By a weak solution of (1.1), then we say that a function

*u* ε

*W*^{1,p}(Ω) such that

${\left[M\left({\int}_{\Omega}{\left|\nabla u\right|}^{p}dx\right)\right]}^{p-1}{\int}_{\Omega}{\left|\nabla u\right|}^{p-2}\nabla u\nabla \phi dx={\int}_{\Omega}f\left(x,u\right)\phi dx,\phantom{\rule{1em}{0ex}}\text{forall}\phantom{\rule{1em}{0ex}}\phi \in {W}^{1,p}\left(\Omega \right)$

So we work essentially in the space

*W*^{1,p}(Ω) endowed with the norm

$\u2225u\u2225={\left({\int}_{\Omega}\left({\left|\nabla u\right|}^{p}+{\left|u\right|}^{p}\right)dx\right)}^{\frac{1}{p}},$

and the space

*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

${W}^{1,p}\left(\Omega \right)={W}_{0}\oplus {W}_{c}.$

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*.

Let us consider the map

*A* :

*W*^{1,p}(Ω) →

*W*^{1,p}(Ω)* corresponding to −Δ

_{
p
} with Neumann boundary data, defined by

$\u27e8A\left(u\right),v\u27e9={\int}_{\Omega}{\left|\nabla u\right|}^{p-2}\nabla u\nabla vdx,\phantom{\rule{1em}{0ex}}\forall u,v\in {W}^{1,p}\left(\Omega \right).$

(2.1)

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,

$\begin{array}{c}J\left(u\right)\le 0for\phantom{\rule{0.3em}{0ex}}u\in {X}_{1},\phantom{\rule{1em}{0ex}}\u2225u\u2225\le r;\\ J\left(u\right)\ge 0for\phantom{\rule{0.3em}{0ex}}u\in {X}_{2},\phantom{\rule{1em}{0ex}}\u2225u\u2225\le r.\end{array}$

*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*
$c=\underset{h\in \Gamma}{inf}\phantom{\rule{0.3em}{0ex}}\underset{u\in \overline{D}}{max}J\left(h\left(u\right)\right).$

*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

$f\left(x,u\right)=\frac{{m}_{0}^{p-1}}{2\eta}{\left|u\right|}^{p-2}u-{\left|u\right|}^{q-2}u,$

hypotheses (

*F*_{0}), (

*F*_{1}), (

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

$f\left(x,u\right)=arctan\phantom{\rule{0.3em}{0ex}}u+\frac{u}{1+{u}^{2}},$

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