))p−1t , for (m1),and don’t require the nondecreasing condition ofM(t), we can get the same results of Theorems 1.1 and 1.2. But condition(m1)′ is far away from the physical sense of the original Kirchhoff equation.

Acknowledgment

Thanks are given to Professor X.L. Fan for his valuable guidance.

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