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

- Yang Yang
^{1}Email author and - Jihui Zhang
^{2}

**2011**:32

https://doi.org/10.1186/1687-2770-2011-32

© Yang and Zhang; licensee Springer. 2011

**Received: **7 January 2011

**Accepted: **11 October 2011

**Published: **11 October 2011

## Abstract

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

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

## Keywords

## 1. Introduction

**R**

^{ N }, 1 <

*p*<

*N*,

*ν*is the unit exterior vector on ∂Ω, Δ

_{ p }is the

*p*-Laplacian operator, that is, Δ

_{ p }

*u*=

*div*(|∇

*u*|

^{p−2}∇

*u*), the function

*M*:

**R**

^{+}→

**R**

^{+}is a continuous function and there is a constant

*m*

_{0}> 0, such that

where *C* denotes a generic positive constant.

*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*|

^{2}d

*x*)Δ

*u*appears in the Kirchhoff equation which arises in nonlinear vibrations, namely

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

*u*ε

*W*

^{1,p}(Ω) such that

*W*

^{1,p}(Ω) endowed with the norm

*W*

^{1,p}(Ω) may be split in the following way. Let

*W*

_{ c }= 〈1〉, that is, the subspace of

*W*

^{1,p}(Ω) spanned by the constant function 1, and ${W}_{0}=\left\{z\in {W}^{1,p}\left(\Omega \right),{\int}_{\Omega}z=0\right\}$, which is called the space of functions of

*W*

^{1,p}(Ω) with null mean in Ω. Thus

As it is well known the Poincar*é*'s inequality does not hold in the space *W*^{1,p}(Ω). However, it is true in *W*_{0}.

**Lemma 2.1**[8] (Poincar*é*-Wirtinger's inequality) *There exists a constant η* > 0 *such that*${\int}_{\Omega}{\left|z\right|}^{p}dx\le \eta {\int}_{\Omega}{\left|\nabla z\right|}^{p}dx$*for all z* ∈ *W*_{0}.

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 {*x*_{
n
} }_{n≥1}⊆ *X* such that *x*_{
n
} ⇀ *x* weakly in *X*, and $lim{sup}_{n\to \infty}\u27e8A\left({x}_{n}\right),{x}_{n}-x\u27e9\le 0$. we have that *x*_{
n
} → *x* in *X*.

*A*:

*W*

^{1,p}(Ω) →

*W*

^{1,p}(Ω)* corresponding to −Δ

_{ p }with Neumann boundary data, defined by

We have the following result:

**Lemma 2.2**[9, 10]*The map A* : *W*^{1,p}(Ω) → *W*^{1,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 x*_{0} ∈ *D and the Fréchet derivative J'* (*x*_{0}) = 0, *then we call that x*_{0}*is a critical point of the functional J and c* = *J*(*x*_{0}) *is a critical value of J*.

**Definition 2.2**. *For* *J* ∈ *C*^{1}(*E*, **R**), *we say J satisfies the Palais-Smale condition (denoted by (PS)) if any sequence* {*u*_{
n
} } ⊂ *E for which J*(*u*_{
n
} ) *is bounded and J'*(*u*_{
n
} ) → 0 *as n* → ∞ *possesses a convergent subsequence*.

**Lemma 2.3**[11]

*Let X be a Banach space with a direct sum decomposition*

*X*=

*X*

_{1}⊕

*X*

_{2},

*with k*=

*dimX*

_{2}< ∞,

*let J be a C*

^{1}

*function on X, satisfying (PS) condition. Assume that, for some r*> 0,

*Assume also that J is bounded below and* inf _{
X
} *J* < 0. *Then J has at least two nonzero critical points*.

**Lemma 2.4**[12]*Let* *X* = *X*_{1} ⊕ *X*_{2}, *where* *X* *is a real Banach space and* *X*_{2} ≠ {0}, *and is finite dimensional. Suppose J* ∈ *C*^{1}(*X*, *R*) *satisfies (PS) and*

*(i) there is a constant α and a bounded neighborhood D of* 0 *in X*_{2}*such that J*| _{
∂D
} *≤ α and*,

*(ii) there is a constant β* > *α such that*${J{\mid}_{X}}_{{}_{1}}\ge \beta $,

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

*where*$\Gamma =\left\{h\in C\left(\overline{D},X\right)\mid h=id\phantom{\rule{2.77695pt}{0ex}}on\phantom{\rule{2.77695pt}{0ex}}\partial D\right\}$.

**Definition 2.3**. *For* *J* ∈ *C*^{1}(*E*, **R**), *we say J satisfies the Cerami condition (denoted by (C)) if any sequence* {*u*_{
n
} } ⊂ *E for which J*(*u*_{
n
} ) *is bounded and* (1 ||*u*_{
n
} ||) *J*'(*u*_{
n
} )|| → 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:*

(*F*_{0}) $0\le {lim}_{\mid u\mid \to 0}\frac{pF\left(x,u\right)}{{\left|u\right|}^{p}}<\frac{{m}_{0}^{p-1}}{\eta}\phantom{\rule{2.77695pt}{0ex}}a.e.\phantom{\rule{2.77695pt}{0ex}}x\in \Omega $, *where*$F\left(x,u\right)={\int}_{0}^{u}f\left(x,s\right)ds$, *η* appears in Lemma 2.1 ;

(*F*_{1}) ${lim}_{\mid u\mid \to \infty}\frac{pF\left(x,u\right)}{{\left|u\right|}^{p}}\le 0\phantom{\rule{2.77695pt}{0ex}}a.e.\phantom{\rule{2.77695pt}{0ex}}x\in \Omega $;

(*F*_{2})${lim}_{\mid u\mid \to \infty}{\int}_{\Omega}F\left(x,u\right)dx=-\infty $.

*Then the problem (1.1) has least three distinct weak solutions in W*^{1,p}(Ω).

**Theorem 2.2**
*If the following hold:*

(*M*_{1}) *The function M that appears in the classical Kirchhoff equation satisfies*$\hat{M}\left(t\right)\le {\left(M\left(t\right)\right)}^{p-1}t$*for all t* ≥ 0, *where*$\hat{M}\left(t\right)={\int}_{0}^{t}{\left[M\left(s\right)\right]}^{p-1}ds$;

(*F*_{3})$f\left(x,u\right)u>0\phantom{\rule{0.3em}{0ex}}for\phantom{\rule{0.3em}{0ex}}all\phantom{\rule{0.3em}{0ex}}u\ne 0$;

(*F*_{4})${lim}_{{}_{\mid u\mid \to \infty}}\frac{pF\left(x,u\right)}{{\left|u\right|}^{p}}=0\phantom{\rule{2.77695pt}{0ex}}a.e.\phantom{\rule{2.77695pt}{0ex}}x\in \Omega $;

(*F*_{5})${lim}_{\mid u\mid \to \infty}\left(f\left(x,u\right)u-pF\left(x,u\right)\right)=-\infty $.

*Then the problem (1.1) has at least one weak solution in W*^{1,p}(Ω).

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

*F*

_{0}), (

*F*

_{1}), (

*F*

_{2}) and (1.2) are clearly satisfied.

hypotheses (*F*_{3}), (*F*_{4}) and (*F*_{5}) and (1.2) are clearly satisfied.

## 3. Proofs of the theorems

*J*:

*W*

^{1,p}(Ω) →

**R**given by

**Proof of Theorem 2.1** By (*F*_{0}), 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 {

*u*

_{ n }} be a bounded (PS) sequence of

*J*. Passing to a subsequence if necessary, there exists

*u*∈

*W*

^{1,p}(Ω) such that

*u*

_{ n }⇀

*u*. From the subcritical growth of

*f*and the Sobolev embedding, we see that

*J'*(

*u*

_{ n })(

*u*

_{ n }−

*u*) → 0, we conclude that

*M*

_{0}), we have

Using Lemma 2.2, we have *u*_{
n
} → *u* as *n* → ∞. □

**Lemma 3.2** *If condition* (*M*_{0}), (*F*_{1}) *and* (*F*_{2}) *hold, then*${lim}_{\parallel u\parallel \to \infty}J\left(u\right)=+\infty $.

**Proof:** If there are a sequence {*u*_{
n
} } and a constant *C* such that ||*u*_{
n
} || → ∞ as *n* → ∞, and *J*(*u*_{
n
} ) ≤ *C* (*n* = 1, 2 ···), let ${v}_{n}=\frac{{u}_{n}}{\u2225{u}_{n}\u2225}$, then there exist *v*_{0} ∈ *W*^{1,p}(Ω) and a subsequence of {*v*_{
n
} }, we still note by {*v*_{
n
} }, such that *v*_{
n
} ⇀ *v*_{0} in *W*^{1,p}(Ω) and *v*_{
n
} → *v*_{0} in *L*^{
p
} (Ω).

*ε*> 0, by (

*F*

_{1}), there is a

*H*> 0 such that $F\left(x,u\right)\le \frac{\epsilon}{p}{\left|u\right|}^{p}$ for all |

*u|*≥

*H*and a.e.

*x*∈ Ω, then there exists a constant

*C*> 0 such that $F\left(x,u\right)\le \frac{\epsilon}{p}{\left|u\right|}^{p}+C$ for all

*u*∈

*R*, and a.e.

*x*∈ Ω, Consequently

*∫*

_{Ω}|

*v*

_{0}|

^{ p }d

*x ≥*1. On the other hand, by the weak lower semi-continuity of the norm, one has

*|v*

_{0}(

*x*)| =

*constant*≠ 0 a.e.

*x*∈ Ω. By (

*F*

_{2}), ${\mathsf{\text{lim}}}_{\mid {u}_{n}\mid \to \infty}{\int}_{\Omega}F\left(x,{u}_{n}\right)\mathsf{\text{d}}x\to -\infty $. Hence

This is a contradiction. Hence *J* is coercive on *W*^{1,p}(Ω), bounded from below, and satisfies the (PS) condition. □

*J*is coercive on

*W*

^{1,p}(Ω), bounded from below, and satisfies the (PS) condition. From condition (

*F*

_{0}), we know, there exist

*r*> 0,

*ε*> 0 such that

*u*∈

*W*

_{ c }, for ||

*u*|| ≤

*ρ*

_{1}, then |

*u*|

*≤*

*r*, we have

*u*∈

*W*

_{0}, then from condition (

*F*

_{0}) and (1.2), we have

Choose ||*u*|| = *ρ*_{2} small enough, such that *J*(*u*) ≥ 0 for ||*u*|| ≤ *ρ*_{2} and *u* ∈ *W*_{0}.

*ρ*= min{

*ρ*

_{1},

*ρ*

_{2}}, then, we have

If inf{*J*(*u*), *u* ∈ *W*^{1,p}(Ω)} = 0, then all *u* ∈ *W*_{
c
} with ||*u*|| ≤ *ρ* are minimum of *J*, which implies that *J* has infinite critical points. If inf{*J*(*u*), *u* ∈ *W*^{1,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 *W*^{1,p}(Ω), Therefore, problem (1.1) has at least three distinct solutions in *W*^{1,p}(Ω). □

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

**Lemma 3.3** *If condition* (*F*_{3}) *and* (*F*_{5}) *hold, then*$J{\mid}_{{W}_{c}}$*is anticoercive. (i.e. we have that* *J*(*u*) → -∞, *as* |*u*| → ∞, *u* ∈ *R.)*

**Proof:**By virtue of hypothesis (

*F*

_{5}), for any given

*L*> 0, we can find

*R*

_{1}=

*R*

_{1}(

*L*) > 0 such that

*F*

_{3}), we have

*L*> 0 is arbitrary, it follows that

This proves that $J{\mid}_{{W}_{c}}$ is anticoercive. □

**Lemma 3.4** *If hypothesis* (*F*_{4}) *holds, then*$J{\mid}_{{W}_{0}}\ge -\infty $.

**Proof:**For a given $0<\epsilon <{m}_{0}^{p-1}$, we can find

*C*

_{ ε }> 0 such that $F\left(x,u\right)\le \frac{\epsilon}{p\eta}\mid u{\mid}^{p}+{C}_{\epsilon}$ for a.e.

*x*∈ Ω all

*u*∈

**R**. Then

then $J{\mid}_{{W}_{0}}\ge -\infty $. □

**Lemma 3.5** *If condition* (*F*_{4}) (*F*_{5}) *hold, then J satisfies the (C) condition*.

**Proof:**Let {

*u*

_{ n }}

_{n ≥1}⊆

*W*

^{1,p}(Ω) be a sequence such that

*M*

_{1}> 0 and

*u*

_{ n }} is bounded. We argue by contradiction. Suppose that ||

*u*|| → +∞, as

*n*→ ∞, we set ${v}_{n}=\frac{{u}_{n}}{\u2225{u}_{n}\u2225}$, ∀

*n*≥ 1. Then ||

*v*

_{ n }|| = 1 for all

*n*≥ 1 and so, passing to a subsequence if necessary, we may assume that

*h*∈

*W*

^{1,p}(Ω)

with *ε*_{
n
} ↓ 0.

*h*=

*v*

_{ n }−

*v*∈

*W*

^{1,p}(Ω), note that by virtue of hypothesis (

*F*

_{4}), we have

where $\frac{1}{p}+\frac{1}{{p}^{\prime}}=1$.

*M*(

*t*) >

*m*

_{0}for all

*t*≥ 0, so we have

*S*

_{+}) property, we have

*v*

_{ n }→

*v*in

*W*

^{1,p}(Ω) with ||

*v*|| = 1, then

*v*≠ 0. Now passing to the limit as

*n*→ ∞ in (3.3), we obtain

then *v* = ξ ∈ *R*. Then |*u*_{
n
} (*x*)| → +∞ as *n* → +∞. Using hypothesis (*F*_{5}), we have *f*(*x*, *u*_{
n
} (*x*))*u*_{
n
} (*x*) - *pF*(*x*, *u*_{
n
} (*x*)) → -∞ for a.e *x* ∈ Ω.

*ε*

_{ n }↓ 0. So choosing

*h*=

*u*

_{ n }∈

*W*

^{1,p}(Ω), we obtain

*t*≥ 0, we obtain

comparing (3.4) and (3.7), we reach a contradiction. So {*u*_{
n
} }in bounded in *W*^{1,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.

## Declarations

### 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).

## Authors’ Affiliations

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