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Existence results for a class of nonlocal problems involving p-Laplacian

Yang Yang1* and Jihui Zhang2

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1 School of Science, Jiangnan University, Wuxi, 214122, People's Republic of China

2 Institute of Mathematics, School of Mathematics Science, Nanjing Normal University, Nanjing, 210097, People's Republic of China

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Boundary Value Problems 2011, 2011:32  doi:10.1186/1687-2770-2011-32


The electronic version of this article is the complete one and can be found online at: http://www.boundaryvalueproblems.com/content/2011/1/32


Received:7 January 2011
Accepted:11 October 2011
Published:11 October 2011

© 2011 Yang and Zhang; licensee Springer.

This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

This paper is concerned with the existence of solutions to a class of p-Kirchhoff type equations with Neumann boundary data as follows:

- M Ω u p d x p - 1 Δ p u = f ( x , u ) , in Ω ; u υ = 0 , on Ω .

By means of a direct variational approach, we establish conditions ensuring the existence and multiplicity of solutions for the problem.

Keywords:
Nonlocal problems; Neumann problem; p-Kirchhoff's equation

1. Introduction

In this paper, we deal with the nonlocal p-Kirchhoff type of problem given by:

- M Ω u p d x p - 1 Δ p u = f ( x , u ) , in Ω ; u υ = 0 , on Ω (1.1)

where Ω is a smooth bounded domain in RN, 1 < p < N, ν is the unit exterior vector on ∂Ω, Δp is the p-Laplacian operator, that is, Δpu = div(|∇u|p−2u), the function M : R+ R+ is a continuous function and there is a constant m0 > 0, such that

( M 0 ) M ( t ) m 0 for all t 0 .

f ( x , t ) : Ω ¯ × R R is a continuous function and satisfies the subcritical condition:

f ( x , t ) C ( t q - 1 + 1 ) , for some p < q < p * = N p N - p , N 3 ; + , N = 1 , 2 . (1.2)

where C denotes a generic positive constant.

Problem (1.1) is called nonlocal because of the presence of the term M, which implies that the equation is no longer a pointwise identity. This provokes some mathematical difficulties which makes the study of such a problem particulary interesting. This problem has a physical motivation when p = 2. In this case, the operator M(∫Ω|∇u|2dxu appears in the Kirchhoff equation which arises in nonlinear vibrations, namely

u t t - M ( Ω u 2 d x ) Δ u = f ( x , u ) , in Ω × ( 0 , T ) ; u = 0 , on Ω × ( 0 , T ) ; u ( x , 0 ) = u 0 ( x ) , u t ( x , 0 ) = u 1 ( x ) .

P-Kirchhoff problem began to attract the attention of several researchers mainly after the work of Lions [1], where a functional analysis approach was proposed to attack it. The reader may consult [2-8] and the references therein for similar problem in several cases.

This work is organized as follows, in Section 2, we present some preliminary results and in Section 3 we prove the main results.

2. Preliminaries

By a weak solution of (1.1), then we say that a function u ε W1,p(Ω) such that

M Ω u p d x p - 1 Ω u p - 2 u φ d x = Ω f ( x , u ) φ d x , for all φ W 1 , p ( Ω )

So we work essentially in the space W1,p(Ω) endowed with the norm

u = Ω ( u p + u p ) d x 1 p ,

and the space W1,p(Ω) may be split in the following way. Let Wc = 〈1〉, that is, the subspace of W1,p(Ω) spanned by the constant function 1, and W 0 = { z W 1 , p ( Ω ) , Ω z = 0 } , which is called the space of functions of W1,p(Ω) with null mean in Ω. Thus

W 1 , p ( Ω ) = W 0 W c .

As it is well known the Poincaré's inequality does not hold in the space W1,p(Ω). However, it is true in W0.

Lemma 2.1 [8] (Poincaré-Wirtinger's inequality) There exists a constant η > 0 such that Ω z p d x η Ω z p d x for all z W0.

Let us also recall the following useful notion from nonlinear operator theory. If X is a Banach space and A : X X* is an operator, we say that A is of type (S+), if for every sequence {xn}n≥1 X such that xn x weakly in X, and lim sup n A ( x n ) , x n - x 0 . we have that xn x in X.

Let us consider the map A : W1,p(Ω) → W1,p(Ω)* corresponding to −Δp with Neumann boundary data, defined by

A ( u ) , v = Ω u p - 2 u v d x , u , v W 1 , p ( Ω ) . (2.1)

We have the following result:

Lemma 2.2 [9,10]The map A : W1,p(Ω) → W1,p(Ω)* defined by (2.1) is continuous and of type (S+).

In the next section, we need the following definition and the lemmas.

Definition 2.1. Let E be a real Banach space, and D an open subset of E. Suppose that a functional J : D R is Fréchet differentiable on D. If x0 D and the Fréchet derivative J' (x0) = 0, then we call that x0 is a critical point of the functional J and c = J(x0) is a critical value of J.

Definition 2.2. For J C1(E, R), we say J satisfies the Palais-Smale condition (denoted by (PS)) if any sequence {un} ⊂ E for which J(un) is bounded and J'(un) → 0 as n → ∞ possesses a convergent subsequence.

Lemma 2.3 [11]Let X be a Banach space with a direct sum decomposition X = X1 X2, with k = dimX2 < ∞, let J be a C1 function on X, satisfying (PS) condition. Assume that, for some r > 0,

J ( u ) 0 f o r u X 1 , u r ; J ( u ) 0 f o r u X 2 , u r .

Assume also that J is bounded below and infX J < 0. Then J has at least two nonzero critical points.

Lemma 2.4 [12]Let X = X1 X2, where X is a real Banach space and X2 ≠ {0}, and is finite dimensional. Suppose J C1(X, R) satisfies (PS) and

(i) there is a constant α and a bounded neighborhood D of 0 in X2 such that J|∂D ≤ α and,

(ii) there is a constant β > α such that J X 1 β ,

then J possesses a critical value c ≥ β, moreover, c can be characterized as

c = inf h Γ max u D ¯ J ( h ( u ) ) .

where Γ = { h C ( D ¯ , X ) h = i d o n D } .

Definition 2.3. For J C1(E, R), we say J satisfies the Cerami condition (denoted by (C)) if any sequence {un} ⊂ E for which J(un) is bounded and (1 ||un||) J'(un)|| → 0 as n → ∞ possesses a convergent subsequence.

Remark 2.1 If J satisfies the (C) condition, Lemma 2.4 still holds.

In the present paper, we give an existence theorem and a multiplicity theorem for problem (1.1). Our main results are the following two theorems.

Theorem 2.1 If following hold:

(F0) 0 lim u 0 p F ( x , u ) u p < m 0 p - 1 η a . e . x Ω , where F ( x , u ) = 0 u f ( x , s ) d s , η appears in Lemma 2.1;

(F1) lim u p F ( x , u ) u p 0 a . e . x Ω ;

(F2) lim u Ω F ( x , u ) d x = - .

Then the problem (1.1) has least three distinct weak solutions in W1,p(Ω).

Theorem 2.2 If the following hold:

(M1) The function M that appears in the classical Kirchhoff equation satisfies M ^ ( t ) ( M ( t ) ) p - 1 t for all t ≥ 0, where M ^ ( t ) = 0 t [ M ( s ) ] p - 1 d s ;

(F3) f ( x , u ) u > 0 f o r a l l u 0 ;

(F4) lim u p F ( x , u ) u p = 0 a . e . x Ω ;

(F5) lim u ( f ( x , u ) u - p F ( x , u ) ) = - .

Then the problem (1.1) has at least one weak solution in W1,p(Ω).

Remark 2.2 We exhibit now two examples of nonlinearities that fulfill all of our hypotheses

f ( x , u ) = m 0 p - 1 2 η u p - 2 u - u q - 2 u ,

hypotheses (F0), (F1), (F2) and (1.2) are clearly satisfied.

f ( x , u ) = a r c t a n u + u 1 + u 2 ,

hypotheses (F3), (F4) and (F5) and (1.2) are clearly satisfied.

3. Proofs of the theorems

Let us start by considering the functional J : W1,p(Ω) → R given by

J ( u ) = 1 p M ^ Ω u p d x - Ω F ( x , u ) d x .

Proof of Theorem 2.1 By (F0), we know that f(x, 0) = 0, and hence u(x) = 0 is a solution of (1.1).

To complete the proof we prove the following lemmas.

Lemma 3.1 Any bounded (PS) sequence of J has a strongly convergent subsequence.

Proof: Let {un} be a bounded (PS) sequence of J. Passing to a subsequence if necessary, there exists u W1,p(Ω) such that un u. From the subcritical growth of f and the Sobolev embedding, we see that

Ω f ( x , u n ) ( u n - u ) d x 0 .

and since J'(un)(un u) → 0, we conclude that

M Ω u n p d x p - 1 Ω u n p - 2 u n ( u n - u ) d x 0 .

In view of condition (M0), we have

Ω u n p - 2 u n ( u n - u ) d x 0 .

Using Lemma 2.2, we have un u as n → ∞. □

Lemma 3.2 If condition (M0), (F1) and (F2) hold, then lim u J ( u ) = + .

Proof: If there are a sequence {un} and a constant C such that ||un|| → ∞ as n → ∞, and J(un) ≤ C (n = 1, 2 ···), let v n = u n u n , then there exist v0 W1,p(Ω) and a subsequence of {vn}, we still note by {vn}, such that vn v0 in W1,p(Ω) and vn v0 in Lp(Ω).

For any ε > 0, by (F1), there is a H > 0 such that F ( x , u ) ε p u p for all |u| H and a.e. x ∈ Ω, then there exists a constant C > 0 such that F ( x , u ) ε p u p + C for all u R, and a.e. x ∈ Ω, Consequently

C u n p J ( u n ) u n p = 1 u n p 1 p M ^ Ω u n p d x - Ω F ( x , u n ) d x 1 p m 0 p - 1 Ω v n p d x - ε p Ω v n p d x - C Ω u n p = 1 p m 0 p - 1 - 1 p m 0 p - 1 + ε p Ω v n p d x - C Ω u n p .

It implies Ω|v0|pdx ≥ 1. On the other hand, by the weak lower semi-continuity of the norm, one has

v 0 lim inf n v n = 1 .

Hence Ω v 0 p d x = 0 , so |v0(x)| = constant ≠ 0 a.e. x ∈ Ω. By (F2), lim u n Ω F ( x , u n ) d x - . Hence

C J ( u n ) = 1 p M ^ Ω u n p d x - Ω F ( x , u n ) d x - Ω F ( x , u n ) d x + a s n .

This is a contradiction. Hence J is coercive on W1,p(Ω), bounded from below, and satisfies the (PS) condition. □

By Lemma 3.1 and 3.2, we know that J is coercive on W1,p(Ω), bounded from below, and satisfies the (PS) condition. From condition (F0), we know, there exist r > 0, ε > 0 such that

0 F ( x , u ) m 0 p - 1 p η - ε u p , f o r u r .

If u Wc, for ||u|| ≤ ρ1, then |u| r, we have

J ( u ) = 1 p M ^ Ω u p d x - Ω F ( x , u ) d x = - Ω F ( x , u ) d x 0 .

If u W0, then from condition (F0) and (1.2), we have

F ( x , u ) m 0 p - 1 p η - ε u p + C u q , f o r u R , q ( p , p * ) .

Noting that

Ω u p d x η Ω u p d x , u W 0 ,

we can obtain

J ( u ) = 1 p M ^ Ω u p d x - Ω F ( x , u ) d x 1 p m 0 p - 1 Ω u p d x - m 0 p - 1 p η Ω u p d x + ε Ω u p d x - C Ω u q d x C ε u p - C C 1 u q .

Choose ||u|| = ρ2 small enough, such that J(u) ≥ 0 for ||u|| ≤ ρ2 and u W0.

Now choose ρ = min{ρ1, ρ2}, then, we have

J ( u ) 0 f o r u W c , u ρ ;

J ( u ) 0 f o r u W 0 , u ρ .

If inf{J(u), u W1,p(Ω)} = 0, then all u Wc with ||u|| ≤ ρ are minimum of J, which implies that J has infinite critical points. If inf{J(u), u W1,p(Ω)} < 0 then by Lemma 2.3, J has at least two nontrivial critical points. Hence problem (1.1) has at least two nontrivial solutions in W1,p(Ω), Therefore, problem (1.1) has at least three distinct solutions in W1,p(Ω). □

Proof of Theorem 2.2. We divide the proof into several lemmas.

Lemma 3.3 If condition (F3) and (F5) hold, then J W c is anticoercive. (i.e. we have that J(u) → -∞, as |u| → ∞, u R.)

Proof: By virtue of hypothesis (F5), for any given L > 0, we can find R1 = R1(L) > 0 such that

F ( x , u ) 1 p L + 1 p f ( x , u ) u , f o r a . e . x Ω , u > R 1 .

Thus, using hypothesis (F3), we have

F ( x , u ) 1 p L - C , f o r a . e . x Ω u R

So

Ω F ( x , u ) d x 1 p L Ω - C Ω .

Since L > 0 is arbitrary, it follows that

Ω F ( x , u ) d x , a s u ,

and so

J ( u ) W C = - Ω F ( x , u ) d x - , a s u .

This proves that J W c is anticoercive. □

Lemma 3.4 If hypothesis (F4) holds, then J W 0 - .

Proof: For a given 0 < ε < m 0 p - 1 , we can find Cε > 0 such that F ( x , u ) ε p η u p + C ε for a.e. x ∈ Ω all u R. Then

J ( u ) u W 0 = 1 p M ^ Ω u p d x - Ω F ( x , u ) d x 1 p m 0 p - 1 Ω u p d x - m 0 p - 1 p η Ω u p d x - C Ω - C Ω .

then J W 0 - . □

Lemma 3.5 If condition (F4) (F5) hold, then J satisfies the (C) condition.

Proof: Let {un}n ≥1 W1,p(Ω) be a sequence such that

J ( u n ) M 1 , n 1 . (3.1)

with some M1 > 0 and

( 1 + u n ) J ( u n ) 0 , in W 1 , p ( Ω ) * a s n . (3.2)

We claim that the sequence {un} is bounded. We argue by contradiction. Suppose that ||u|| → +∞, as n → ∞, we set v n = u n u n , ∀n ≥ 1. Then ||vn|| = 1 for all n ≥ 1 and so, passing to a subsequence if necessary, we may assume that

v n v in W 1 , p ( Ω ) ;

v n v in L p ( Ω ) .

from (3.2), we have ∀h W1,p(Ω)

M Ω u n p d x p - 1 Ω v n p - 2 v n h d x - Ω f ( x , u n ) h u n p - 1 d x ε n 1 + u n h u n p - 1 (3.3)

with εn ↓ 0.

In (3.3), we choose h = vn v W1,p(Ω), note that by virtue of hypothesis (F4), we have

f ( x , u n ) u n p - 1 0 in L p ( Ω ) ,

where 1 p + 1 p = 1 .

So we have

M Ω u n p d x p - 1 Ω v n p - 2 v n ( v n - v ) d x 0 .

Since M(t) > m0 for all t ≥ 0, so we have

Ω v n p - 2 v n ( v n - v ) d x 0 .

Hence, using the (S+) property, we have vn v in W1,p(Ω) with ||v|| = 1, then v ≠ 0. Now passing to the limit as n → ∞ in (3.3), we obtain

Ω v p - 2 v h d x 0 , h W 1 , p ( Ω ) ,

then v = ξ ∈ R. Then |un(x)| → +∞ as n → +∞. Using hypothesis (F5), we have f(x, un(x))un(x) - pF(x, un(x)) → -∞ for a.e x ∈ Ω.

Hence by virtue of Fatou's Lemma, we have

Ω f ( x , u n ) u n - p F ( x , u n ) d x - , a s n + . (3.4)

From (3.1), we have

M ^ Ω u n p d x - p Ω F ( x , u n ) d x - p M 1 , n 1 . (3.5)

From (3.2), we have

M Ω u n p d x p - 1 Ω u n p - 2 u n h d x - Ω f ( x , u n ) h d x ε n h 1 + u n h W 1 , p ( Ω ) .

With εn ↓ 0. So choosing h = un W1,p(Ω), we obtain

- M ( Ω u n p d x ) p - 1 Ω u n p d x + Ω f ( x , u n ) u n d x - ε n . (3.6)

Adding (3.5) and (3.6), noting that M ^ ( t ) ( M ( t ) ) p - 1 t for all t ≥ 0, we obtain

Ω ( f ( x , u n ) u n - p F ( x , u n ) ) d x - M 2 , n 1 , (3.7)

comparing (3.4) and (3.7), we reach a contradiction. So {un}in bounded in W1,p(Ω). Similar with the proof of Lemma 3.1, we know that J satisfied the (C) condition. □

Sum up the above fact, from Lemma 2.4 and Remark 2.1, Theorem 2.2 follows from the Lemma 3.3 to 3.5.

Competing interests

The authors declare that they have no competing interests.

Authors' contributions

All authors read and approved the final manuscript.

Acknowledgements

The authors would like to thank the referees for their valuable comments and suggestions.

This study was supported by NSFC (No. 10871096), the Fundamental Research Funds for the Central Universities (No. JUSRP11118).

References

  1. Lions, JL: On some equations in boundary value problems of mathematical physics. Contemporary developments in Continuum Mechanics and Partial Differential equations (Proc. Internat. Sympos., Inst. Mat., Univ. fed. Rio de Janeiro, Riio de Janeiro, 1977), North-Holland Mathematics Studies, pp. 284–346. North-Holland, Amsterdam (1978)

  2. Alves, CO, Corrêa, FJSA, Ma, TF: Positive solutions for a quasilinear elliptic equation of Kirchhoff type. Comput Math Appl. 49(1), 85–93 (2005). Publisher Full Text OpenURL

  3. Ma, TF, Rivera, JEM: Positive solutions for a nonlinear elliptic transmission problem. Appl Math Lett. 16(2), 243–248 (2003). Publisher Full Text OpenURL

  4. Corrêa, FJSA, Figueiredo, GM: On an elliptic equation of p-Kirchhoff type via variational methods. Bull Austral Math Soc. 74, 263–277 (2006). Publisher Full Text OpenURL

  5. Perera, K, Zhang, ZT: Nontrivial solutions of Kirchhoff-type problems via the Yang-index. J Differ Equ. 221(1), 246–255 (2006). Publisher Full Text OpenURL

  6. Zhang, ZT, Perera, K: Sign-changing solutions of Kirchhoff type problems via invariant sets of descent flow. J Math Anal Appl. 317(2), 456–463 (2006). Publisher Full Text OpenURL

  7. Mao, AM, Zhang, ZT: Sign-changing and multiple solutions of Kirchhoff type problems without the P.S. condition. Nonlinear Anal. 70, 1275–1287 (2009). Publisher Full Text OpenURL

  8. Corrêa, FJSA, Nascimento, RG: On a nonlocal elliptic system of p-Kirchhoff type under Neumann boundary condition. Math Comput Model. 49, 598–604 (2009). Publisher Full Text OpenURL

  9. Gasiński, L, Papageorgiou, NS: Nonsmooth Critical Point Theory and Nonlinear Boundary Value Problems. Chapman and hall/CRC Press, Boca Raton (2005)

  10. Gasiński, L, Papageorgiou, NS: Nontrivial solutions for a class of resonant p-Laplacian Neumann problems. Nonlinear Anal. 71, 6365–6372 (2009). Publisher Full Text OpenURL

  11. Brezis, H, Nirenberg, L: Remarks on finding critical points. Commun Pure Appl Math. 44, 939–963 (1991). Publisher Full Text OpenURL

  12. Rabinowitz, PH: Minimax methods in critical point theory with applications to differential equations. CBMS Regional Conference Series in Mathematics, American Mathematical Soceity, Providence (1986)