**2. Preliminaries**

In this section we give the variational formulation of the problem and present two critical point theorems.

Let *X*1 be the Sobolev's space

$$X\_1 := \{ u \in X : \int\_{\mathbb{R}} \left( \left| u''(t) \right|^p - w \left| u'(t) \right|^p + V(t) \left| u(t) \right|^p \right) dt < \infty \},$$

equipped by the norm

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

Denote

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

and *v*0 = min{1, *<sup>v</sup>*1}. The next lemma shows that under condition (*V*) for *w* < *<sup>v</sup>*0*w*<sup>∗</sup> the norms ||.|| and ||.||*<sup>X</sup>* are equivalent and *X* = *X*1.

**Lemma 1.** *Let w* < *<sup>v</sup>*0*w*<sup>∗</sup>. *Then, there exists a constant C* > 0 *such that*

$$\int\_{\mathbb{R}} \left( |u''(t)|^p - w \left| u'(t) \right|^p + V(t) \left| u(t) \right|^p \right) dt \ge \mathbb{C} \left\| u \right\|\_{X}^p, \forall u \in \mathcal{X}. \tag{4}$$

**Proof of Lemma 1.** In view of Lemma 4.10 in [14], there exists a positive constant *K* = *<sup>K</sup>*(*p*) depending only on *p* such that

$$\int\_{\mathbb{R}} \left| \mu'(t) \right|^p dt \le \mathcal{K} \int\_{\mathbb{R}} \left( \left| \mu''(t) \right|^p + \left| \mu(t) \right|^p \right) dt.$$

Then

$$\frac{1}{K} \le w^\* = \inf\_{\mu \ne 0} \frac{\int\_{\mathbb{R}} \left( |u''(t)|^p + |u(t)|^p \right) dt}{\int\_{\mathbb{R}} |u'(t)|^p \, dt}.$$

Let

$$\mathcal{C}\_0 = \frac{v\_0 w^\* - w}{(K+1)v\_0 w^\*}$$

and *C* = *<sup>v</sup>*0*C*0. We have

$$\begin{split} &\int\_{\mathbb{R}} \left( |u''(t)|^{p} - w \left| u'(t) \right|^{p} + V(t) \left| u(t) \right|^{p} \right) dt \\ &\geq \quad v\_{0} \int\_{\mathbb{R}} \left( |u''(t)|^{p} - \frac{w}{v\_{0}} \left| u'(t) \right|^{p} + |u(t)|^{p} \right) dt \\ &= \quad v\_{0} ((1 - \frac{w}{v\_{0}w^{\*}}) \int\_{\mathbb{R}} \left( |u''(t)|^{p} + |u(t)|^{p} \right) dt \\ &\quad + \frac{w}{v\_{0}w^{\*}} \int\_{\mathbb{R}} \left( |u''(t)|^{p} - w^{\*} \left| u'(t) \right|^{p} + |u(t)|^{p} \right) dt \right) \\ &\geq \quad v\_{0} (1 - \frac{w}{v\_{0}w^{\*}}) \int\_{\mathbb{R}} \left( |u''(t)|^{p} + |u(t)|^{p} \right) dt \\ &= \quad v\_{0}C\_{0}(K+1) \int\_{\mathbb{R}} \left( |u''(t)|^{p} + |u(t)|^{p} \right) dt \\ &\geq \quad v\_{0}C\_{0} \left( \left| u''(t) \right|^{p} + |u'(t)|^{p} + |u(t)|^{p} \right) dt = C||u||\_{X'}^{p} .\end{split}$$

which completes the proof.

> By Brezis [15], Theorem 8.8 and Corollary 8.9 for *u* ∈ *X* and *s* > *p*

$$||u||\_{\infty} \quad : \quad = ||u||\_{L^{\infty}(\mathbb{R})} \le \mathbb{C}\_{1} ||u||\_{X'} $$

$$\int\_{\mathbb{R}} |u(t)|^{s} dt \quad \le \quad ||u||\_{\infty}^{s-p} ||u||\_{X'}^{p}$$

and lim |*t*|→∞ *u*(*t*) = 0.

We consider the functional *I* : *X* → R

$$I(u) = \int\_{\mathbb{R}} (\Phi\_p(u''(t)) - w\Phi\_p(u'(t)) + V(t)\Phi\_p(u(t)))dt - \int\_{\mathbb{R}} a(t)F(t, u(t))dt,\tag{5}$$

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

One can show that 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

$$
\begin{split}
\left< \left< I'(u), v \right> = \int\_{\mathbb{R}} \left( \varrho\_{p} \left( u'' \left( t \right) \right) v''(t) - u \varrho\_{p} \left( u' \left( t \right) \right) v'(k) \right) dt + V(t) \varrho\_{p} \left( u(t) \right) v(t) dt \\ & \qquad - \int\_{\mathbb{R}} a(t) f \left( t, u(t) \right) v(t) dt. \tag{6}
$$

Let *Lpa* (R), *p* ≥ 1 be the weighted Lebesque space of functions *u* : R → R with norm ||*u*||*<sup>p</sup>*,*<sup>a</sup>* := R *<sup>a</sup>*(*t*)|*u*(*t*)|*pdt*1/*p*. We have

**Lemma 2.** *Assume that the assumptions* (*A*) *and* (*V*) *hold. Then, the inclusion X* ⊂ *Lpa* (R) *is continuous and compact.*

**Proof of Lemma 2.** The embedding *X* ⊂ *Lpa* (R) is continuous by the boundedness of the function *a* by (*A*). We show that the inclusion is compact. Let *uj* ⊂ *X* be a sequence such that ||*uj*|| ≤ *M* and *uj u* weakly in *X*. We'll show that *uj* → *u* strongly in *Lpa* (R). Without loss of generality we can assume that *u* = 0, considering the sequence *uj* − *u* . By (*A*) for any *ε* > 0, there exists *R* > 0, such that for |*t*| ≥ *R*

$$0 \le a(t) \le \frac{\varepsilon}{2(1 + M^p)}.$$

> Then

$$\int\_{|t| \ge R} a(t) |u\_j(t)|^p dt \le \frac{\varepsilon M^p}{2(1 + M^p)}.$$

By Sobolev's imbedding theorem *uj* → 0 strongly in *<sup>C</sup>*([−*R*, *R*]) and there exists *j*0 such that for *j* > *j*0 : 

$$\int\_{|t| \le R} a(t) |u\_j(t)|^p dt < \frac{\varepsilon}{2(1 + M^p)}.$$

Then, for *j* > *j*0 we have R *a*(*t*)|*uj*(*t*)|*pdt* < *ε*, which shows that *uj* → 0 strongly in *Lpa* (R).

**Lemma 3.** *Let assumptions* (*A*),(*<sup>F</sup>*1) − (*<sup>F</sup>*3) *and* (*V*) *hold. If uj u weakly in X*, *there exists a subsequence of the sequence uj* , *still denoted by uj such that f*(*<sup>t</sup>*, *uj*) → *f*(*<sup>t</sup>*, *u*) *in Lpa* (R).

**Proof of Lemma 3.** Let *uj u* weakly in *X*. By Banach-Steinhaus theorem there exists *M*1 > 0, such that ||*uj*|| ≤ *M*1 and ||*u*|| ≤ *M*1. By the elementary inequality for *a* > 0, *b* > 0, *p* > 1

$$(a+b)^p \le \mathcal{2}^{p-1}(a^p+b^p)\_r$$

and (*<sup>F</sup>*2) we have

$$\begin{aligned} |f(t, \boldsymbol{\mu}\_{j}) - f(t, \boldsymbol{\mu})|^{p} &\leq 2^{p-1} (|f(t, \boldsymbol{\mu}\_{j})|^{p} + |f(t, \boldsymbol{\mu})|^{p}) \\ &\leq 2^{p-1} |b(t)|^{p} (|\boldsymbol{\mu}\_{j}|^{p(q-1)} + |\boldsymbol{\mu}|^{p(q-1)}). \end{aligned}$$

Let 0 < *a*(*t*) ≤ *A*. Then, by Hölder inequality and *b* ∈ *L p* 2−*q* (R) it follows that

$$\begin{aligned} &\int\_{\mathbb{R}} a(t) |f(t, u\_j(t)) - f(t, u(t))|^p dt \\ \leq & 2^{p-1} A \int\_{\mathbb{R}} |b(t)|^p (|u\_j|^{p(q-1)} + |u|^{p(q-1)}) dt \\ \leq & 2^{p-1} A (\int\_{\mathbb{R}} |b(t)|^{\frac{p}{2-q}})^{2-q} (\left(\int\_{\mathbb{R}} |u\_j(t)|^p dt\right)^{q-1} + (\int\_{\mathbb{R}} |u(t)|^p dt)^{q-1}) \\ \leq & 2^p A ||b||\_{L^{\frac{p}{2-q}}(\mathbb{R})}^p M\_1^{p(q-1)} .\end{aligned}$$

By Lemma 2, *uj u* weakly in *X* implies that there exists a subsequence {*uj*}, such that *uj* → *u* strongly in *Lpa* (R). By analogous way as above we have that there exists *B* > 0, such that

$$\int\_{\mathbb{R}} |f(t, u\_j(t)) - f(t, u(t))|^p dt \le B.$$

Let *ε* > 0, *R* > 0 are s.t. 0 < *a*(*t*) < *ε*2*B* for |*t*| ≥ *R* by (*A*). Then

$$\int\_{||t||\ge R} a(t)|f(t, u\_j(t)) - f(t, u(t))|^p dt < \frac{\varepsilon}{2}.\tag{7}$$

Let 0 < *aR* < *a*(*t*) ≤ *A* for |*t*| ≤ *R*. By *uj* → *u* strongly in *Lpa* (R) it follows that

$$\int\_{|t| \le R} a(t)|u\_j(t) - u(t)|^p dt \ge a\_R \int\_{|t| \le R} |u\_j(t) - u(t)|^p dt \to 0$$

and *uj*(*t*) − *u*(*t*) → 0 a.e. in |*t*| ≤ *R*. Then, by Lebesque's dominated convergence theorem

$$I\_{\mathbb{R}} := \int\_{|t| \le R} a(t) |f(t, u\_j(t)) - f(t, u(t))|^p dt \to 0.$$

Let *j*0 is sufficiently large, such that for *j* > *j*0, 0 ≤ *IR* < *ε*2. Then by (7) for *j* > *j*0 we have

$$\int\_{\mathbb{R}} a(t)|f(t, u\_j(t)) - f(t, u(t))|^p dt < \varepsilon\_\prime$$

which completes the proof.

> Next we have:

**Lemma 4.** *Under assumptions* (*A*),(*<sup>F</sup>*1) − (*<sup>F</sup>*3),(*V*) *the functional I* ∈ *<sup>C</sup>*<sup>1</sup>(*<sup>X</sup>*, R) *and the identity (6) holds for all u*, *v* ∈ *X*. *holds.*

It can be proved in a standard way using Lemma 3 (see Yang [8], Tersian, Chaparova [6]).

**Lemma 5.** *Under assumptions* (*A*),(*<sup>F</sup>*1) − (*<sup>F</sup>*3) *and* (*V*) *the functional I satisfies the* (*PS*) *condition.*

**Proof of Lemma 5.** Let {*uj*} be a sequence such that {*I*(*uj*)} is bounded in *X* and *I* (*uj*) → 0 in *X*<sup>∗</sup>. Then, there exists a constant *C*1 > 0, s.t.

$$||I(u\_j)|| \le \mathcal{C}\_{1\prime} \quad ||I'(u\_j)||\_{X^\*} \le \mathcal{C}\_1.$$

By (*<sup>F</sup>*2) we have

$$\begin{aligned} \left| C\_1 + \frac{C\_1}{q} ||u\_j|| \right| &\geq \quad \frac{1}{q} < I'(u\_j), u\_j > -I(u\_j) \\ &= \quad \left( \frac{1}{q} - \frac{1}{p} \right) ||u\_j||^p + \int\_{\mathbb{R}} a(t) (F(t, u\_j(t)) - \frac{1}{q} f(t, u\_j(t))u\_j(t)) dt \\ &\geq \quad \left( \frac{1}{q} - \frac{1}{p} \right) ||u\_j||^p. \end{aligned}$$

Then, {*uj*} is a bounded sequence in *X* and up to a subsequence, still denoted by {*uj*}, *uj u* weakly in *X*. There exists *M*2 > 0, such that ||*uj*|| ≤ *M*2, ||*u*|| ≤ *M*2. By Lemma 2, *um* → *u* in *<sup>L</sup>*2*a*(R) and by Lemma 3, *f*(*<sup>t</sup>*, *um*(*t*)) → *f*(*<sup>t</sup>*, *u*(*t*)) in *<sup>L</sup>*2*a*(R) . By Hölder inequality we have:

$$\begin{aligned} I\_{\bar{\jmath}} &:= \int\_{\mathbb{R}} a(t) (f(t, u\_{\bar{\jmath}}(t)) - f(t, u(t))) (u\_{\bar{\jmath}}(t) - u(t)) dt \\ &= \int\_{\mathbb{R}} a^{\frac{p-1}{p}}(t) (f(t, u\_{\bar{\jmath}}(t)) - f(t, u(t))) a^{\frac{1}{p}}(t) (u\_{\bar{\jmath}}(t) - u(t)) dt \\ &\leq \quad A^{\frac{p-1}{p}} \int\_{\mathbb{R}} a(t) |u\_{\bar{\jmath}}(t) - u(t)|^p dt \left( \int\_{\mathbb{R}} |f(t, u\_{\bar{\jmath}}(t)) - f(t, u(t))|^{\frac{p}{p-1}} dt \right)^{\frac{p-1}{p}} \end{aligned}$$

.

As in the proof of Lemma 3, by assumption (*<sup>F</sup>*2), *b* ∈ *L p p*−*q* (R) and Hölder inequality we have for *p*1 = *p p*−1 > 1:

$$\begin{aligned} &\int\_{\mathbb{R}} |f(t, u\_j(t)) - f(t, u(t))|^{p\_1} dt \\ \leq & \quad 2^{p\_1 - 1} \int\_{\mathbb{R}} |b(t)|^{p\_1} |\left( |u\_j(t)|^{(q-1)p\_1} + |u(t)|^{(q-1)p\_1} \right) dt \\ \leq & \quad 2^{p\_1 - 1} \left( \int\_{\mathbb{R}} |b|^{\frac{p}{p-q}} dt \right)^{\frac{p-q}{p-1}} \left( \left( \int\_{\mathbb{R}} |u\_j|^p dt \right)^{\frac{q-1}{p-1}} + \left( \int\_{\mathbb{R}} |u|^p dt \right)^{\frac{q-1}{p-1}} \right) \\ \leq & \quad 2^{p\_1} ||b||\_{L^{\frac{p}{p-q}}}^{p\_1} M\_2^{(q-1)p\_1} .\end{aligned}$$

Then, by *uj* → *u* in *<sup>L</sup>*2*a*(R) it follows that *Ij* → 0 as *j* → ∞. Next, we have

$$||u\_j - u||^p \le \epsilon' I'(u\_j) - I'(u), u\_j - u > + I\_{j'}$$

which shows that *uj* → *u* in *X*.

Next, we recall a minimization theorem which will be used in the proof of Theorem 1. (see [16], Theorem 2.7 of [13]).

**Theorem 2.** *(Minimization theorem) Let E be a real Banach space and J* ∈ *<sup>C</sup>*<sup>1</sup>(*<sup>E</sup>*, R) *satisfying* (*PS*) *condition. If J is bounded below, then c* = inf*E I is a critical value of J*.

We will use also the following generalization of Clark's theorem (see Rabinowitz [13], p. 53) due to Z. Liu and Z. Wang [17]:

**Theorem 3.** *(Generalized Clark's theorem, [17]) Let E be a Banach spa ce, J* ∈ *<sup>C</sup>*<sup>1</sup>(*<sup>E</sup>*, <sup>R</sup>)*. Assume that J satisfies the* (*PS*) *condition, it is even, bounded from below and J*(0) = 0*. If for any k* ∈ N*, there exists a k*−*dimensional subspace E<sup>k</sup> of E and ρk* > 0 *such that* sup*Ek*∩*Sρk J* < 0*, where Sρ* = {*u* ∈ *E* , *uE* = *ρ*}*, then at least one of the following conclusions holds:*


Note that Theorem 3 implies the existence of infinitely many pairs of critical points (*uk*, <sup>−</sup>*uk*), *uk* = 0 of *J*, s.t. *J*(*uk*) ≤ 0, lim*k*→+∞ *J*(*uk*) = 0 and lim*k*→+∞ *ukE* = 0.

**Lemma 6.** *Assume that assumptions* (*A*),(*<sup>F</sup>*2) *and* (*V*) *hold. Then the functional I is bounded from below.*

**Proof of Lemma 6.** By (*<sup>F</sup>*2) and the proof of Lemma 3 we have

$$|F(t,\mu)| \le \frac{1}{q}b(t)|\mu|^q.$$

and

$$\begin{aligned} I(u) &= -\frac{1}{p}||u||^p - \int\_{\mathbb{R}} a(t)F(t, u(t))dt \\ &\ge -\frac{1}{p}||u||^p - \frac{A}{q}\int\_{\mathbb{R}} b(t)|u(t)|^q dt \\ &\ge -\frac{1}{p}||u||^p - \frac{A}{q}\left(\int\_{\mathbb{R}} |b(t)|^{\frac{p}{p-q}}dt\right)^{\frac{p-q}{p}}\left(\int\_{\mathbb{R}} |u(t)|^p dt\right)^{\frac{q}{p}} \\ &\ge -\frac{1}{p}||u||^p - \frac{A}{q}||b||\_{L^{\frac{p}{p-q}}}||u||^q. \end{aligned}$$

By *p* > *q* it follows that *I* is bounded from below functional.

#### **3. Proof of the Main Result**

In this section we prove Theorem 1. The proof is based on the minimization Theorem 2 and multiplicity result Theorem 3. Their conditions are satisfied according to Lemmas 1–6.

**Proof of Theorem 1.** The functional *I* satisfies the assumptions of minimization Theorem 2. Let *u*0 be the minimizer of *I*. Since *I*(0) = 0 to show that *u*0 = 0, let us take *v* ∈ *W*2,*<sup>p</sup>* 0 (*J*), where *J* is the interval from condition (*<sup>F</sup>*3). Suppose that ||*v*||∞ ≤ 1. Then for *λ* > 0 by (*<sup>F</sup>*3)

$$\begin{aligned} I(\lambda \boldsymbol{\nu}) &= \frac{\lambda^p}{p} ||\boldsymbol{\nu}||^p - \int\_I a(t) F(t, \lambda \boldsymbol{\nu}(t)) dt \\ &\leq \frac{\lambda^p}{p} ||\boldsymbol{\nu}||^p - c\lambda^q \int\_I a(t) |\boldsymbol{\nu}(t)|^q dt. \end{aligned}$$

By 1 < *q* < *p* and the last inequality it follows for *λ*0 sufficiently small and *λ*0 > *λ* > 0 *<sup>I</sup>*(*<sup>λ</sup>v*) < 0. Then *<sup>I</sup>*(*<sup>u</sup>*0) = min{*I*(*u*) : *u* ∈ *X*} < *<sup>I</sup>*(*<sup>λ</sup>v*) < 0 and *u*0 is a nonzero weak solution. Let the condition (*<sup>F</sup>*4) holds additionally. We show that the functional *I* satisfies the assumptions of Theorem 3. We construct a sequence of finite dimensional subspaces *Xn* ⊂ *X* and spheres *Sn*−<sup>1</sup> *rn* ⊂ *Xn* with sufficiently small radius *rn* > 0 such that *sup*{*I*(*u*) : *u* ∈ *Sn*−<sup>1</sup> *rn* } < 0. Let *J* = (*a*, *b*) ⊂ R and for *k* ∈ {1, 2, ..., *n*} *Jk* = (*xk*−1, *xk*) , where *xk* = *a* + *kn* (*b* − *<sup>a</sup>*). Next, we choose functions *vk* ∈ *C*20 (*Jk*) such that ||*vk*||∞ < ∞ and ||*vk*||*X* = 1.

Let *Xn* be the *n*− dimensional subspace *Xn* := *span*{*<sup>v</sup>*1, ..., *vk*} ⊂ *X* and

$$S\_{\rho}^{n-1} := \{ \mu \in X\_n : ||\mu||\_X = \rho \}.$$

For *u* = ∑*nk*=<sup>1</sup> *ckvk* ∈ *Xn* we have

$$\begin{aligned} ||\boldsymbol{u}||^{p} &= \int\_{\mathbb{R}} \left( |\boldsymbol{u}''(t)|^{p} - \boldsymbol{w} \, |\boldsymbol{u}'(t)|^{p} + V(t) \, |\boldsymbol{u}(t)|^{p} \right) dt \\ &= \sum\_{j=k}^{n} |c\_{k}|^{p} \int\_{\mathbb{R}} (|\boldsymbol{v}\_{k}''(t)|^{p} - \boldsymbol{w} |\boldsymbol{v}\_{k}'(t)|^{p} + V(t) |\boldsymbol{v}\_{k}(t)|^{p}) dt \\ &= \sum\_{k=1}^{n} |c\_{k}|^{p} .\end{aligned}$$

By analogous way for *γk* = *Jk*(|*vk*(*t*)|*qdt* > 0 we have

$$||\boldsymbol{u}||\_{\mathfrak{u}}^{q} = \sum\_{k=1}^{n} \gamma\_{k} |c\_{k}|^{q} \tag{8}$$

The space *Xn* is n-dimensional and the norms ||.|| and ||.||*<sup>n</sup>* are equivalent. There are positive constants *d*1*n* and *d*2*n* s.t.

$$d\_{1n} ||u|| \le ||u||\_n \le d\_{2n} ||u||\_\prime \quad \forall u \in X\_n. \tag{9}$$

Then, for *u* ∈ *Xn Sn*−<sup>1</sup> 1

$$\begin{aligned} I(\lambda u) &= \quad \frac{\lambda^p}{p} ||u||^p - \sum\_{k=1}^n \int\_{I\_k} a(t) F(t, \lambda c\_k \upsilon\_k(t)) dt \\ &\le \quad \frac{\lambda^p}{p} ||u||^p - c\lambda^q \sum\_{k=1}^n |c\_k|^q \int\_{I\_k} a(t) |\upsilon\_k(t)|^q dt \\ &\le \quad \frac{\lambda^p}{p} ||u||^p - c\lambda^q d\_{1n} ||u||^q \end{aligned}$$

By 1 < *q* < *p* and the last inequality it follows that *<sup>I</sup>*(*v*) < 0 for *v* ∈ *<sup>S</sup>ρ<sup>n</sup>*−<sup>1</sup> := {*u* ∈ *Xn* : ||*u*|| = *ρ*}. Finally, all assumptions of Theorem 3 are satisfied and by Remark 1 there exist infinitely many weak solutions {*uj*} of the problem (1), such that *<sup>I</sup>*({*uj*}) ≤ 0 and ||*uj*|| → 0. By imbedding *X* ⊂ *L*∞(R) it follows that ||*uj*||∞ → 0 as *j* → ∞ which completes the proof.
