On nonlinear perturbations of a periodic integrodifferential Kirchhoff equation with critical exponential growth

Abstract

In this paper, we investigate the existence of solutions for a class of integrodifferential Kirchhoff equations. These equations involve a nonlocal operator with a measurable kernel that satisfies “structural properties” that are more general than the standard kernel of the fractional Laplacian operator. Additionally, the potential can be periodic or asymptotically periodic, and the nonlinear term exhibits critical exponential growth in the sense of Trudinger–Moser inequality. To guarantee the existence of solutions, we employ variational methods, specifically the mountain-pass theorem. In this context, it is important to emphasize that we have additional difficulties due to the lack of compactness in our problem, because we deal with critical growth nonlinearities in unbounded domains. Moreover, the Kirchhoff term adds complexity to the problem, as it requires suitable calculations for control the estimate the minimax level, representing the main challenge in this work. Finally, we consider two different approaches to estimate the minimax level. The first approach is based on a hypothesis proposed by D. M. Cao, while the second one involves a slightly weaker assumption addressed by Adimurthi and Miyagaki.

1 Introduction

In this paper, we are concerned with the existence of solutions for a class of integrodifferential Kirchhoff equations

$$\begin{aligned}{}[-m(\Vert u\Vert ^2)]\mathcal {L}_{K}u + V(x)u= f(x,u) \quad \text{ in }\quad \mathbb {R}, \end{aligned}$$

(1.1)

where V and f are functions that satisfy mild conditions, $m:\mathbb {R}_+\rightarrow \mathbb {R}_{+}$ is the Kirchhoff function, $\mathbb {R}_{+}$ denotes $[0,+\infty )$, and $\mathcal {L}_{K}u$ stands for the integrodifferential operator defined by

$$\begin{aligned} -\mathcal {L}_{K}u(x) =2 P.V.\int \limits _{\mathbb {R}}(u(x)-u(y))K(x,y) \, \textrm{d} y.\end{aligned}$$

(1.2)

Here $K(x,y)=K(x-y)$ and belongs to a class of singular symmetric kernels, and P.V. means “in the principal value sense”.

If $K(x) = C_{N,s}|x|^{-(N+2s)}$, where

$$\begin{aligned} C_{N,s} =\left( \ \int \limits _{\mathbb {R}^N}\frac{1-\cos (\zeta _1)}{|\zeta |^{N+2s}}\,\textrm{d} \zeta \right) ^{-1}, \end{aligned}$$

this is, when $-\mathcal {L}_{K}$ is the fractional Laplacian operator $(-\Delta )^s$, $0<s<1$, (see [1]), several papers have studied the existence of solutions for equations of this type,

$$\begin{aligned} m\left( \ \int \limits _{\mathbb {R}^N\times \mathbb {R}^N}\dfrac{|u(x)-u(y)|^2}{|x-y|^{N+2s}}\textrm{d} x \textrm{d} y\right) (-\Delta )^su+ V(x)u=f(x,u)\quad \text{ in }\,\, \mathbb {R}^N, \end{aligned}$$

(1.3)

with $m:\mathbb {R}_+\rightarrow \mathbb {R}_{+}$ the Kirchhoff function, whose prototype, due to Kirchhoff himself, is $m(t) = a + b \gamma t^{\gamma -1}, a, b \ge 0, a + b >0, \gamma \ge 1.$

Moreover, when $s\rightarrow 1^-$ then problem (1.3) formally reduces to the well-known Kirchhoff equation in the literature with $m(t)=a+tb$

$$\begin{aligned} -\left( a+b\int \limits _{\mathbb {R}^N}|\nabla u|^{2}\;\textrm{d} x\right) \Delta u+V(x)u =f(u) \,\text{ in }\, \mathbb {R}^{N}, \end{aligned}$$

which is related to the stationary analogous of the Kirchhoff-type equation

$$\begin{aligned} -\dfrac{\partial ^{2}u}{\partial t^{2}}\left( a+b\int \limits _\Omega |\nabla u|^{2}\;\textrm{d} x\right) \Delta u= f(t, x, u), \,\text{ in }\,\Omega , \end{aligned}$$

where $\Omega$ is a bounded domain in $\mathbb {R}^{N}$, u denotes the displacement, f is the external force, b is the initial tension and a is related to the intrinsic properties of the string. Equations of this type were first proposed by Kirchhoff [2] to describe the transversal oscillations of a stretched string. Besides, we also point out that such nonlocal problems appear in other fields like biological systems, where u describes a process depending on the average of itself. In this direction, we refer readers to Chipot and Lovat [3], Alves and Corrêa [4]. There is extensive literature on this subject, when $a> 0$, that is in the so-called non-degenerate case. We cite e.g. [5,6,7], as well as the references therein. For the degenerate case, there are a few papers, see [8, 9], as well as the references therein. We mention in passing that in [10] variational techniques were used for the first time to handle Kirchhoff elliptic problems.

Fiscella and Valdinoci [6], proposed the following stationary Kirchhoff variational equation with critical growth

$$\begin{aligned} \left\{ \begin{array}{ll} m\left( \,\,\displaystyle \int \limits _{\mathbb {R}^N}\int \limits _{\mathbb {R}^N} \dfrac{|u(x)-u(y)|^2}{|x-y|^{N+2s}}\textrm{d} x \textrm{d} y \right) (-\Delta )^su =\lambda f(x,u)+ |u|^{2^{*}_s-2}u\quad \text {in}\, \,\Omega , \\ u =0 \quad \text {in} \,\, \mathbb {R}^N\setminus \Omega ,\\ \end{array} \right. \end{aligned}$$

(1.4)

where $\Omega \subset \mathbb {R}^N$ is an open bounded set and $2^*_s= \dfrac{2N}{N-2s}$. This equation models nonlocal aspects of the tension arising from measurements of the fractional length of the string. They obtained the existence of nonnegative solutions when m is an increasing and continuous function, there exists $m_0 > 0$ such that $m(t)\ge m_0 = m(0)$ for any $t \in \mathbb {R}_{+}$ and f is a continuous function with subcritical growth satisfying suitable assumptions. Autuori et al. [8] considered the existence and the asymptotic behavior of nonnegative solutions of (1.4) for the degenerate case.

To the best of our knowledge, there are few papers in the literature on fractional Kirchhoff equations in $\mathbb {R}^N$. Recently, Ambrosio and Isernia [11] considered the fractional Kirchhoff problem

$$\begin{aligned} \left( a+b\int \limits _{\mathbb {R}^N}|(-\Delta )^\frac{s}{2}u|^2\,\textrm{d}x \right) (-\Delta )^{s}u =f(u) \quad \text {in }\mathbb {R}^N, \end{aligned}$$

where f is an odd subcritical nonlinearity satisfying the well-known Berestycki and Lions [12] assumptions. By minimax arguments, the authors established a multiplicity result in the radial space $H^{\alpha }_{rad}(\mathbb {R}^N)$ when the parameter $b>0$ is sufficiently small.

Liu et al. [7] ensure the existence of positive ground state solutions to the following fractional Kirchhoff equation with the Berestycki–Lions type conditions of critical type

$$\begin{aligned} \left( a+b\int \limits _{\mathbb {R}^N}|(-\Delta )^\frac{s}{2}u|^2\,\textrm{d}x \right) (-\Delta )^{s}u +V(x)u =f(u) \quad \text {in }\mathbb {R}^N, \end{aligned}$$

where $u\in H^s(\mathbb {R}^N)$, $u>0$, a, b are positive constants and $N>2s$.

Motivated by some of these works de Albuquerque et al. [5], studied the existence of bound and ground state solutions for fractional Kirchhoff equations of the form

$$\begin{aligned} \left( a+b[u]_{1/2}^{2}\right) (-\Delta )^{1/2}u + V(x)u=f(x,u) \quad \text{ in }\,\,\mathbb {R}, \end{aligned}$$

(1.5)

where $a>0$, $b\ge 0$, $(-\Delta )^{1/2}$ denotes the square root of the Laplacian and the term

$$\begin{aligned}{}[u]_{1/2}=\left( \ \int \limits _{\mathbb {R}^{2}}\frac{|u(x)-u(y)|^{2}}{|x-y|^{2}}\;\textrm{d} x\textrm{d} y\right) ^{1/2} \end{aligned}$$

is the so-called Gagliardo semi-norm of the function u, V is a bounded potential which may change the sign and the nonlinear term f(x, u) has the critical exponential growth in the sense of Trudinger–Moser, generalizing the results in [13] by de Souza and Araújo, who address the problem of the fractional Schrödinger equations. We emphasize that the results in [13] were also improved by Barboza et al. [14], once, who treated a problem with a more general non-local operator. More specifically, they studied the following integrodifferential Schrödinger equation

$$\begin{aligned} \mathcal {L}_{K}u + V(x)u=f(x,u)\, \text{ in }\, \mathbb {R}^N, \end{aligned}$$

(1.6)

when V is a nonnegative and bounded potential, and the nonlinear term f(x, u) has critical exponential growth with respect to the Trudinger–Moser inequality. This problem is a version of (1.5) in case $a=1$ and $b=0$ but generalizing the operator.

We highlight that, when $\mathcal {L}_{K}$ is the fractional Laplacian operator, (1.6) had been studied by other authors under many different assumptions on the potential V(x) and nonlinearity f(x, u). Almost all works and therein references considered nonlinearities involving polynomial growth of subcritical type in terms of the Sobolev embedding when $N>2s$. In the borderline case $N=2s$, this is, $N=1$ and $s=1/2$, Sobolev embedding states that $H^{1/2}(\mathbb {R}) \hookrightarrow L^{q}(\mathbb {R})$ for any $q\in [2,+\infty )$, but $H^{1/2}(\mathbb {R})$ is not continuous embedded in $L^{\infty }(\mathbb {R})$; for details see [1, 15]. In this case, the maximal growth which allows us to treat this problem type variationally in $H^{1/2}(\mathbb {R})$ is motivated by the Trudinger–Moser inequality proved by Ozawa [15] and improved by Iula [16], Kozono et al. [17] and Takahashi [18]. Precisely, by combining some of the results contained in previous studies, it is established

$$\begin{aligned} \sup _{\begin{array}{c} u \in H^{1/2}(\mathbb {R})\\ \Vert u\Vert _{1/2} \le 1 \end{array}}\int \limits _{\mathbb {R}}(e^{\alpha u^{2}}-1) \, \textrm{d} x < \infty \,; \alpha \in [0, \pi ], \end{aligned}$$

(1.7)

where

$$\begin{aligned} \Vert u\Vert _{{1/2}}:=\left( \dfrac{1}{2\pi }\int \limits _{\mathbb {R}^{2}}\dfrac{|u(x)-u(y)|^{2}}{|x-y|^{2}}\, \textrm{d} x \,\textrm{d} y+\Vert u\Vert _{2}^{2}\right) ^{{1}/{2}}. \end{aligned}$$

Moreover, it holds

$$\begin{aligned} \int \limits _{\mathbb {R}}(e^{\alpha u^{2}}-1)\, \textrm{d} x \le C(\alpha )\Vert u\Vert _{2}^{2}, \quad \text{ for } \text{ all } \quad 0<\alpha \le \pi , \end{aligned}$$

(1.8)

for details see [15, Theorem 1] and [18, Proposition 1.1].

Therefore, in order to deal with this class of problem in a variational approach in $H^{1/2}(\mathbb {R})$, the maximal growth on the nonlinearity f(x, u) is given by $e^{ \pi u^2}$ when $|u| \rightarrow +\infty$ (see also the pioneering works [19, 20]).

Recently, the borderline case was approached by Miyagaki and Pucci [21]; more specifically, in this paper, they deal with the existence of solutions for a class of nonlinear elliptic equations, involving a nonlocal Kirchhoff term and possibly Trudinger–Moser critical growth nonlinearities in an unbounded domain. In this context, in order to overcome the lack of compactness of the associated energy functional, it is usually assumed a hypothesis under the nonlinearity which helps to estimate the minimax level.

In general, when we consider a nonlinearity f(t) with critical exponential growth, there exist two kinds of assumptions for this purpose. The first one is due to D. M Cao (see [22]) and is widely used in literature. See, for instance, [5, 13] and reference in therein. Precisely, it is supposed that there are constants $p>2$ and $C_{p}>0$ such that

$$\begin{aligned} {f}(t)\ge C_{p}t^{p-1}, \quad \text {for all}~t~ \text{ in } \text{ domain } \text{ of }~f, \end{aligned}$$

where $C_p$ is chosen suitably. For this case, it is crucial to show that the embedding constant of solutions space into appropriated Lebesgue space is attained and a suitable version of Lions’ Lemma plays a main role to prove it. As far as we know, almost all papers that are concerned with Kirchhoff equations and nonlinearity with exponential critical growth have assumed this type of hypothesis. In the second case, it is considered a heavy dependence on the asymptotic behavior of $h(t)={{f}(t)t}/{e^{\alpha _0t^2}}$ at infinity, as in the pioneer works [23, 24]. This asymptotic behavior can appear in different ways. For example, in [24] the authors considered an equation involving the Laplacian operator with f satisfying (among other conditions) that $\lim _{t\rightarrow \infty }h(t)= C(r)$, where r is the radius of the largest open ball in the domain. In [25], it was assumed that $\lim _{t\rightarrow \infty }h(t)= \infty$ for a equation with the 1/2-Laplacian operator. In papers that use this approach, the Moser functions are used to estimate the minimax level of functional associated to a problem with an exponential nonlinearity. For other works with this kind of conditions, see [26, 27].

Our purpose is to generalize the results in [5], for this, we study (1.5) when the fractional Laplacian operator is replaced by a more general integrodifferential operator $-\mathcal {L}_{K}$ where $K:\mathbb {R}\setminus \{0\}\rightarrow \mathbb {R}_+$ is a measurable kernel which satisfies “structural properties” and the Kirchhoff function is more general. We also improve some results in [14, 21], in different perspectives. In these papers, the potential V is assumed nonnegative; in [14], the Kichhorff function is equal to 1, and in [21], it is considered a weight to control nonlinearity behavior at infinity to recover the compactness of energy functional associated to problem. Here in this paper, we will assume the two types of hypotheses and neglect the weight in the nonlinearity, which requires different techniques to estimate the minimax level. This fact and assumptions more general hypotheses under the potential V, the Kichhorff function m and the integrodifferential kernel K improve some results in [5, 13, 14, 21], more specifically, we obtain new versions of the results in these papers.

As expected, our main difficulties are related to unbounded domains and nonlinearities with critical growth. These difficulties become harder due to the presence of the general Kirchhoff term, once to control it, we need to make some suitable calculations in order to recover the lack of compactness.

For easy reference, we record problems, assumptions, and the main results.

1.1 A periodic problem

Here we present the periodic problem for a positive bounded potential and a nonlinearity with critical exponential growth. For this matter, initially, we will study the following problem

$$\begin{aligned} \left\{ \begin{array}{ll} [-m(\Vert u\Vert ^2)]\mathcal {L}_{K}u +V_0(x)u=f_{0}(x,u)\quad \text{ in }\quad \mathbb {R},\\ u\in X_{0}\quad \text{ and }\quad u \ge 0,\\ \end{array} \right. \end{aligned}$$

(1.9)

where $-\mathcal {L}_{K}u$ is given in (1.2) and we assume that $K:\mathbb {R}\setminus \{0\}\rightarrow \mathbb {R}_+$ is a measurable function with the properties

($K_1$):

$\gamma K \in L^1(\mathbb {R})$, where $\gamma (x)=min\{1, |x|^2\}$;

($K_2$):

there exists $\lambda >0$ such that $K(x)\ge \lambda |x|^{-2}$, for all $x\in \mathbb {R}\setminus \{0\}$;

($K_3$):

$K(x)=K(-x),$ $\forall x\in \mathbb {R}\setminus \{0\}$.

These hypotheses allow us to obtain a wide range of nonlocal integrodifferential operators of the fractional, that order is different from $s=1/2$. For example, K given by

$$\begin{aligned} K(x)= \left\{ \begin{array}{clccc} C_1|x|^{-r}&{} \text{ with }&{} 1<r\le 2&{} \text{ if }&{} |x|\ge 1; \\ C_2|x|^{-q}&{} \text{ with }&{} 2\le q< 3&{} \text{ if }&{} |x|\le 1, \end{array} \right. \end{aligned}$$

satisfies ($K_1$)–($K_3$). Moreover, we can chose r and q in the intervals (1, 2] and [2, 3), respectively, and if $C_1\ne C_2$, K is not continuous in $\mathbb {R}\setminus \{0\}$. For more details, see [14].

Moreover, we assume $m:\mathbb {R}_{+}\rightarrow \mathbb {R}_+$ is a continuous and nondecreasing function that satisfies

($m_1$):

there exists $\sigma \in [1,+\infty )$ such that $tm(t)\le \sigma M(t) \, \forall t\in \mathbb {R}_+$ where $M(t)=\int _{0}^{t}m(\tau )\,\textrm{d}{\tau }$;

($m_2$):

for all $\tau >0$ there exists $\eta (\tau )=\eta >0$ such that $m(t)\ge \eta \, \,\forall t\ge \tau$;

($m_3$):

$t\mapsto \sigma M(t)-m(t)t$ is nondecreasing in $\mathbb {R}_{+}.$

A typical example for m is given by $m(t) = m_0 +b t^{\sigma -1}$ with $m_0, b \ge 0$, $m_0+b>0$ and $1\le \sigma <+\infty$.

Note that ($m_1$) implies, in particular, that

$$\begin{aligned} M(t)\ge M(1)t^\sigma \, \text {for all} \, t\in [0,1], \end{aligned}$$

(1.10)

and

$$\begin{aligned} M(t)\le \dfrac{M(t_0)}{t_0^{\sigma }}t^\sigma \, \text {for all} \, t\ge t_0\quad \text{ for } \text{ all }\quad t_0>0. \end{aligned}$$

(1.11)

Moreover, ($m_2$) yields that $M(t)>0$ for all $t>0$ as in the Kirchhoff model.

We suppose that the function $V_{0}:\mathbb {R} \rightarrow \mathbb {R}$ is a continuous $1-$periodic function satisfying:

$(V_{0,1})$:

there exists a positive constant $\upsilon _0$ such that $V_0(x)\ge -\upsilon _0$ for all $x\in \mathbb {R}$;

$(V_{0,2})$:

The infimum

$$\begin{aligned} \xi _0:= \inf _{\begin{array}{c} u\in X_0\\ \Vert u\Vert _{2}=1 \end{array}}\left( \,\,\int \limits _{\mathbb {R}^2}[u(x)-u(y)]^2K(x,y)\, \textrm{d}x\textrm{d}y + \int \limits _{\mathbb {R}}V_0(x)u^2(x)\, \textrm{d}x \right) \end{aligned}$$

is positive.

Moreover, we consider $f_{0}:\mathbb {R}\times \mathbb {R}\rightarrow \mathbb {R}$ is a continuous $1-$periodic function in x, which has critical exponential growth in s, that is,

$$\begin{aligned} \lim _{|s|\rightarrow +\infty }f_{0}(x,s)e^{-\alpha s^{2}}={\left\{ \begin{array}{ll} 0, &{} \text{ for } \text{ all }\,\, \alpha > \pi , \\ +\infty , &{} \text{ for } \text{ all }\,\, \alpha < \pi ,\end{array}\right. } \end{aligned}$$

uniformly in $x \in \mathbb {R}$.

We call the attention that this notion of criticality is driven by (1.7) and it has been used in several papers involving exponential growth, see for instance [13, 28]. Since we are interested in the existence of nonnegative solutions, we set $f_{0}(x,s)=0$ for all $(x,s)\in \mathbb {R}\times (-\infty ,0]$.

We also assume that the nonlinearity $f_0(x,u)$ satisfies the conditions

$(f_{0,1})$:

$0 \le {\lim _{t\rightarrow 0} \dfrac{f_0(x,t)}{t^{2\sigma -1}}}< M(1)$ uniformly in $x\in \mathbb {R}$;

$(f_{0,2})$:

there exists a constant $\theta >2\sigma$ such that

$$\begin{aligned} 0 < \theta F_{0}(x,s): = \theta \int \limits _0^s f_0(x,t)\, \textrm{d} t \le sf_{0}(x,s) \quad \text {for all}\quad (x,s)\in \mathbb {R}\times (0,+\infty ); \end{aligned}$$

$(f_{0,3})$:

for each fixed $x\in \mathbb {R}$, the function $f_{0}(x,s)/s^{2\sigma -1}$ is increasing with respect to $s \in \mathbb {R}$;

$(f_{0,4})$:

there are constants $p>2\sigma$ and $C_{p}>0$ (to be shown precisely later) such that

$$\begin{aligned} f_{0}(x,s)\ge C_{p}s^{p-1}, \quad \text {for all}\quad (x,s)\in \mathbb {R}\times [0,+\infty ). \end{aligned}$$

Here, we define

$$\begin{aligned} X_0:=\left\{ u\in L^2(\mathbb {R});\ (u(x)-u(y))K(x,y)^{\frac{1}{2}}\in L^2(\mathbb {R}^2) \right\} \end{aligned}$$

which is endowed with norm

$$\begin{aligned} \Vert u\Vert _{X_{0}} =\left( [u]_{1/2,K}^2+ \int \limits _{\mathbb {R}}V_{0}(x)u(x)^2 \,\textrm{d}x\right) ^{1/2} \end{aligned}$$

where

$$\begin{aligned}_{1/2,K}^{2}=\int \limits _{\mathbb {R}^{2}}(u(x)-u(y))^2K(x,y) \, \textrm{d}x \, \textrm{d}y.\end{aligned}$$

We would like to point out that space $X_0$ has suitable properties which give to problem (1.9) a variational framework. More specifically, in light of results proved in [14], even with more general hypotheses under the potential, $X_0$ is uniformly convex Banach space and therefore is a reflexive space. Moreover, $C^{\infty }_{0}(\mathbb {R})$ is dense in $X_0$.

Throughout this paper, we say that $u\in X_{0}$ is a weak solution for (1.9) if the following equality holds:

$$\begin{aligned} m(\Vert u\Vert _{X_0}^{2})\langle u,v \rangle _{0}=\int \limits _{\mathbb {R}} f_{0}(x,u)v \, \textrm{d}x, \end{aligned}$$

for all $v\in X_{0}$ with

$$\begin{aligned} \langle u,v\rangle _{0}:=\int \limits _{\mathbb {R}^{2}}(u(x)-u(y))(v(x)-v(y))K(x,y) \, \textrm{d}x \, \textrm{d}y + \int \limits _{\mathbb {R}} V_{0}(x)uv \, \textrm{d}x. \end{aligned}$$

The main results of this subsection are presented in the following. The first result of the paper involves a classical assumption under the nonlinearity (see assumption $(f_{0,4})$ above) which was first introduced by Cao [22].

Theorem 1.1

Assume that ($m_1$)–($m_3$), ($K_1$)–($K_3$), $(V_{0,1})$,$(V_{0,2})$ and $(f_{0,1})$–$(f_{0,4})$ hold. Then (1.9) has a nonnegative and nontrivial solution.

The second theorem of the paper also deals with the critical growth nonlinearity but involves a little weaker assumption addressed by Adimurthi and Miyagaki [24, 29] instead of Cao assumption $(f_{0,4})$. We assume that

$({\tilde{f}_{0,4}})$:

$\lim _{t \rightarrow \infty }\dfrac{f_0(x,t)t}{\exp ({\pi t^{2}})}=+\infty ,\ \text {uniformly in}\ x.$

In order to establish an existence result of a solution to problem (1.9), without the hypothesis $(f_{0,4})$, in addition to the hypothesis $({\tilde{f}_{0,4}})$ we need the following hypothesis additional under kernel

$(K_4)$:

There are $x_0,r, \lambda _0\in \mathbb {R}$ such that if $x\in (x_0-r,x_0+r)$ then $K(x)\le \dfrac{1}{\lambda _0}|x|^{-2}.$

The second result says that

Theorem 1.2

Assume that ($m_1$)–($m_3$), ($K_1$)-$(K_1)$, $(V_{0,1})$,$(V_{0,2})$, $(f_{0,1})$–$(f_{0,3})$ and $({\tilde{f}_{0,4}})$ hold. Then (1.9) has a nonnegative and nontrivial solution.

It is important to stress that it is possible to obtain different solutions for (1.9) which depend on the kind of hypothesis assumed to estimate the minimax level of functional associated to this problem. For details, see Remark 5.4.

1.2 A nonperiodic problem

The second problem that we will study in this paper is the following,

$$\begin{aligned} \left\{ \begin{array}{ll} [-m(\Vert u\Vert ^2)]\mathcal {L_{K}}u +V(x)u=f(x,u)\quad \text{ in }\quad \mathbb {R},\\ u\in X_{1}\quad \text{ and }\quad u \ge 0.\\ \end{array} \right. \end{aligned}$$

(1.12)

Note that the terms V(x) and f(x, u) are not necessarily periodic anymore. Here we deal with the class of asymptotically periodic functions that were introduced by Lins and Silva [30]. Precisely, we introduce the set

$$\begin{aligned} \mathcal {F}:=\{g\in C(\mathbb {R})\cap L^{\infty }(\mathbb {R}):\left| \{x\in \mathbb {R}:|g(x)|\ge \varepsilon \}\right| <\infty , \text{ for } \text{ all } \varepsilon >0\}, \end{aligned}$$

where |A| denotes the Lebesgue measure of a set A. In order to deal with the difficulties imposed by the lack of periodicity, we require assumptions that compare the periodic terms with the asymptotically periodic terms. On the potential V(x), we assume that:

$(v_1)$:

$V_{0}-V\in \mathcal {F}$ and $V_0(x)\ge V(x)\ge -v_{0}$, for all $x\in \mathbb {R}$;

$(v_2)$:

The infimum

$$\begin{aligned} \xi _1:= \inf _{\begin{array}{c} u\in X_{1} \\ \Vert u \Vert _2=1 \end{array}}\left( \,\, \int \limits _{\mathbb {R}^2}|u(x)-u(y)|^2 K(x,y)\,\textrm{d}x\textrm{d}y +\int \limits _{\mathbb {R}} V(x)u^2\,\textrm{d}x \right) \end{aligned}$$

is positive.

We assume that the nonlinearity $f:\mathbb {R}\times \mathbb {R}\rightarrow \mathbb {R}$ is a continuous function that have critical exponential growth, $f(x,s)=0$ for all $(x,s)\in \mathbb {R}\times (-\infty ,0]$ and satisfies the following conditions

$(f_1)$:

$f(x,s)\ge f_{0}(x,s)$ for all $(x,s)\in \mathbb {R}\times [0,+\infty )$, and for all $\varepsilon >0$, there exists $\nu >0$ such that for $s\ge 0$ and $|x|\ge \nu$,

$$\begin{aligned} |f(x,s)-f_{0}(x,s)|\le \varepsilon e^{\alpha _{0} s^{2}}; \end{aligned}$$

$(f_2)$:

$0 \le {\lim _{t\rightarrow 0} \dfrac{f(x,t)}{t^{2\sigma -1}}}< M(1)$ uniformly in $x\in \mathbb {R}$;

$(f_3)$:

there exists a constant ${\tilde{\theta }}\ge \theta > 2{\sigma }$ such that

$$\begin{aligned} 0 < {\tilde{\theta }} F(x,s): = {\tilde{\theta }}\int \limits _0^s f(x,t)\, \textrm{d} t \le sf(x,s), \quad \text {for all}\quad (x,s)\in \mathbb {R}\times (0,+\infty ); \end{aligned}$$

$(f_4)$:

for each fixed $x\in \mathbb {R}$, the function $f(x,s)/s^{2\sigma -1}$ is increasing with respect to $s \in \mathbb {R}$;

$(f_5)$:

at least one of the nonnegative continuous functions $V_{0}(x)-V(x)$ and $f(x,s)-f_{0}(x,s)$ is positive on a set of positive measure.

In order to define the weak solution to problem (1.12), as in problem (1.9), we consider

$$\begin{aligned} X_1:=\left\{ u\in L^2(\mathbb {R});\ (u(x)-u(y))K(x,y)^{\frac{1}{2}}\in L^2(\mathbb {R}^2) \right\} \end{aligned}$$

which is a Hilbert space endowed with the inner product

$$\begin{aligned} \langle u,v \rangle _{1}:=\int \limits _{\mathbb {R}^{2}}(u(x)-u(y))(v(x)-v(y))K(x,y) \, \textrm{d}x \, \textrm{d}y + \int \limits _{\mathbb {R}} V(x)uv \, \textrm{d}x \end{aligned}$$

and the correspondent induced norm $\Vert u\Vert _{X_1}^{2}=\langle u,u\rangle$.

We would like to point out that space $X_1$ also has suitable properties which give to problem (1.12) a variational framework.

Throughout this paper, we say that $u\in X_1$ is a weak solution for (1.12) if the following equality holds:

$$\begin{aligned} m(\Vert u\Vert _{X_1}^{2})\langle u,v \rangle _{1}=\int \limits _{\mathbb {R}} f(x,u)v \, \textrm{d}x, \end{aligned}$$

for all $v\in X_1$.

Considering the functions $V_0, f_0$ and m, K as in Theorems 1.1 and 1.2, the main result of this subsection is the following.

Theorem 1.3

Assume that $(v_1)$,$(v_2)$ and $(f_1)$–$(f_5)$ hold. Then (1.12) has a nonnegative and nontrivial solution.

Remark 1.4

As mentioned earlier, the results of this paper were motivated by the works [5, 13, 14, 21]. Particularly, our Theorems 1.1–1.3 are generalization of Theorems 1.1 and 1.2 of [5], in the sense of the operator, the Kirchhoff term and the nonlinearity. Consequently, we improve the results in [13]. Moreover, Theorems 1.1–1.2 are versions of Theorem 1.1 in [14] for a integrofferential Kirchhoff equation and we consider a nonlinearity with critical growth and two kinds of assumptions. We also improve some results in [21], because we assume more general hypotheses in order overcome the loss of compactness.

The outline of this paper is as follows: Sect. 2 contains some preliminary results necessary to obtain suitable properties for the solutions spaces. In Sects. 3–5, we approach results related to the periodic problem. More specifically, in Sect. 3, we work with its variational formulation. In Sect. 4, we estimate the minimax level of associated functional, and in Sect. 5 deal with the proof of the main results. Lastly, in Sect. 6, we are concerned with the results related to the nonperiodic problem for the proof of the main theorem.

2 Some preliminary results

We recall the definition of the fractional Sobolev space

$$\begin{aligned} H^{1/2}(\mathbb {R})=\left\{ u\in L^{2}(\mathbb {R}):\int \limits _{\mathbb {R}^{2}}\frac{|u(x)-u(y)|^{2}}{|x-y|^{2}}\;\textrm{d} x\textrm{d} y<\infty \right\} , \end{aligned}$$

which is endowed with the natural norm

$$\begin{aligned} \Vert u\Vert _{1/2}=\left( [u]_{1/2}^{2}+\int \limits _{\mathbb {R}}u^{2}\;\textrm{d} x\right) ^{1/2}. \end{aligned}$$

Lemma 2.1

Assume the conditions $(V_{0,1})$–$(V_{0,2})$ or $(v_1)$–$(v_2)$ and ($K_1$)–($K_3$). The space $X_{i}$ is embedded in $H^{1/2}(\mathbb {R})$ and there exists $C(\lambda ,\xi _i)>0$ such that

$$\begin{aligned}\Vert u\Vert _{1/2}\le \sqrt{C(\lambda ,\xi _i)} \Vert u\Vert _{X_{i}},\ \forall \ u\in X_i\quad \text{ with } \quad i=0,1.\end{aligned}$$

Proof

By ($K_2$), given $u\in X_i$ we have

$$\begin{aligned} \dfrac{1}{2\pi }\int \limits _{\mathbb {R}^{2}}\dfrac{|u(x)-u(y)|^{2}}{|x-y|^{2}}\, \textrm{d}x\, \textrm{d}y \le \dfrac{1}{\lambda }\int \limits _{\mathbb {R}^{2}}(u(x)-u(y))^{2}K(x,y)\,\textrm{d}x \, \textrm{d}y. \end{aligned}$$

(2.1)

Let $u\in X_i$ with $u\ne 0$, then by $(V_{0,2})$ or $(v_2)$ we have

$$\begin{aligned} \xi _i\le \left\| \frac{u}{\Vert u\Vert _2} \right\| _{X_i}^{2} = \frac{\Vert u\Vert _{X_i}^{2}}{\Vert u\Vert _{2}^{2}}. \end{aligned}$$

So

$$\begin{aligned} \Vert u\Vert _2^2\le \frac{1}{\xi _i}\Vert u\Vert _{X_i}^{2},\quad \text {for all}\ u\in X_i. \end{aligned}$$

(2.2)

From the estimates (2.1) and (2.2) we obtain

$$\begin{aligned} \Vert u\Vert _{1/2}^{2}\le C(\lambda ,\xi _i)\Vert u\Vert _{X_i}^{2},\quad \text {for all}\ u\in X_i, \end{aligned}$$

where $C(\lambda ,\xi _i) =\dfrac{1}{\xi _i} +\dfrac{1}{\lambda }$. $\square$

Corollary 2.2

Let $q\in [2,+\infty )$, then the embedding $X_i\hookrightarrow L^q(\mathbb {R})$ is continuous with $i=0,1$. Moreover, if $q\in [1,2]$ the embedding $X_i\hookrightarrow L^q_{loc}(\mathbb {R})$ is compact with $i=0,1$.

As a consequence of Corollary 2.2, the norms $\Vert \cdot \Vert _{X_0}$ and $\Vert \cdot \Vert _{X_1}$ are equivalent.

Since we have the above results hold, following the same ideas as in [14, Lemma 2.4], we can obtain that $C_{0}^{\infty }(\mathbb {R})$ is dense in $X_i$ for $i=0,1$.

Now we show a suitable version of Trudinger–Moser inequality for $X_i$ with $i=0,1.$

Lemma 2.3

Assume ($K_1$)–($K_3$) and $(V_{0,1})$–$(V_{0,2})$ or $(v_1)$–$(v_2)$, then there exists $\omega$ such that if $0< \alpha \le \omega$, then one has a constant $C=C(\omega )>0$, such that

$$\begin{aligned} \sup _{\begin{array}{c} u \in X_i\\ \Vert u\Vert _{X_i} \le 1 \end{array}}\int \limits _{\mathbb {R}}(e^{\alpha u^{2}}-1) \, \textrm{d} x \le C(\omega ) \quad \text{ for }\quad i=0,1. \end{aligned}$$

(2.3)

Moreover, for any $\alpha >0$ and $u \in X_i$, for $i=0,1$, we have

$$\begin{aligned} \int \limits _{\mathbb {R}}(e^{\alpha u^{2}}-1) \, \textrm{d} x < \infty . \end{aligned}$$

Proof

First of all, fix $u\in X_i$ with $\Vert u\Vert _{X_i}\le 1.$ Now, consider $C(\lambda ,\xi _i)$, given in Lemma 2.1, and define

$$\begin{aligned} v=\dfrac{u}{\sqrt{C(\lambda ,\xi _i)}}, \end{aligned}$$

consequently, v is in $X_i$ and $H^{{1}/{2}}(\mathbb {R})$. So, using Lemma 2.1

$$\begin{aligned} \Vert v\Vert _{1/2}= \dfrac{\Vert u\Vert _{1/2}}{\sqrt{C(\lambda ,\xi _i)}} \le \Vert u\Vert _{X_{i}}\le 1. \end{aligned}$$

(2.4)

Applying (2.4) in (1.8)

$$\begin{aligned} \int \limits _{\mathbb {R}}(e^{\alpha v^{2}}-1)\, \textrm{d} x \le C(\alpha )\Vert v\Vert _{2}^{2}, \quad \text{ for } \text{ all } \quad 0<\alpha \le \pi . \end{aligned}$$

Set $0<\alpha \le {\omega _i}=\dfrac{\pi }{C(\lambda ,\xi _i)}$ and ${\tilde{\alpha }}=\alpha C(\lambda ,\xi _i)$, and notice that $0<{\tilde{\alpha }}\le \pi .$ By Corollary 2.2 and (2.4)

$$\begin{aligned} \int \limits _{\mathbb {R}}(e^{\alpha u^{2}}-1)\, \textrm{d} x = \int \limits _{\mathbb {R}}(e^{{\tilde{\alpha }} v^{2}}-1)\, \textrm{d} x \le C({\tilde{\alpha }})\Vert v\Vert _{2}^{2} \le C(\alpha , \lambda )\Vert v\Vert ^2_{X_i}\le C(\alpha , \lambda ). \end{aligned}$$

So we obtain

$$\begin{aligned} \sup _{\begin{array}{c} u \in X_i\\ \Vert u\Vert _{X_i} \le 1 \end{array}}\int \limits _{\mathbb {R}}(e^{\alpha u^{2}}-1) \, \textrm{d} x \le C({\omega _i}). \end{aligned}$$

Choosing $\omega = \min \{\omega _0,\omega _1\}$, we obtain (2.3).

Now let us take $\alpha >0, u\in X_i$ and $\varepsilon >0$. There exists $\phi \in C_0^{\infty }(\mathbb {R})$ such that $\Vert u-\phi \Vert _{X_i}<\varepsilon$. Observe that

$$\begin{aligned} e^{\alpha |u|^2}-1\le e^{2\alpha |u-\phi |^2} e^{2\alpha |\phi |^2}-1\le \frac{1}{2}\left( e^{4\alpha |u-\phi |^2}-1\right) + \left( \frac{1}{2} e^{4\alpha |\phi |^2}-1\right) . \end{aligned}$$

Then,

$$\begin{aligned} \int \limits _{\mathbb {R}}(e^{\alpha |u|^2}-1) \textrm{d} x\le \frac{1}{2}\int \limits _{\mathbb {R}}(e^{4\alpha \Vert u-\phi \Vert _{X_i}^2}-1) \textrm{d} x+ \frac{1}{2}\int \limits _{\mathbb {R}} (e^{4\alpha |\phi |^2}-1) \textrm{d} x. \end{aligned}$$

(2.5)

Choosing $\varepsilon >0$ such that $4\alpha \varepsilon ^2<\omega$, we have $4\alpha \Vert u-\phi \Vert _{X_i}^2<\omega$. By (2.5) and (2.3), we conclude that

$$\begin{aligned} \int \limits _{\mathbb {R}}(e^{\alpha |u|^2}-1) \textrm{d} x\le \dfrac{C}{2}+ \frac{1}{2}\int \limits _{Supp(\phi )} (e^{4\alpha |\phi |^2}-1) \textrm{d} x < \infty . \end{aligned}$$

$\square$

Lemma 2.4

If $\alpha >0$, $q>2$, $v \in X_i$ and $\Vert v\Vert _{X_i}\le D$ with $\alpha D^{2}<\omega$, then there exists $C=C(\alpha ,D,q)>0$, such that

$$\begin{aligned} \int \limits _\mathbb {R}(e^{\alpha v^{2}}-1)|v|^{q}\textrm{d}x \le C\Vert v\Vert _{X_i}^{q}\quad \text{ for }\quad i=0,1. \end{aligned}$$

Proof

Consider $r > 1$ sufficiently close to 1 such that $\alpha r D^{2}<\omega$ and $r'q \ge 2$, where $r'={r}/{(r-1)}$. Using Hölder’s inequality, we have

$$\begin{aligned} \int \limits _\mathbb {R}(e^{\alpha v^{2}}-1)|v|^{q}\textrm{d}x \le \left( \,\,\int \limits _{\mathbb {R}}(e^{\alpha v^{2}}-1)^{r}\textrm{d} x \right) ^{1/r}\Vert v\Vert _{r'q}^{q}. \end{aligned}$$

(2.6)

Notice that given $\beta >r$ there exists $C=C(\beta ) >0$ such that for all $s \in \mathbb {R}$,

$$\begin{aligned} (e^{\alpha s^{2}}-1)^{r} \le C(e^{\alpha \beta s^{2}}-1). \end{aligned}$$

(2.7)

Hence, from (2.6) and (2.7) we get

$$\begin{aligned} \int \limits _{\mathbb {R}}(e^{\alpha v^{2}}-1)|v|^{q}\textrm{d}x\le & {} C \left( \int \limits _\mathbb {R}(e^{\alpha \beta v^{2}}-1)\textrm{d} x\right) ^{1/r}\Vert v\Vert _{r'q}^{q}\\\le & {} C \left( \,\,\int \limits _{\mathbb {R}}\left[ e^{\alpha \beta D^{2} \left( \frac{v}{\Vert v\Vert _{X_i}}\right) ^{2}}-1\right] \,\textrm{d}x \right) ^{1/r} \Vert v\Vert _{r'q}^q. \end{aligned}$$

By choosing $\beta >r$ close to r, in such way that $\alpha \beta D^{2}<\omega$, it follows from (2.3) and the continuous embedding $X_i \hookrightarrow L^{r'q}(\mathbb {R})$ that

$$\begin{aligned} \int \limits _{\mathbb {R}}(e^{\alpha v^{2}}-1)|v|^{q}\,\textrm{d}x \le C\Vert v\Vert _{X_i}^q, \end{aligned}$$

which proves the lemma. $\square$

3 A functional setting for the periodic problem

In order to use a variational framework considering the space $X_0$, we assume suitable conditions such that weak solutions of (1.9) become critical points of the Euler functional $I_{0}: X_0 \rightarrow \mathbb {R}$ defined by

$$\begin{aligned} I_{0}(u)=\frac{1}{2}M(\Vert u\Vert _{X_0}^{2})-\int \limits _{\mathbb {R}} F_{0}(x,u)\, \textrm{d} x \end{aligned}$$

(3.1)

where ${F_{0}(x,t)=\int _0^t f_{0}(x,\tau )\;\textrm{d} \tau }$. Notice that by the condition $(f_{0,1})$ and the fact that $f_0(x,s)$ has critical exponential growth, for each $\alpha >\pi$, $q>2$ and $\varepsilon >0$ there exists $C_{\varepsilon }>0$ such that

$$\begin{aligned} f_{0}(x,t)\le (M(1)-\varepsilon )t{^{2\sigma -1}}+C_{\varepsilon }(e^{\alpha t^{2}}-1)|t|^{q-1}, \quad \text{ for } \text{ all } \hspace{0,2cm} (x,t)\in \mathbb {R}^{2}, \end{aligned}$$

(3.2)

which implies that

$$\begin{aligned} F_{0}(x,t)\le \frac{(M(1)-\varepsilon )}{2}|t|^{2{\sigma }}+C_{\varepsilon }(e^{\alpha t^{2}}-1)|t|^{q}, \quad \text{ for } \text{ all } \hspace{0,2cm} (x,t)\in \mathbb {R}^{2}. \end{aligned}$$

(3.3)

By using the above estimate jointly with the continuous embedding $X_{0}\hookrightarrow L^{q}(\mathbb {R})$, we can conclude that $I_{0}$ is well defined. Moreover, using standard arguments we can check that $I_{0}\in C^{1}(X_{0},\mathbb {R})$ with the derivative given by

$$\begin{aligned} I^{\prime }_{0}(u)v=m(\Vert u\Vert _{X_0}^{2})\langle u,v\rangle _{0} -\int \limits _{\mathbb {R}}f_{0}(x,u)v\,\textrm{d}x,\quad \forall v\in X_0. \end{aligned}$$

Thus, critical points of $I_{0}$ are weak solutions of problem (1.9) and conversely.

3.1 The geometric condition

Next using the hypotheses $(f_{0,1})$ and $(f_{0,2})$, we prove some facts about the geometric structure of $I_0$ required by the minimax procedure.

Lemma 3.1

There exist $\mu >0$ and $\varrho >0$ such that $I_{0}(u)\ge \mu$, provided that $\Vert u\Vert _{X_0}=\varrho$.

Proof

We can use (3.3) to get

$$\begin{aligned} \int \limits _{\mathbb {R}}F_{0}(x,u)\, \textrm{d}x \le \frac{(M(1)-\varepsilon )}{2}\int \limits _{\mathbb {R}}|u|^{2{\sigma }}\, \textrm{d}x +C_{\varepsilon }\int \limits _{\mathbb {R}}(e^{\alpha u^{2}}-1)|u|^{q} \, \textrm{d}x. \end{aligned}$$

By continuous embedding $X_0\hookrightarrow L^{q}(\mathbb {R})$ for all $q\in [2,+\infty )$ and Hölder inequality, for $\Vert u\Vert _{X_{0}}\le \rho _0<1$ we obtain

$$\begin{aligned} \int \limits _{\mathbb {R}}F_{0}(x,u)\, \textrm{d}x \le \frac{(M(1)-\varepsilon )}{2}\Vert u\Vert _{X_0}^{2{\sigma }}+C_{\varepsilon }\Vert u\Vert _{X_0}^{q}\left[ \int \limits _{\mathbb {R}}(e^{2\alpha \rho _0^2 u^{2}/\Vert u\Vert _{X_{0}}^{2}}-1)\, \textrm{d}x\right] ^{1/2}. \end{aligned}$$

(3.4)

From (1.10) and (3.4), we have

$$\begin{aligned} I_0(u)\ge \displaystyle \dfrac{M(1)}{2}\Vert u\Vert _{X_0}^{2\sigma }- \frac{(M(1)-\varepsilon )}{2}\Vert u\Vert _{X_0}^{2{\sigma }}-C_{\varepsilon }\Vert u\Vert _{X_0}^{q}\left[ \int \limits _{\mathbb {R}}(e^{2\alpha \rho _0^2 u^{2}/\Vert u\Vert _{X_{0}}^{2}}-1)\, \textrm{d}x\right] ^{1/2}. \end{aligned}$$

If $\rho _0< \min \{1,\sqrt{\omega / 2\alpha }\}$, we obtain

$$\begin{aligned} \begin{array}{lcr} I_0(u)\ge & {} \displaystyle \frac{\varepsilon }{2}\Vert u\Vert _{X_0}^{2{\sigma }}-C_{2}\Vert u\Vert _{X_0}^{q}. \end{array} \end{aligned}$$

Choosing $q> 2\sigma$ and $\varepsilon >0$ small enough, we may choose $0<\varrho < \rho _0$ such that

$$\begin{aligned} \displaystyle \frac{\varepsilon }{2}\varrho ^{2{\sigma }}-C_{2}\varrho ^{q}= \mu >0.\end{aligned}$$

(3.5)

$\square$

Lemma 3.2

There exists $e\in X_{0}$ with $\Vert e\Vert _{X_0}>\varrho$ such that $I_0(e)<0$.

Proof

Let $u\in C_0^\infty (\mathbb {R})\backslash \{0\}$ with support $\Omega$. By $(f_{0,2})$ there exist $C_1,C_2>0$ such that

$$\begin{aligned} F_{0}(x,u)\ge C_{1}|u|^{\theta }-C_{2}\quad \text{ for } \text{ all }\quad x\in \Omega . \end{aligned}$$

Consequently, by (1.11), for $t>0$ such that $\Vert tu\Vert ^2_{X_0}\ge 1$, we have the following estimate

$$\begin{aligned} I_{0}(tu)\le & {} \dfrac{t^{2}M(1)}{2}\Vert u\Vert _{X_0}^{2\sigma }- C_{1} t^{\theta } \int \limits _{\Omega }|u|^{\theta } \, \,d x + C_{2} \int \limits _{\Omega } \,d x. \end{aligned}$$

Since $\theta >2\sigma$, we obtain $I_{0}(tu) \rightarrow - \infty$ as $t \rightarrow \infty$. Setting $e=tu$ with t large enough, the proof is finished.

$\square$

3.2 Palais–Smale sequence

By using the mountain-pass theorem without the (PS) condition (see [31]), there exists a sequence $(u_k)$ in $X_0$ satisfying

$$\begin{aligned} I_{0}(u_{k})\rightarrow c_{0} \quad \text {and} \quad I'_{0}(u_{k})\rightarrow 0, \end{aligned}$$

(3.6)

where

$$\begin{aligned} c_0=\inf _{g\in \Gamma }\max _{t\in [0,1]} I_0(g(t)) \end{aligned}$$

and $\Gamma =\{g\in C([0,1], X_0): g(0)=0\,\, \text{ and }\,\, g(1)=e\}$.

Lemma 3.3

Suppose that $(f_{0,1})$ and $(f_{0,2})$ hold. Then, the sequence $(u_{k})$ is bounded in $X_0$.

Proof

Using well-known arguments, it is not difficult to check that $(u_{k})$ is a bounded sequence in $X_0$. Indeed, by ($m_1$) and $(f_{0,2})$ we have

$$\begin{aligned} \begin{aligned} I_{0}(u_{k})-\dfrac{1}{\theta }I'_{0}(u_{k})u_{k} \ge \left( \dfrac{1}{2}-\dfrac{\sigma }{\theta }\right) M(\Vert u_{k}\Vert _{X_0}^{2}). \end{aligned} \end{aligned}$$

Now we have to consider two cases. Either $\inf _{k\in \mathbb {N}}\Vert u_k\Vert _{X_0}=d>0$ or $\inf _{k\in \mathbb {N}}\Vert u_k\Vert _{X_0}= 0$. If $\inf _{k\in \mathbb {N}}\Vert u_k\Vert _{X_0}=d>0$, we may assume that d does not depend on Palais–Smale sequence considered, from ($m_1$) and ($m_2$), with $\tau = d^2$, there exists $\eta _0>0$ such that

$$\begin{aligned} M(\Vert u_k\Vert _{X_0}^2)\ge \eta _0 \Vert u_k\Vert _{X_0}^2 \quad \text{ for } \text{ all } \quad k \in \mathbb {N}. \end{aligned}$$

So we have

$$\begin{aligned} I_{0}(u_{k})-\dfrac{1}{\theta }I'_{0}(u_{k})u_{k}\ge \eta _0 \left( \dfrac{1}{2}-\dfrac{\sigma }{\theta }\right) \Vert u_k\Vert _{X_0}^2 \quad \text{ for } \text{ all } \quad k \in \mathbb {N}. \end{aligned}$$

(3.7)

By (3.6) and (3.7), there exists $k_0 \in \mathbb {N}$ such that for all $k \ge k_0$, it holds

$$\begin{aligned} \left( \dfrac{1}{2}-\dfrac{\sigma }{\theta }\right) \eta _0\Vert u_{k}\Vert _{X_0}^{2} \le C+\Vert u_{k}\Vert _{X_0}. \end{aligned}$$

Since $\theta > 2\sigma$, this implies that $\Vert u_{k}\Vert _{X_0}\le C_1$. If $\inf _{k\in \mathbb {N}}\Vert u_k\Vert _{X_0}=0$, when 0 is an accumulation point for real sequence $(\Vert u_k\Vert _{X_0})$, we may conclude that $u_0=0$, so $0=I_0(u_0)=c>0$, which is impossible. Consequently, 0 is an isolated point of sequence $(\Vert u_k\Vert _{X_0})$, then there exists a subsequence, denoted also by $(\Vert u_k\Vert _{X_0})$, such that $\inf _{k\in \mathbb {N}}\Vert u_k\Vert _{X_0}=d>0$ and we may proceed as before. Thus, in both cases, we have that this sequence is bounded. $\square$

4 Minimax level for the periodic problem

As already mentioned, we highlight that the main difficulty in our work is the lack of compactness typical for elliptic problems in unbounded domains with nonlinearities with critical growth. To recover this, we will make use of assumptions $(f_{0,4})$ or $({\tilde{f}_{0,4}})$ together with $(K_1)$ to control the minimax level in a suitable range where we are able to recover some compactness. For this purpose, in the first case, we need a version of Lions’s lemma. In the second case, let us consider the Moser’s functions sequence supported in a ball with an appropriated radius, which depends on $(K_1)$. Besides this, we lead with a general Kirchhoff function M(t), which become this difficulty harder. To overcome this obstacle, we need to make some estimates depending on this term.

For this, observe that, by definition, $M(0)=0$, and since m is a continuous function, it follows that M is continuous. Thus, by theses facts, together with Lemma 3.1, there exists $0< \varrho '<\min \{1, \rho \}$ such that

$$\begin{aligned} I_0 (t)\le \dfrac{\mu }{2} \quad \text{ for } \text{ all } \quad 0\le t<\rho '. \end{aligned}$$

(4.1)

By ($m_1$),

$$\begin{aligned} M(t) \le C_{\sigma } t^{\sigma } \quad \text{ for } \text{ all } \quad t\ge \rho ', \quad \text{ where }\quad C_{\sigma }=\dfrac{M(\rho ')}{(\rho ')^{\sigma }}. \end{aligned}$$

(4.2)

Now, observe that as a consequence of Lemmas 3.1 and 3.2, the minimax level

$$\begin{aligned} c_0=\inf _{g\in \Gamma }\max _{t\in [0,1]} I_0(g(t)) \end{aligned}$$

is positive.

Moreover, we may compare the minimax levels relying on $(f_{0,4})$ or $({\tilde{f}_{0,4}})$ and $(K_1)$ and show that they are real numbers different. So we may obtain distinct solutions for Problem (1.9).

4.1 Minimax estimative of Theorem 1.1

In order to provide an estimate to the minimax level of the functional associated to (1.9), in [14], the authors proved a version of a Lions’s result (see Lions [32]) for critical growth in $\mathbb {R}$, more specifically in [14, Lemma 3.1], for a bounded sequence in a suitable space they guarantee that if

$$\begin{aligned} \lim _{n \rightarrow \infty }\sup _{y\in \mathbb {R}}\int \limits _{B_{R}(y)}|u_{n}(x)|^{2}\, \textrm{d}x=0, \end{aligned}$$

(4.3)

for some $R>0$, then $u_{n}\rightarrow 0$ strongly in $L^{q}(\mathbb {R})$ for $2<q<\infty$. This result is also available for bounded sequence in $X_0$.

Now we consider the embedding constant, given by

$$\begin{aligned} S_{p}:=\inf _{u\in X_0\atop {\Vert u\Vert _p=1}}\left( \ \displaystyle {\int \limits _{\mathbb {R}^{2}}{|u(x)-u(y)|^{2}}K(x,y)\, \textrm{d}x \textrm{d}y+ \int \limits _{\mathbb {R}} V_0(x) u^{2}(x)\, \textrm{d}x}\right) ^{1/2}, \end{aligned}$$

which is achieved by a nonnegative function $u_p$ in $X_0$. For more details see [14, Lemma 3.2]. From this, we may estimate the level.

Proposition 4.1

Suppose that $(f_{0,4})$ holds. Then

$$\begin{aligned} \mu \le c_0< \frac{\eta _0 (\theta -2\sigma ) w}{2\theta \pi }. \end{aligned}$$

Proof

Let $u_p\in X_0$ such that $\Vert u_p\Vert _{X_0}^2=S_p$ and $\Vert u_p\Vert _p=1$. Then

$$\begin{aligned} c_0\le \max _{t\ge 0}I_0(tu_p) = \max _{t\ge 0}\left\{ \frac{1}{2}M(t^2\Vert u_p\Vert _{X_0}^2)-\int \limits _{\mathbb {R}}F_0(x,tu_p)\textrm{d} x \right\} . \end{aligned}$$

When

$$\begin{aligned} t < \frac{\sqrt{\rho '}}{S_p}=\rho _p, \end{aligned}$$

(4.4)

we have $\Vert t u_p\Vert _{X_0}^2<\rho '.$ So, using the estimate (4.1), we get

$$\begin{aligned} I_0(t u_p)\le \frac{\mu }{2}. \end{aligned}$$

By (4.2) and $(f_{0,4})$, we obtain

$$\begin{aligned} \begin{aligned} c_0&\le \max _{t\ge \rho _p}\left\{ \frac{1}{2}M(t^2\Vert u_p\Vert _{X_0}^2)-\int \limits _{\mathbb {R}}F_0(x,tu_p)\textrm{d} x\right\} \\&\le \max _{t\ge \rho _p}\left\{ \frac{1}{2}C_{\sigma }t^{2\sigma }S_p^{2\sigma }-\frac{C_p}{p}t^p\right\} .\end{aligned} \end{aligned}$$

Observe that $t_0=\left( \frac{\sigma C_{\sigma }S_{p}^{2\sigma }}{C_p}\right) ^{\frac{1}{p-2\sigma }}$, where the maximum is achieved, satisfies the estimate (4.4) if $C_p<\dfrac{\sigma C_{\sigma }S_{p}^p}{(\rho ')^{\frac{p-2\sigma }{2}}}$, what occurs as $\rho ' < C_{\sigma }^{\frac{1}{\sigma }}\left( \dfrac{pw\eta _0(\theta -2\sigma )}{(p-2\sigma )\theta \pi }\right) ^{\frac{1}{\sigma }}$. Consequently,

$$\begin{aligned} \begin{aligned} c_0&\le \frac{1}{2}C_{\sigma }S_{p}^{2\sigma }\left[ \frac{\sigma C_{\sigma } S_{p}^{2\sigma }}{C_p}\right] ^{\frac{2\sigma }{p-2\sigma }} - \frac{C_p}{p}\left[ \frac{\sigma C_{\sigma } S_{p}^{2\sigma }}{C_p}\right] ^{\frac{p}{p-2\sigma }}\\&= \frac{C_{\sigma }^{\frac{p}{p-2\sigma }}S_{p}^{\frac{2p\sigma }{p-2\sigma }}\sigma ^{\frac{2\sigma }{p-2\sigma }}(p-2\sigma )}{2pC_p^{\frac{2\sigma }{p-2\sigma }}}. \end{aligned} \end{aligned}$$

Then, taking

$$\begin{aligned} C_p\ge C_{\sigma }^{\frac{p}{2\sigma }}S_{p}^p\sigma \left[ \frac{(p-2\sigma )\theta }{p\eta _0(\theta -2\sigma )w}\right] ^{\frac{p-2\sigma }{2\sigma }}, \end{aligned}$$

we have

$$\begin{aligned} c_0< \frac{\eta _0 (\theta -2\sigma ) w}{2\theta \pi }. \end{aligned}$$

$\square$

4.2 Minimax estimative of Theorem 1.2

In this section, in order to estimate the minimax level replacing $(f_{0,4})$ by $({\tilde{f}_{0,4}})$, we need the additional hypothesis $(K_1)$ under the kernel.

In order to control the minimax level when we assume a hypothesis kind of $({\tilde{f}_{0,4}})$, it is usual to consider the following sequence of nonnegative functions given by

$$\begin{aligned} v_n(y)=\left\{ \begin{array}{ll} (\ln {n})^{1/2},&{}\quad 0 \le \left| y\right| < \frac{1}{n},\\ \frac{\ln {\frac{1}{|y|}}}{(\ln {n})^{1/2}},&{}\quad \frac{1}{n}\le \left| {y}\right| \le 1,\\ 0,&{}\quad \left| {y}\right| \ge 1, \end{array}\right. \end{aligned}$$

well-known as Moser’s sequence.

By changing of variable, we obtain the following sequence of functions supported in $(x_0-r_0,x_0+r_0)$ given by

$$\begin{aligned} u_n(x)=\left\{ \begin{array}{ll} (\ln {n})^{1/2},&{}\quad 0 \le \left| {x-x_0}\right| < \frac{r_0}{n},\\ \frac{\ln {\frac{r_0}{|x-x_0|}}}{(\ln {n})^{1/2}},&{}\quad \frac{r_0}{n}\le \left| {x-x_0}\right| \le r_0,\\ 0,&{}\quad \left| {x-x_0}\right| \ge r_0, \end{array}\right. \end{aligned}$$

with $r_0=\min \left\{ r,\dfrac{\lambda _0}{2\pi }, \dfrac{\lambda _0 \eta _0(\theta -2\sigma )\omega }{2\theta C_{\sigma } \pi ^2}\right\}$. Notice that the restriction of $u_n$ to $(x_0-r_0,x_0+r_0)$ belongs to $H^{1/2}((x_0-r_0,x_0+r_0))$ (see [33]). The following lemma deals with the asymptotic estimate on Moser’s sequence.

Lemma 4.2

Suppose that $(K_1)$ holds, then there exist convergente sequences $\tilde{C}_n$ and $\delta _n$ such that

$$\begin{aligned} \Vert u_n\Vert ^2_{X_0}\le 2\pi \tilde{C}_n +\delta _n. \end{aligned}$$

Proof

Note that $\Vert (-\Delta )^{1/4} u_{n}\Vert ^2_{2}= r_0\Vert (-\Delta )^{1/4} v_{n}\Vert ^2_{2}$, so by Takahashi [18], we have

$$\begin{aligned} \Vert (-\Delta )^{1/4} u_{n}\Vert ^2_{2}\le \pi r_0\left( 1 +\frac{1}{C\ln (n)}\right) . \end{aligned}$$

From $(K_1)$,

$$\begin{aligned} \dfrac{1}{2\pi }[u_n]_{1/2,K} \le \frac{1}{\lambda _0}\Vert (-\Delta )^{1/4} u_{n}\Vert ^2_{2}\le \frac{\pi r_0}{\lambda _0}\left( 1 +\frac{1}{C\ln (n)}\right) := \widetilde{C}_n. \end{aligned}$$

(4.5)

Thus, for n large enough, we have

$$\begin{aligned} \begin{aligned} \Vert u_{n}\Vert _{X_0}^{2}&\le 2\pi \widetilde{C}_n +2{V r_0}\left[ \int \limits _{-\frac{1}{n}}^{\frac{1}{n}}\ln (n)\,\textrm{d}x+\frac{1}{\ln (n)}\int \limits _{-1}^{-\frac{1}{n}}(\ln |x|)^{2}\,\textrm{d}x\right] \\&\quad +2{V r_0}\left[ \frac{1}{\ln (n)}\int \limits _{\frac{1}{n}}^{1}(\ln |x|)^{2}\,\textrm{d}x\right] , \end{aligned} \end{aligned}$$

where $V:= \max _{x\in \mathbb {R}} V_0(x)$, which implies that $\Vert u_{n}\Vert _{X_0}^{2} {\le }2\pi \widetilde{C}_n+\delta _{n}$, with

$$\begin{aligned} \delta _{n}:=4 Vr_0\left( \frac{n-1-\ln (n)}{n\ln (n)} \right) . \end{aligned}$$

Notice that

$$\begin{aligned} \delta _n \rightarrow 0 \quad \text{ and } \quad \tilde{C}_n \rightarrow \frac{\pi r_0}{\lambda _0} \quad \text{ as } \hspace{0,2cm} n\rightarrow +\infty . \end{aligned}$$

(4.6)

$\square$

Now we considered $\omega _n=\frac{u_{n}}{\sqrt{2\pi \widetilde{C}_n+\delta _{n}}}$, so by Lemma 4.2$\Vert \omega _n\Vert _{X_0}\le 1$, which will help us to estimate the minimax level.

Proposition 4.3

Suppose that $(K_1)$ and $({\tilde{f}_{0,4}})$ are satisfied, then

$$\begin{aligned} \mu \le {c}_0 < \frac{\pi r_0 }{\lambda _0 } {C_{\sigma }}, \end{aligned}$$

where $C_{\sigma }$ is given in (4.2).

Proof

By applying Lemma 3.1, we have that ${c}_{0}\ge \mu$. In order to get an upper estimate, it is enough to prove that there exists a function $\omega \in X_0$, $\Vert \omega \Vert _{X_0}\le 1$, such that

$$\begin{aligned} \max _{t\in [0,1]} I_0(t\omega )<\frac{\pi r_0 }{\lambda _0} {C_{\sigma }}. \end{aligned}$$

Let us argue by contradiction and suppose that for all $n\in \mathbb {N}$ there exists $t_n>0$ such that

$$\begin{aligned} I_0(t_n\omega _n)=\max _{t\in [0,+\infty )} I_0(t\omega _n) \ge \frac{\pi r_0}{\lambda _0} {C_{\sigma }}. \end{aligned}$$

(4.7)

Note still, that $t_n\ge \varrho '$ for all $n\in \mathbb {N}$, with $\rho '$ is given in (4.1), because, otherwise, there exists $n_0\in \mathbb {N}$ such that $t_{n_0}<\varrho '\le 1$, then

$$\begin{aligned} \Vert t_{n_0}\omega _{n_0}\Vert ^2_{X_0}=t^2_{n_0}\Vert \omega _{n_0}\Vert ^2_{X_0}\le t^2_{n_0}< t_{n_0}<\varrho '. \end{aligned}$$

Consequently, by (4.1), $I_0(t_{n_0}\omega _{n_0})< \dfrac{\mu }{2}$. On another hand, as $c_0 \ge \mu$, we have $I_0(t_{n_0}\omega _{n_0}) \ge \mu ,$ which is a contradiction. Since $t_n\ge \varrho '$ for all $n\in \mathbb {N}$, from (4.7), we have

$$\begin{aligned} \max _{t\in [0,+\infty )} I_0(t\omega _n)= \max _{t\in [\varrho ',+\infty )} I_0(t\omega _n).\end{aligned}$$

So, by (4.2), for n sufficiently large we obtain

$$\begin{aligned} \frac{C_{\sigma }}{2}\Vert t_n\omega _n \Vert _{X_0}^{2\sigma }-\int \limits _\mathbb {R}F_0(x,t_n \omega _n)\,\textrm{d}x \ge \frac{\pi r_0}{\lambda _0 }C_{\sigma }. \end{aligned}$$

As $r_0 \le \dfrac{\lambda _0}{2\pi },$ we obtain

$$\begin{aligned} t_{n}^{2}\ge \left( \frac{2\pi r_0 }{\lambda _0}\right) ^{1/\sigma } \ge \frac{2\pi r_0}{\lambda _0}. \end{aligned}$$

(4.8)

Since $t_n$ satisfies

$$\begin{aligned} \left. \frac{d}{dt}I_0(t\omega _n)\right| _{t=t_n}=0, \end{aligned}$$

by using ($m_1$), it follows that

$$\begin{aligned} \sigma C_{\sigma } t_{n}^{2\sigma }{\ge }\int \limits _{\mathbb {R}}t_n\omega _nf_0(x,t_n\omega _n). \end{aligned}$$

(4.9)

For $n\in \mathbb {N}$ large enough, we can use (4.9) in order to obtain

$$\begin{aligned} \begin{alignedat}{2} t_{n}^{2\sigma }&\ge \frac{1}{\sigma C_\sigma }\displaystyle \int \limits _{x_0-\frac{r_0}{n}}^{x_0+\frac{r_0}{n}}\frac{t_n\sqrt{\ln n}}{\sqrt{2\pi \widetilde{C}_n+\delta _{n}}}f_0\left( x,\frac{ t_n\sqrt{\ln n}}{\sqrt{2\pi \widetilde{C}_n+\delta _{n}}}\right) \,\textrm{d}x\\&\ge \frac{2r_0}{\sigma C_\sigma }\exp \left[ \left( \frac{\pi t_n^{2}}{2\pi \widetilde{C}_n+\delta _{n} }-1\right) \ln (n)\right] \frac{f_0\left( x_1,\frac{ t_n\sqrt{\ln n}}{\sqrt{2\pi \widetilde{C}_n+\delta _{n}}}\right) }{\exp \left( \frac{\pi t_n^{2}\ln n}{2\pi \widetilde{C}_n+\delta _{n} }\right) },\\ \end{alignedat} \end{aligned}$$

(4.10)

where $x_1$ is a minimum point of $f_0$ in $[x_0-\frac{r_0}{n},x_0+\frac{r_0}{n}]$. This implies that $t_{n}^{2}$ is bounded. Moreover, from (4.6) and (4.8), we get

$$\begin{aligned} t_n^{2}\rightarrow \frac{2\pi r_0}{\lambda _0}. \end{aligned}$$

(4.11)

Thus, (4.10) together with assumption $({\tilde{f}_{0,4}})$ contradict Eq. (4.8). $\square$

Remark 4.4

It is important to point out that as $r_0 \le \dfrac{\lambda _0 \eta _0(\theta -2\sigma )\omega }{2\theta \pi ^{2} C_{\sigma }}$, we have

$$\begin{aligned} \dfrac{\pi r_0}{\lambda _0}C_{\sigma }\le \frac{\eta _0 (\theta -2\sigma ) w}{2\theta \pi }. \end{aligned}$$

So in both cases, in Propositions 4.1 and 4.3, we have

$$\begin{aligned} c_0< \frac{\eta _0 (\theta -2\sigma ) w}{2\theta \pi }.\end{aligned}$$

(4.12)

5 Existence of a solution for the periodic problem

In the Sect. 3.2, we guarantee that a Palais Smale sequence is bounded in $X_0$. Since $X_{0}$ is a Hilbert space, up to a subsequence, we can assume that there exists $u_{0}\in X_{0}$ such that

$$\begin{aligned} \left\{ \begin{array}{ll} u_{k}\rightharpoonup u_{0}\,\,\, \hbox {weakly in}\,\, X_{0}, \\ u_{k}\rightarrow u_{0} \,\,\,\hbox {in}\,\, L_{loc}^q(\mathbb {R})\,\,\,\text{ for } \text{ all }\,\, q\ge 1, \\ u_{k}(x)\rightarrow u_{0}(x) \,\,\,\hbox {almost everywhere in }\,\, \mathbb {R}. \\ \end{array} \right. \end{aligned}$$

In order to ensure that the weak limit of the Palais Smale sequence is a solution of (1.9), we need the following auxiliary results, which hold under all hypotheses already cited.

Lemma 5.1

$$\begin{aligned} \int \limits _{\mathbb {R}}f_{0}(x,u_{k})v\, \textrm{d}x \rightarrow \int \limits _{\mathbb {R}}f_{0}(x,u_{0})v\, \textrm{d}x, \quad \text{ for } \text{ all }\quad v\in C^{\infty }_{0}(\mathbb {R}). \end{aligned}$$

(5.1)

Proof

Note that combining (3.6) and (3.7), we reach

$$\begin{aligned} c_{0}\ge & {} \dfrac{(\theta -2\sigma )\eta _0}{2\theta }\,\limsup \Vert u_{k}\Vert _{X_0}^{2}. \end{aligned}$$

Thus, by Proposition 4.1 (or Proposition 4.3 ) and (4.12) we obtain

$$\begin{aligned} \limsup \Vert u_{k}\Vert _{X_0}^{2} <\dfrac{\omega }{\pi }. \end{aligned}$$

This implies $\pi \Vert u_k\Vert _{X_0}^2 < \omega$ for k enough large. Hence, we can choose $p>1$ sufficiently close to 1 and $\delta >0$ small enough such that $p (\pi + \delta ) \Vert u_k\Vert _{X_0}^2 < \omega$ for k sufficiently large. Consequently, by (2.3) there exists $C>0$ such that

$$\begin{aligned} \int \limits _{\mathbb {R}}(e^{p(\alpha _{0}+\delta )\Vert u_k\Vert _{X_0}^2 \left( \frac{u_{k}}{\Vert u_k\Vert _{X_0}}\right) ^{2}}-1)\, \textrm{d}x \le C. \end{aligned}$$

(5.2)

Combining (3.2) and Hölder’s inequality for $p'=p/(p-1) > 2$, we get

$$\begin{aligned} \begin{aligned} \int \limits _{\mathbb {R}}f_{0}(x,u_{k})u_{k}\, \textrm{d}x&\le ( M(1)-\varepsilon ) \int \limits _{\mathbb {R}} u_{k}^{2\sigma }\, \textrm{d}x+ C_{\varepsilon } \int \limits _{\mathbb {R}}(e^{(\alpha _{0}+\delta ) u_{k}^{2}}-1)|u_{k}|^{q}\, \textrm{d} x \\&\le ( M(1)- \varepsilon ) C\\&\quad + C_\varepsilon \Vert u_k\Vert _{p'q}^q\left( \,\,\int \limits _{\mathbb {R}}(e^{p(\alpha _{0}+\delta )\Vert u_k\Vert _{X_0}^2 \left( \frac{u_{k}}{\Vert u_k\Vert _{X_0}}\right) ^{2}}-1)\, \textrm{d}x\right) ^{1/p}. \end{aligned} \end{aligned}$$

(5.3)

Hence, by (5.2), we have

$$\begin{aligned} \int \limits _{\mathbb {R}}f_{0}(x,u_{k})u_{k}\, \textrm{d}x \le C. \end{aligned}$$

Consequently, thanks to Lemma 2.1 in [24], we reach

$$\begin{aligned} f_{0}(x,u_{k})\rightarrow f_{0}(x,u_{0}) \,\,\, \hbox {in}\,\,\, L_{loc}^1(\mathbb {R}), \end{aligned}$$

which implies (5.1). $\square$

Lemma 5.2

Assume $(f_{0,3})$, then for all $x\in \mathbb {R}$

$$\begin{aligned}f_0(x,t)t - 2\sigma F_0(t,x) \quad \text{ is } \text{ increasing } \text{ for } \quad t>0 \quad \text{ and } \text{ decreasing } \text{ for } \quad t<0.\end{aligned}$$

Proof

Let $0<t_1<t_2$ be fixed. By $(f_{0,3})$, it follows

$$\begin{aligned} f_0(x,t_1)t_1 - 2\sigma F_0(x,t_1)< \dfrac{f_0(x,t_2)}{t_2^{2\sigma -1}} t_1 ^{2\sigma } - 2\sigma F_0(x,t_2)+ 2\sigma \int \limits _{t_1}^{t_2}f_0(x,\kappa ) \textrm{d} \kappa .\end{aligned}$$

(5.4)

Again from $(f_{0,3})$, we obtain

$$\begin{aligned} 2\sigma \int \limits _{t_1}^{t_2}f_0(x,\kappa )\textrm{d}\kappa < 2\sigma \dfrac{f_0(x,t_2)}{t_2^{2\sigma -1}}\int \limits _{t_1}^{t_2}\kappa ^{2\sigma - 1}\textrm{d}\kappa = \dfrac{f_0(x,t_2)}{t_2^{2\sigma -1}}(t_2^{2\sigma }- t_1^{2\sigma }). \end{aligned}$$

(5.5)

Combining (5.4) and (5.5), we get

$$\begin{aligned}f_0(x,t_1)t_1- 2\sigma F_0(x, t_1)< f_0(x,t_2)t_2- 2\sigma F_0(x, t_2).\end{aligned}$$

Analogously, we obtain the result for $t<0$. $\square$

Proposition 5.3

Let $(u_k)$ Palais Smale sequence for $I_0$. Then there exists $u_0 \in X_0$ such that $m(\Vert u_k\Vert ^2) \rightarrow m(\Vert u_0\Vert ^2)$. In particular, $I'_{0}(u_0)=0$, this is, $u_0$ is a weak solution of $(P_0)$.

Proof

By Lemma 3.3, we have that $(u_k)$ is a bounded sequence in $X_0$. Since $X_{0}$ is a reflexive Hilbert space, up to a subsequence, we can assume that there exists $u_{0}\in X_{0}$ such that

$$\begin{aligned} \left\{ \begin{array}{ll} u_{k}\rightharpoonup u_{0}\,\,\, \hbox {weakly in}\,\, X_{0}, \\ u_{k}\rightarrow u_{0} \,\,\,\hbox {in}\,\, L_{loc}^q(\mathbb {R})\,\,\,\text{ for } \text{ all }\,\, q\ge 1, \\ u_{k}(x)\rightarrow u_{0}(x) \,\,\,\hbox {almost everywhere in }\,\,\mathbb {R}. \\ \end{array} \right. \end{aligned}$$

On other hand, $(\Vert u_k\Vert _{X_0})_k$ is a bounded sequence in $\mathbb {R}$. Thus, up a subsequence, we have that there exists $L\in \mathbb {R}$ such that $\Vert u_k\Vert _{X_0}\rightarrow L$. Moreover, as the norm is weakly lower semicontinuous, we have

$$\begin{aligned} \Vert u_0\Vert _{X_0}^2 \le \lim \inf \Vert u_k\Vert _{X_0}^2. \nonumber \\ \end{aligned}$$

Since m is nondecreasing, we get

$$\begin{aligned} m(\Vert u_0\Vert _{X_0}^2) \le m(\lim \inf \Vert u_k\Vert _{X_0}^2), \nonumber \\ \end{aligned}$$

consequently,

$$\begin{aligned} m(\Vert u_0\Vert _{X_0}^2) \le m(L^2). \end{aligned}$$

Since $I'_0(u_k) \rightarrow 0$ we obtain that

$$\begin{aligned}m(\Vert u_k\Vert _{X_0}^2)\langle u_k,v\rangle - \int \limits _{\mathbb {R}}f_{0}(x,u_k)v\textrm{d}x\rightarrow 0 \quad \forall v\in C_0^{\infty }(\mathbb {R}).\end{aligned}$$

Consequently, by Lemma 5.1,

$$\begin{aligned} m(L^2)\langle u_0,v\rangle - \int \limits _{\mathbb {R}}f_0(x,u_0)v=0 \quad \forall v\in C_0^{\infty }(\mathbb {R}). \end{aligned}$$

By density arguments, in particular, for $v=u_0$ we have

$$\begin{aligned}m(L^2)\Vert u_0\Vert _{X_0}^2 - \int \limits _{\mathbb {R}}f_0(x,u_0)u_0=0.\end{aligned}$$

Note that

$$\begin{aligned} I'_0(u_0)u_0=(m(\Vert u_0\Vert _{X_0}^2)-m(L^2))\Vert u_0\Vert _{X_0}^2. \end{aligned}$$

(5.6)

We claim that $m(\Vert u_0\Vert _{X_0}^2)=m(L^2).$ In order to show this, we suppose, by contradiction, that $m(\Vert u_0\Vert _{X_0}^2)<m(L^2)$. Thus, it follows from (5.6) that $I'_0(u_0)u_0<0.$ However, by using $(f_{0,1})$ and critical exponential growth, proceeding as in the demonstration of the Lemma 3.1, we have that

$$\begin{aligned} I'_0(t_0u_0)t_0u_0>0\end{aligned}$$

for t sufficiently small. Therefore, $I'_0(u_0)u_0<0$ e $I'_0(tu_0)tu_0>0.$ Then there exists $t_0\in (0,1)$ such that

$$\begin{aligned} I'_0(t_0u_0)t_0u_0=0 \quad \text {and} \quad \max _{t\in [0,1]} I_0(tu_0)=I_0(t_0u_0).\end{aligned}$$

By Lemma 5.2, ($m_3$) and Fatou’s lemma, we conclude

$$\begin{aligned} c_0\le & {} I_0(t_0u_0) - \frac{1}{2\sigma }I'_0(t_0u_0)t_0u_0\\= & {} \frac{1}{2}M(t_0^2\Vert u_0\Vert _{X_0}^2) - \int \limits _{\mathbb {R}}F(x,t_0u_0)\textrm{d} x\\{} & {} - \frac{1}{2\sigma }\left[ m(t_0^2\Vert u_0\Vert _{X_0}^2)t_0^2\Vert u_0\Vert _{X_0}^2 + \int \limits _{\mathbb {R}}f_0(x,t_0u_0)t_0u_0\textrm{d} x\right] \\= & {} \frac{1}{2\sigma }\left[ \sigma M(t_0^2\Vert u_0\Vert _{X_0}^2)-m(t_0^2\Vert u_0\Vert _{X_0}^2)t_0^2\Vert u_0\Vert _{X_0}^2\right] \\{} & {} + \frac{1}{2\sigma }\int \limits _{\mathbb {R}}\left[ f(x,t_0u_0)t_0u_0 - 2\sigma F(x,t_0u_0)\right] \textrm{d} x\\< & {} \frac{1}{2\sigma }\left[ \sigma M(\Vert u_0\Vert _{X_0}^2)-m(\Vert u_0\Vert _{X_0}^2)\Vert u_0\Vert _{X_0}^2\right] + \frac{1}{2\sigma }\int \limits _{\mathbb {R}}\left[ f(x,u_0)u_0 - 2\sigma F(x,u_0)\right] \textrm{d} x\\\le & {} \frac{1}{2\sigma }\left[ \sigma M(\liminf \Vert u_k\Vert _{X_0}^2)-m(\liminf \Vert u_k\Vert _{X_0}^2)\liminf \Vert u_0\Vert _{X_0}^2\right] \\{} & {} + \frac{1}{2\sigma }\int \limits _{\mathbb {R}}\left[ f(x,u_0)u_0 - 2\sigma F(x,u_0)\right] \textrm{d} x\\= & {} \liminf \left[ I_0(u_k)-\frac{1}{2\sigma }I'_0(u_k)u_0\right] = c_0. \end{aligned}$$

But this is a contradiction. Therefore $m(\Vert u_0\Vert _{X_0}^2)=m(L^2)$. So we conclude that $u_0$ is a weak solution of (1.9).

$\square$

Remark 5.4

By (3.5), we can take $0< {\tilde{\varrho }}< \rho '$ such that $I_0(u)\ge {\tilde{\mu }}$ when $\Vert u\Vert _{X_0}= {\tilde{\rho }}$ for some ${\tilde{\mu }}<\min \left\{ { \dfrac{\pi r_0}{\lambda _0}\tilde{C}_{\sigma },}\, \mu \right\}$, where ${\rho '}$ and ${\mu }$ are given in (4.1). Moreover, as $M(0)=0$ and $I_0$ is continuous, we have

$$\begin{aligned} I_0 (t)\le \dfrac{{\tilde{\mu }}}{2} \quad \text{ for } \text{ all } \quad 0\le t<{\tilde{\rho }}. \end{aligned}$$

By $(m_1)$,

$$\begin{aligned} M(t) \le \tilde{C}_{\sigma } t^{\sigma } \quad \text{ for } \text{ all } \quad t\ge {\tilde{\rho }}, \quad \text{ where }\quad \tilde{C}_{\sigma }=\dfrac{M({\tilde{\rho }})}{({\tilde{\rho }})^{\sigma }}. \end{aligned}$$

So we may define

$$\begin{aligned} \tilde{c}_0=\inf _{g\in \Gamma }\max _{t\in [0,1]} I_0(g(t)), \end{aligned}$$

where $\Gamma =\{g\in C([0,1], X_0): g(0)=0\,\, \text{ and }\,\, g(1)=e\}$ and we conclude, as in Proposition 4.3, that

$$\begin{aligned} {\tilde{\mu }} \le \tilde{c}_0 < \frac{\pi r_0 }{\lambda _0 } {\tilde{C}_{\sigma }}, \end{aligned}$$

Since here ${\tilde{\mu }}< \mu \le c_0$, we can take $\dfrac{{\tilde{\mu }}\lambda _0}{\pi \tilde{C}_{\sigma }}<r_0 < \dfrac{c_0\lambda _0}{\pi \tilde{C}_{\sigma }}$, and, consequently, we get

$$\begin{aligned}\tilde{c}_0< c_0.\end{aligned}$$

Thus, we observe that solutions which will obtained in Theorems 1.1 and 1.2 can be different.

5.1 Proof of Theorems 1.1 and 1.2

Using Proposition 5.3 we have that $u_0$ is a weak solution of (1.9), thus if $u_0$ is nontrivial the theorem is proved. If $u_0 = 0$, we have the following claim: There exist $(y_k)\subset \mathbb {R}$ and $R,a>0$ such that

$$\begin{aligned} \liminf \sup _{y_k\in \mathbb {R}}\int \limits _{B_{R}(y_k)}|u_{k}|^{2}\, \textrm{d}x>a. \end{aligned}$$

(5.7)

Indeed, assume that (5.7) does not hold, then for all sequence $(y_k)\subset \mathbb {R}$ and $R>0$, we have

$$\begin{aligned} \liminf \sup _{y_k\in \mathbb {R}}\int \limits _{B_{R}(y_k)}|u_{k}|^{2}\, \textrm{d}x=0. \end{aligned}$$

(5.8)

Using (5.8) and the version of Lions’ lemma for $X_0$, we obtain that $u_{k}\rightarrow 0$ in $L^{t}(\mathbb {R})$ for $2<t<\infty$. Thus, by applying (5.2) and (5.3) we reach

$$\begin{aligned} \int \limits _{\mathbb {R}}f_{0}(x,u_{k})u_{k}\, \textrm{d}x\rightarrow 0. \end{aligned}$$

Combining this estimate together with (3.6), we get that $\Vert u_{k}\Vert _{X_0} \rightarrow 0$. Furthermore, in view of assumption $(f_{0,2})$ we conclude that

$$\begin{aligned} \int \limits _{\mathbb {R}}F_{0}(x,u_{k})\, \textrm{d}x \rightarrow 0. \end{aligned}$$

(5.9)

Combining the convergence $\Vert u_{k}\Vert _{X_0} \rightarrow 0$, (5.9) and (3.6), we get that $c_{0}=0$, but this is impossible. Thus, (5.7) holds.

We may assume, without loss of generality, that $(y_k)\subset \mathbb {Z}$. Letting $w_{k}(x)=u_{k}(x-y_k)$, since $V_{0}(\cdot )$, $f_{0}(\cdot ,s)$ and $F_{0}(\cdot ,s)$ are 1-periodic functions, by a careful calculation we obtain

$$\begin{aligned} \Vert u_{k}\Vert _{X_0}=\Vert w_{k}\Vert _{X_0},\,\, m(\Vert u_{k}\Vert _{X_0})=m(\Vert w_{k}\Vert _{X_0}),\,\, \end{aligned}$$

$$\begin{aligned} I_{0}(u_{k})=I_{0}(w_{k}) \rightarrow c_0\quad \text {and} \quad I'_{0}(w_{k})\rightarrow 0. \end{aligned}$$

Consequently, by similar arguments made in the previous sections, we obtain that $(w_{k})$ is bounded in $X_{0}$ and there exists $w_{0}\in X_{0}$ such that $w_{k}\rightharpoonup w_{0}$ weakly in $X_{0}$ and $w_0$ is a weak solution of the problem (1.9). Moreover, by (5.7), taking a subsequence and R sufficiently large, we get

$$\begin{aligned} a^{1/2}\le \Vert w_{k}\Vert _{L^{2}(B_{R}(0))}\le \Vert w_k-w_{0}\Vert _{L^{2}(B_{R}(0))}+\Vert w_{0}\Vert _{L^{2}(B_{R}(0))}. \end{aligned}$$

(5.10)

Thus, by using Corollary 2.2 we conclude that $w_{0}$ is nontrivial.

To finalize, notice that if u is a weak solution of (1.9), since $f_{0}(x,s)=0$ for all $s\le 0$ and $I'_{0}(u)v=0$ for all $v\in X_{0}$, choosing the test function $v=-u^{-}$, by using the following inequality $|u^{-}(x)-u^{-}(y)|^{2}\le (u(x)-u(y))(u^{-}(y)-u^{-}(x))$ and the fact that m is nondecreasing we get that $\Vert u^{-}\Vert _{X_0}\le 0$. Thus, u is a nonnegative function. This completes the proof of Theorems 1.1 and 1.2.

6 Existence of a solution for the nonperiodic problem

In the section, we are concerned to find a nonnegative and nontrivial solution for (1.12). For this, we consider the functional $I: X_{1}\rightarrow \mathbb {R}$ given by

$$\begin{aligned} I(u)=\dfrac{1}{2}M(\Vert u\Vert _{X_1}^{2})-\int \limits _{\mathbb {R}}F(x,u)\, \textrm{d}x. \end{aligned}$$

From $(f_1)$ - $(f_2)$, Lemmas 2.3 and 2.4, similarly to Sect. 3, we can see that I is well defined and by using standard arguments $I \in C^1(X_1,\mathbb {R})$ with

$$\begin{aligned} I^{\prime }(u)v=m(\Vert u\Vert _{X_1}^{2})\langle u,v\rangle _{1} -\int \limits _{\mathbb {R}}f(x,u)v\,\textrm{d}x. \end{aligned}$$

for all $v\in X_{1}$. Thus, a critical point of I is a weak solution of (1.12) and reciprocally. Moreover, I has the geometry of the mountain-pass theorem, by analogous steps to Propositions 3.1 and 3.2, we obtain

Proposition 6.1

If $(f_2)$-$(f_3)$ and $(v_1)$ hold, then

• (i) there exist $\sigma _1,\,\rho _1>0$ such that $I\left( u\right) \ge \sigma _1$ if  $\Vert u\Vert _{X_1}=\rho _1$;

• (ii) there exists $e_1 \in E$, with $\Vert e_1\Vert _{X_1} > \rho _1$, such that $I(e_1) < 0$.

As a consequence of Proposition 6.1, the minimax level

$$\begin{aligned} c_1:=\inf _{\gamma \in \Gamma }\max _{t\in [0,1]} I(\gamma (t)) \end{aligned}$$

is positive, where $\Gamma =\{\gamma \in C([0,1],X_1): \gamma (0)=0\,\, \text{ and }\,\, \gamma (1)=e_1\}$.

Moreover, by applying the mountain-pass theorem without the (PS) condition (see [31]), there exists a sequence $(v_{k})\subset X_1$ such that

$$\begin{aligned} I(v_{k})\rightarrow c_{1} \quad \text{ and } \quad I'(v_{k})\rightarrow 0. \end{aligned}$$

Thus, similarly to Lemma 3.3, we obtain that $(v_{k})$ is a bounded sequence in $X_1$. Moreover, by using the arguments made in Proposition 5.3, we get the following result:

Proposition 6.2

If $(f_{0,4})$ (or $({\tilde{f}_{0,4}})$ and $(K_1)$), $(f_1)$-$(f_3)$ and $(v_1)$ hold, then $v_{k}\rightharpoonup v_{0}$ weakly in $X_1$ and $v_{0}$ is a critical point of functional I.

Now, in order to prove that there exists a nontrivial critical point of I we need some auxiliary results, among then a lemma of convergence. More specifically, assuming for the sake of contradiction that $v_0$ is trivial, following the same steps of Lemma 4.3 in [13], we obtain the following result.

Lemma 6.3

If $(v_1)$, $(f_{0,1})$-$(f_{0,2})$ and $(f_1)$-$(f_3)$ hold, then

• (i) ${\int \limits _{\mathbb {R}}[f_{0}(x,v_{k})-f(x,v_{k})]v_{k}\, \textrm{d}x \rightarrow 0}$;

• (ii) ${\int \limits _{\mathbb {R}}[F_{0}(x,v_{k})-F(x,v_{k})]\, \text {d}x \rightarrow 0}$

• (iii) ${\int \limits _{\mathbb {R}}[V_{0}(x)-V(x)]v_{k}^{2}\, \text {d}x\rightarrow 0}$.

6.1 Proof of Theorem 1.3

Assuming for the sake of contradiction that $v_0$ is trivial, since m is a continuous function, as a consequence of Lemma 6.3, it follows that

$$\begin{aligned} |I_{0}(v_{k})-I(v_{k})|\rightarrow 0 \quad \text {and} \quad \Vert I'_{0}(v_{k})-I'(v_{k})\Vert _*\rightarrow 0. \end{aligned}$$

(6.1)

Hence,

$$\begin{aligned} I_{0}(v_{k})\rightarrow c_{1} \quad \text {and} \quad I'_{0}(v_{k})\rightarrow 0. \end{aligned}$$

In addition, we obtained a version of Lions’s result for a sequence in $X_1$ as in (4.3). From this, we conclude that there exist $(y_{k})\subset \mathbb {Z}$ and $R, a>0$ such that

$$\begin{aligned} \liminf \sup _{y_{k}\in \mathbb {R}}\int \limits _{B_{R}(y_{k})}|v_{k}|^{2}\, \textrm{d}x>a. \end{aligned}$$

Now consider $w_k(x)=v_{}(x-y_{k})$, since $V_{0}(x), f_{0}(x,s)$ and $F_{0}(x,s)$ are $1-$periodic functions in x, we get

$$\begin{aligned} \Vert v_{k}\Vert _{X_0}=\Vert w_{k}\Vert _{X_0},\,\, m(\Vert v_{k}\Vert _{X_0})=m(\Vert w_{k}\Vert _{X_0}),\,\, \end{aligned}$$

$$\begin{aligned}I_{0}(v_{k})=I_{0}(w_{k}) \rightarrow c_1\quad \text {and} \quad I'_{0}(w_{k})\rightarrow 0.\end{aligned}$$

Then, there exists $w_0\in X_{0}$ such that $w_k\rightharpoonup w_0$ weakly in $X_{0}$ and $I'_{0}(w_0)=0$. Moreover, using (6.1) and Fatou’s lemma, we have

$$\begin{aligned} I_{0}(w_0)= & {} I_{0}(w_0)-\frac{1}{2}I'_{0}(w_0)w_0\\= & {} \dfrac{1}{2}\int \limits _{\mathbb {R}}[f_{0}(x,w_0)w_0-2F_{0}(x,w_0)]\, \textrm{d}x\\\le & {} \liminf \dfrac{1}{2}\int \limits _{\mathbb {R}}[f_{0}(x,w_k)w_k-2F_{0}(x,w_k)]\, \textrm{d} x\\ {}= & {} \lim [I_{0}(w_k)-\frac{1}{2}I'_{0}(w_k)w_k]= c_{1}. \end{aligned}$$

Arguing as in (5.10) we conclude that $w_0$ is nontrivial. Now, by $(f_{0,3})$, we have that $\max \{I_{0}(t w_0 )\,:\, t \ge 0\}$ is unique and then

$$\begin{aligned} c_{0}\le \max _{t\ge 0}I_{0}(tw_0)=I_{0}(w_0)\le c_{1}. \end{aligned}$$

(6.2)

On the other hand, considering $u_0$ the solution obtained in Theorem  1.1 (or Theorem1.2), as m is nondecreasing, so from $(v_1)$, $(f_1)$, $(f_5)$, $(f_4)$ and $(f_{0,3})$, we have

$$\begin{aligned} c_{1}\le \max _{t\ge 0}I(tu_{0})=I(t_{1}u_{0})<I_{0}(t_{1}u_{0})\le \max _{t\ge 0}I_{0}(tu_{0})=I_{0}(u_{0})=c_{0}, \end{aligned}$$

that is, $c_{1}<c_{0}$, which is a contradiction with (6.2). Therefore, $v_{0}$ is nontrivial.

To finalize, notice that similarly to proof of Theorems 1.1 and 1.2 if we have a weak solution of (1.12), then it is a nonnegative function. This completes the proof of Theorem 1.3.