**1. Introduction**

In this paper, we study the existence of infinitely many nonzero solutions homoclinic solutions for the fourth-order p-Laplacian differential equation

$$\left(\varphi\_{\mathcal{P}}\left(\boldsymbol{u}^{\prime\prime}(t)\right)\right)^{\prime\prime} + w\left(\varphi\_{\mathcal{P}}\left(\boldsymbol{u}^{\prime}(t)\right)\right)^{\prime} + V(t)\varphi\_{\mathcal{P}}\left(\boldsymbol{u}(t)\right) = a(t)f(t,\boldsymbol{u}(t)),\tag{1}$$

where *t* ∈ R, *w* is a constant, *<sup>ϕ</sup>p*(*t*) = |*t*|*<sup>p</sup>*−<sup>2</sup> *t*, for *p* ≥ 2, *V* is a positive bounded function, *a* is a positive continuous function and *f* ∈ *<sup>C</sup>*<sup>1</sup>(<sup>R</sup>, R) satisfies some growth conditions with respect to *p*. As usual, we say that a solution *u* of (1) is a nontrivial homoclinic solution to zero solution of (1) if

$$
\mu \neq 0, \mu(t) \to 0, \qquad |t| \to \infty. \tag{2}
$$

They are known in phase transitions models as ground states or pulses (see [1]). The existence of homoclinic and heteroclinic solutions of fourth-order equations is studied by various authors (see [2–12] and references therein). Sun and Wu [4] obtained existence of two homoclinic solutions for a class of fourth-order differential equations:

$$u^{(4)} + wu^{\prime\prime} + a(t)u = f(t, u) + \lambda h(t)|u|^{p-2}u, \; t \in \mathbb{R}\_{\star}$$

where *w* is a constant, *λ* > 0, 1 ≤ *p* < 2, *a* ∈ *C* (<sup>R</sup>, R+) and *h* ∈ *L* 2 2−*p* (R) by using mountain pass theorem.

Yang [8] studies the existence of infinitely many homoclinic solutions for a the fourth-order differential equation:

$$u^{(4)} + wu'' + a(t)u = f(t, u), \; t \in \mathbb{R},$$

where *w* is a constant, *a* ∈ *C* (R) and *f* ∈ *C* (R × R, <sup>R</sup>). A critical point theorem, formulated in the terms of Krasnoselskii's genus (see [13], Remark 7.3), is applied, which ensures the existence of infinitely many homoclinic solutions.

We suppose the following conditions on the functions *a*, *f* and *V*. (*A*) *a* ∈ *<sup>C</sup>*(<sup>R</sup>, R+) and *a*(*t*) → 0 as |*t*| → +<sup>∞</sup>.

(*<sup>F</sup>*1) There are numbers *p* and *q* s.t. 1 < *q* < 2 ≤ *p* and for *f* ∈ *<sup>C</sup>*<sup>1</sup>(<sup>R</sup>, R)

$$uf(t, \mathfrak{u}) \le qF(t, \mathfrak{u}), \,\forall \mathfrak{u} \in \mathbb{R}, \mathfrak{u} \ne 0,$$

where *<sup>F</sup>*(*<sup>t</sup>*, *u*) = *u*0*f*(*<sup>t</sup>*, *<sup>x</sup>*)*dx*.

 (*<sup>F</sup>*2) | *f*(*<sup>t</sup>*, *u*)| ≤ *<sup>b</sup>*(*t*)|*u*|*<sup>q</sup>*−1, ∀(*<sup>t</sup>*, *u*) ∈ R × R, where *b* is a positive function, s.t. *b* ∈ *Lr*(R) *L p* 2−*q* (R), where *r* = *p p*−*q*.

(*<sup>F</sup>*3) There exists an interval *J* ⊂ R and a constant *c* > 0 s. t. *<sup>F</sup>*(*<sup>t</sup>*, *u*) ≥ *<sup>c</sup>*|*u*|*<sup>q</sup>*, ∀(*<sup>t</sup>*, *u*) ∈ *J* × R. (*<sup>F</sup>*4) *<sup>F</sup>*(*<sup>t</sup>*, −*<sup>u</sup>*) = *<sup>F</sup>*(*<sup>t</sup>*, *u*) for all (*t*, *u*) ∈ R × R.

(*V*) There exist positive constants *v*1 and *v*2 such that 0 < *v*1 ≤ *V*(*t*) ≤ *v*2, ∀*t* ∈ R. Let 

$$w^\* = \inf\_{\boldsymbol{\mu}\neq 0} \frac{\int\_{\mathbb{R}} \left( |\boldsymbol{\mu}^{\prime\prime}(t)|^p + |\boldsymbol{\mu}(t)|^p \right) dt}{\int\_{\mathbb{R}} |\boldsymbol{\mu}^{\prime}(t)|^p dt}$$

.

.

Denote by *X* the Sobolev's space

$$X := \mathcal{W}^{2,p}\left(\mathbb{R}\right) = \{ \mu \in L^p(\mathbb{R}) \, : \, \mu' \in L^p(\mathbb{R}), \mu'' \in L^p(\mathbb{R}) \},$$

equipped by the usual norm

$$||u||\_X := \left(\int\_{\mathbb{R}} \left( |u''(t)|^p + |u'(t)|^p + |u(t)|^p \right) dt\right)^{1/p}$$

The functional *I* : *X* → R is defined as follows

$$I(u) = \int\_{\mathbb{R}} (\Phi\_p(u''(t)) - w\Phi\_p(u'(t)) + V(t)\Phi\_p(u(t)))dt - \int\_{\mathbb{R}} a(t)F(t, u(t))dt,\tag{3}$$

where Φ(*t*) = |*t*|*<sup>p</sup> p* for *p* ≥ 2.

Under conditions (*A*),(*<sup>F</sup>*1) − (*<sup>F</sup>*3) and *V* the functional *I* is differentiable and for all *u*, *v* ∈ *X* we have

$$\begin{aligned} \left< \left< I'(u), v \right> \right> &= \int\_{\mathbb{R}} \left( \left. \varrho\_{\mathcal{P}} \left( u''(t) \right) v''(t) - w \varrho\_{\mathcal{P}} \left( u'(t) \right) v'(k) \right) dt + V(t) \, \varrho\_{\mathcal{P}} \left( u(t) \right) v(t) dt \\ &- \int\_{\mathbb{R}} a(t) f \left( t, u(t) \right) v(t) dt. \end{aligned}$$

where ., . means the duality pairing between *X* and it's dual space *X*<sup>∗</sup>. The homoclinic solutions of the Equation (1) are the critical points of the functional *I*, i.e., *u*0 is a homoclinic solution of the problem if *I* (*<sup>u</sup>*0), *v* = 0 for every *v* ∈ *X* (see [6,11,12]).

Let *v*0 = min{1, *<sup>v</sup>*1}, where *v*1 is the positive constant from condition (*V*). Our main result is:

**Theorem 1.** *Let p* ≥ 2, *w* < *<sup>v</sup>*0*w*<sup>∗</sup> *and the functions a*, *f and V satisfy the assumptions* (*A*)*,* (*<sup>F</sup>*1) − (*<sup>F</sup>*3) *and* (*V*) *. Then the Equation (1) has at least one nonzero homoclinic solution u*0 ∈ *X*. *Additionally if* (*<sup>F</sup>*4) *holds, the Equation (1) has infinitely many nonzero solutions uj such that* ||*uj*||∞ → 0 *as j* → ∞.

**Remark 1.** *An example of a function f*(*<sup>t</sup>*, *<sup>u</sup>*)*, which satisfies the assumptions* (*<sup>F</sup>*1) − (*<sup>F</sup>*4) *is as follows. Let p* = 3, *q* = 32*and f*(*<sup>t</sup>*, *u*) = *α*(*t*)|*u*|1/2*u, where*

$$\mathfrak{a}(t) = \begin{cases} \frac{3-t^2}{2}, & |t| \le 1, \\\frac{1}{|t|}, & |t| \ge 1. \end{cases}$$

*We have that r* = *p p*−*q* = 2*, p* 2−*q* = 6 *and b*(*t*) = *α*(*t*) ∈ *L*<sup>2</sup>(R) ∩ *<sup>L</sup>*<sup>6</sup>(R)*, because* ∞ 1 1 *t*2 *dt* = 1 *and* ∞ 1 1 *t*6 *dt* = 15 . *Moreover α*(*t*) ≥ 1 *if t* ∈ (−1, 1) = *J*. *Next, we have*

$$\begin{array}{rcl}|f(t,u)| &=& a(t)|u|^{3/2},\\ \boldsymbol{F}(t,u) &=& \frac{2}{5}a(t)|u|^{5/2},\end{array}$$

*and <sup>F</sup>*(*<sup>t</sup>*, *u*) ≥ 25 |*u*|5/2*, t* ∈ *J* = (−1, <sup>1</sup>).

As an open problem we state the existence of weak solutions of the problem when 1 < *q* < *p* < 2. This paper is organized as follows. In Section 2 we present the variational formulation of the problem and critical point theorems used in the proof of the main result. In Section 3, we give the proof of Theorem 1.
