Introduction

We consider the discrete anisotropic problem governed by the p(k)-Laplacian-like operators with Dirichlet-type boundary condition given as follows:

$$\begin{aligned} {(P_\lambda )} {\left\{ \begin{array}{ll} -\Delta \left( \left( 1+\phi _c\left( \vert \Delta u(t-1) \vert ^{p (t-1)}\right) \right) \vert \Delta u(t-1) \vert ^{p (t-1)-2}\Delta u(t-1) \right) \\ ~~~~~~=\lambda f(t,u(t)), ~ t\in [1,N]_{\mathbb {N}}, \\ u(0)=u(N+1)=0 , \end{array}\right. } ~~ \end{aligned}$$

where \(N \ge 2\) is a positive integer, \([1,N]_{\mathbb {N}}:=\lbrace 1,2, \ldots ,N\rbrace\) is the discrete interval, \(\Delta\) denotes the forward difference operator defined by \(\Delta u (t)=u (t+1)-u (t)\). For every fixed \(t \in [0,N]_{\mathbb {N}}\), the function \(f(t,.): \mathbb {R} \rightarrow \mathbb {R}\) is continuous and \(p:[0,N+1]_\mathbb {N}\rightarrow [2,+\infty )\) is a given bounded function. \(\phi _c\) is the so-called mean curvature operator [19] defined by

$$\phi _c(s)= \frac{s}{\sqrt{1+s^2}}, s\in \mathbb {R}.$$

For any bounded function \(h: [0,N+1]_{\mathbb {N}} \rightarrow \mathbb {R}\), we consider the symbols

$$h^{+}:=\max _{t \in [0, N+1]_\mathbb {N}} h(t), ~~~ h^{-}:=\min _{t \in [0, N+1]_\mathbb {N}} h(t).$$

Furthermore, we will consider the following assumptions:

\((H_1)\)::

There exist \(C>0\) and \(2\le \alpha (t)< p^-\) for all \(t \in [0,N+1]_{\mathbb {N}}\) such that

$$\vert f(t,x)\vert \le C(1+\vert x \vert ^{\alpha (t)-1}) ~~ \text {for all} ~(t,x) \in [0,N]_{\mathbb {N}} \times \mathbb {R}.$$
\((H_2)\)::

\(f(r, 0) \ne 0\) for any \(r \in [1, N]_\mathbb {N}\).

\((H_3)\)::

There exists \(\kappa\) such that \(\kappa > \frac{2^{p^+ +1}}{p^-} \left( N+1\right) ^{\frac{p^+}{2}}\) and

$$\liminf _{\vert x\vert \rightarrow \infty } \frac{F(r, x)}{\vert x\vert ^{p^+}} \ge \kappa , \text { for all } r\in [1, N]_{\mathbb {N}},$$

where \(F(r,x)= \int _0^x f(r,s)ds.\)

In recent years, a great deal of work has been devoted to the study of difference equations, because it is an interesting topic and plays an important role in various research domains, such as the dynamic problems of combustible gas [28], electrorheological fluids [27] and image restoration [9].

Problem \((P_\lambda )\) makes a part of discrete nonlinear problems involving the p-Laplacian-like operator and the specified mean curvature operator, which has its origin from the study of capillary phenomena. These phenomena can be briefly explained by considering the effects of two opposing forces: cohesion, i.e., the attractive force between the molecules of the liquid; and adhesion, i.e., the attractive (or repulsive) force between the molecules of the liquid and those of the container. The study of these phenomena has gained more attention recently, motivated by its importance in applied fields such as industrial, biomedical, economic systems [11, 26], the analysis of capillary surfaces [4, 13, 15] and the capillarity problem in hydrodynamics [21]. We refer the reader to [1, 10, 11, 16, 18] and references therein.

For the continuous counterpart of problem \((P_\lambda )\), we refer the reader to recent works (see [12, 24, 25]). For instance, in [24], Rodrigues has studied the existence and multiplicity of solutions for the following problem involving the p(x)-Laplacian-like operators arising from capillary phenomena:

$$\left\{ \begin{array}{l} -{\text {div}}\left( \left( 1+\frac{\vert \nabla u\vert ^{p(x)}}{\sqrt{1+\vert \nabla u\vert ^{2 p(x)}}}\right) \vert \nabla u\vert ^{p(x)-2} \nabla u\right) =\lambda f(x, u),~ x \in \Omega , \\ u=0, ~ x \in \partial \Omega , \end{array}\right.$$

where \(\lambda >0\), \(\Omega \subset \mathbb {R}^N\) is a bounded domain with smooth boundary \(\partial \Omega\), \(p \in C(\overline{\Omega })\) and \(p(x)>2\), \(\forall x \in \Omega\). Based on the mountain pass theorem, the author showed that the problem has at least one nontrivial solution. The author also proved the existence of a sequence of solutions using the Fountain theorem. The problem \((P_\lambda )\) may be regarded as the discrete analog of the above problem.

Despite the interest mentioned above, the number of works dealing with problems involving p-Laplacian-like in the discrete case remains relatively small. For some related papers in the area of discrete problems, we can mention, for instance, [5, 17, 29]. In [29], the authors have treated the following discrete problem involving the mean curvature operator with Dirichlet boundary conditions:

$$\left\{ \begin{array}{l} -\Delta \phi _c(\Delta u(k-1))=\lambda f(k, u(k)), ~~ k \in \mathbb {Z}(1, N),\\ u(0)=u(N+1)=0, \end{array}\right.$$

where \(\lambda >0\), \(N \in \mathbb {Z}^{+}, ~ \mathbb {Z}(1, N)=\{1,2, \cdots , N\}\) and for each \(k \in \mathbb {Z}(1, N)\), \(f(k, \cdot ) \in C(\mathbb {R}, \mathbb {R})\). By using the critical point theory, they proved the existence of infinitely many positive solutions under certain sufficient conditions.

In this work, we are inspired by the above-mentioned works and the ideas introduced in [17], where the author studied the problem

$$\begin{aligned} -\Delta \left( \phi _{p_1(k-1)} (\Delta u(k-1))\right) -\Delta \left( \phi _{p_2(k-1)} (\Delta u(k-1))\right) = \lambda f(k,u(k)),\\ \text { for all } k\in [1,N]_\mathbb {N}, \\ u(0)=u(T+1)=0, \end{aligned}$$

where \(\phi _{p_i(k)} (t)= \vert t \vert ^{p_i(k) -2}t\) for \(i=1, 2\) and \(t\in \mathbb {R}\). The author proved the existence and multiplicity results by using some critical point theorems and based on the hypotheses \((H_1)\), \((H_2)\) and the so-called Ambrosetti-Rabinowitz condition ((AR)-condition for short [3]) given as:

(AR)::

There exist two positive constants \(\theta >p^{+}\)and \(t^*>0\) such that for every \(t \in [1, N]_\mathbb {N}\) and \(\vert s \vert \ge t^*\)

$$0<\theta F(t, s) \le s f(t, s).$$

It is well known that the (AR)-condition has a crucial role in the application of variational methods; it is extensively used to ensure the boundedness of the Palais-Smale sequences of the energy functional. Despite this, it is still restrictive and eliminates many nonlinearities. For example, the function

$$f(t, x) = \vert x \vert ^{p^+-2}x\ln (1+\vert x\vert ) + \frac{1}{p^+} \frac{ \vert x \vert ^{p^+-1}x}{1+\vert x\vert }, ~ t\in [1, N]_\mathbb {N}, ~x\in \mathbb {R}$$

with a primitive

$$F(t,x) =\frac{1}{p^+}\vert x \vert ^{p^+}\ln (1+\vert x\vert ), ~ t\in [1, N]_\mathbb {N},~ x\in \mathbb {R},$$

does not satisfy the (AR)-condition. However, it satisfies the hypothesis \((H_3)\), which is weaker and covers a wider functional space. For this reason, in our work, we have taken hypothesis \((H_3)\) instead of the (AR)-condition.

The rest of this article is structured as follows. In the “Preliminaries” section, we present some basic preliminaries and provide several inequalities useful in our approach. In the “Variational framework” section, we give the variational framework corresponding to the problem (P). Our main results will be established in the “Main results and proofs” section, where some examples are also provided.

Preliminaries

Consider E the N-dimensional Hilbert space [2]:

$$E= \Big \{ u:[0,N+1]_\mathbb {N}\rightarrow \mathbb {R}\mid ~ u(0)=u(N+1)=0 \Big \},$$

endowed with the inner product

$$<u,v>= \displaystyle \sum _{t=1}^{N+1} \Delta u(t-1)\Delta v(t-1), ~~ \text {for all} ~u,v \in E.$$

The associated norm is defined by

$$\Vert u\Vert = \left( \displaystyle \sum _{t=1}^{N+1} \vert \Delta u(t-1)\vert ^2 \right) ^\frac{1}{2}.$$

Also, for \(r\ge 2\), we define the norm

$$\begin{aligned} \vert u\vert _r =\left( \sum _{t=1}^{N}\vert u(t)\vert ^r\right) ^{\frac{1}{r}} , \hbox {~~} \forall u\in E . \end{aligned}$$

Since E is a finite-dimensional space, all norms are equivalent.

For our purpose, it is useful to use the following inequalities.

Proposition 1

([14, 20])

  1. (a)

    Let \(u \in E\) and \(\Vert u\Vert >1\). Then

    $$\begin{aligned} \sum _{t=1}^{N+1} \frac{\vert \Delta u(t-1)\vert ^{p(t-1)}}{p(t-1)} \ge N^{\frac{2-p^- }{2}}\Vert u\Vert ^{p^{-}}-N. \end{aligned}$$
  2. (b)

    Let \(u \in E\) and \(\Vert u\Vert <1\). Then

    $$\begin{aligned} \sum _{t=1}^{N+1} \frac{\vert \Delta u(t-1)\vert ^{p(t-1)}}{p(t-1)} \ge N^{\frac{p^+ -2}{2}}\Vert u\Vert ^{p^{+}}. \end{aligned}$$
  3. (c)

    For all \(u\in E\), we have

    $$\begin{aligned} \Vert u\Vert _{\infty }:=\max _{t\in [1,N]_\mathbb {N}}\vert u(t)\vert \le \sqrt{N+1} \Vert u \Vert . \end{aligned}$$
  4. (d)

    For all \(u\in E\), we have

    $$\sum _{t=1}^{N+1} \vert \Delta u(t-1)\vert ^{p^+} \le (N+1)\Vert u \Vert ^{p^+}.$$
  5. (e)

    For all \(u\in E\), \(m\ge 2\) we have

    $$\sum _{t=1}^{N} \vert u(t)\vert ^{m} \ge 2^{-m}\left( N+1\right) ^{\frac{2-m}{2}}\Vert u \Vert ^{m}.$$

In this article, we are based on three local minimum theorems due to Bonanno and D’Agui [6, 7], which are a simple extension of the Ricceri’s variational principle [23].

Let E be a finite-dimensional Banach space and let \(F, G: E\rightarrow \mathbb {R}\) two functions of class \(C^1\) on E with F be coercive.

Theorem 2.1

(see [6]) Suppose that there exist \(r \in \mathbb {R}\) and \(\omega \in E\), with \(0<F(\omega )<r\) such that

$$\frac{G(\omega )}{F(\omega )} >\frac{\sup _{u\in F^{-1}([0, r])} G(u)}{r}.$$

Then, for each

$$\lambda \in \Lambda :=\left]\frac{F(\omega )}{G(\omega )}, \frac{r}{\sup _{u\in F^{-1}([0, r])} G(u)}\right[,$$

the function \(I_\lambda =F-\lambda G\) admits at least one local minimum \(u_0 \in E\backslash \{ 0\}\) such that \(F(u_0)<r, I_\lambda (u_0) \le I_\lambda (u)\) for all \(u \in F^{-1}([0, r])\) and \(I_\lambda ^{\prime }(u_0)=0\).

Theorem 2.2

(see [6]) Let \(r>0\). Suppose that the function \(I_\lambda =F-\lambda G\) satisfies the (PS)-condition and is unbounded from below for all

$$\lambda \in \Lambda _2:=\bigg ] 0, \frac{r}{\sup _{u\in F^{-1}([0, r])} G(u)}\bigg [.$$

Then, for any \(\lambda \in \Lambda _2\), the function \(I_\lambda\) has at least two distinct critical points.

We will also use this result, which guarantees the existence of at least three critical points.

Theorem 2.3

[7, 23] Let W be a reflexive real Banach space, \(F: W \rightarrow \mathbb {R}\) be a continuously Gâteaux differentiable, coercive and sequentially weakly lower semicontinuous functional whose Gâteaux derivative admits a continuous inverse on \(W^*\), \(G: W \rightarrow \mathbb {R}\) be a continuously Gâteaux differentiable functional whose Gâteaux derivative is compact, moreover

$$F(0)=G(0)=0.$$

Assume that there exist \(r \in \mathbb {R}\) and \(\omega \in W\), with \(0<r<F(\omega )\), such that

  1. 1.
    $$\frac{\sup _{ F^{-1}(]-\infty , r ] )} G}{r}<\frac{G(\omega )}{F(\omega )}.$$
  2. 2.

    The functional \(I_\lambda = F-\lambda G\) is coercive for any

    $$\lambda \in \Lambda :=\bigg ] \frac{F(\omega )}{G(\omega )}, \frac{r}{\sup _{u \in \Phi ^{-1}(]-\infty , r ])} G(u)}\bigg [.$$

Then, for any \(\lambda \in \Lambda\), the functional \(I_\lambda\) admits at least three distinct critical points in W.

Variational framework

Let \(T_\lambda : E \rightarrow \mathbb {R}\) the functional associated with problem \((P_\lambda )\) defined in the following way

$$\begin{aligned} T_\lambda (u) = A(u)- \lambda B(u), ~~ \forall u \in E, \end{aligned}$$
(1)

with

$$A(u) = \sum _{t=1}^{N+1} \frac{1}{p (t-1)}\left( \vert \Delta u(t-1) \vert ^{p (t-1)} + \sqrt{1+ \vert \Delta u(t-1) \vert ^{2p (t-1)} } -1 \right) ,$$
$$B(u)= \sum _{t=1}^{N} F(t,u(t)), \text { where } F(t,x)= \int _0^x f(t,s)ds.$$

The functional \(T_\lambda\) is well defined, of class \(C^1\) and its Gâteaux derivative is given by:

$$\begin{aligned} (T_\lambda '(u), v )&= \sum _{i=1}^{N+1} \left( 1+\phi _c\left( \vert \Delta u(i-1) \vert ^{p (i-1)}\right) \right) \vert \Delta u(i-1) \vert ^{p (i-1)-2}\Delta u(i-1) \Delta v(i-1)\nonumber \\&- \lambda \sum _{i=1}^{N} f(i,u(i))v(i) , \end{aligned}$$
(2)

for all \(u,v \in E\).

By a standard argument, using summation by parts, it can be shown that the critical points of the functional \(T_\lambda\) are exactly the solutions of problem \((P_\lambda )\).

In the remainder of the article, we put

$$R(\xi ) = \left( \frac{\xi }{\sqrt{N+1}}\right) ^{p^-}\frac{2N^{\frac{2-p^- }{2} }}{p^+} -2N -\frac{N+1}{p^-},$$

for any \(\xi \in \mathbb {R}\).

Main results and proofs

Before giving our main results, we give and demonstrate some lemmas.

Lemma 4.1

The functional A is coercive on E.

Proof

We recall that for all \(s\ge 0\), we have \(\max {(0, s-1)} \le \sqrt{1+s^2}-1 \le s\).

Let \(u\in E\) with \(\Vert u \Vert\) be large enough such that \(\vert \Delta u(t-1)\vert \ge 1\) for all \(k\in [1,N]_\mathbb {N}\), in view of (a), we have

$$\begin{aligned} A(u)&= \sum _{t=1}^{N+1} \frac{1}{p (t-1)}\left( \vert \Delta u(t-1) \vert ^{p (t-1)} + \sqrt{1+ \vert \Delta u(t-1) \vert ^{2p (t-1)} }-1 \right) \nonumber \\&\ge \sum _{t=1}^{N+1} \frac{2}{p (t-1)} \vert \Delta u(t-1) \vert ^{p (t-1)} - \sum _{t=1}^{N+1} \frac{1}{p (t-1)} \nonumber \\&\ge \frac{2}{p^{+}(\sqrt{N})^{p^{-}-2}}\Vert u\Vert ^{p^{-}}-2N - \frac{N+1}{p^-} \longrightarrow \infty ~\text {as} ~ \Vert u\Vert \longrightarrow \infty . \end{aligned}$$
(3)

So, A is coercive. \(\square\)

Lemma 4.2

Suppose that condition \((H_3)\) is verified, then the functional \(T_\lambda\) is unbounded from below and satisfies the (PS)-condition, i.e., any sequence \((u_n)\subset E\) such that \((T_\lambda (u_n))\) is bounded and \((T_\lambda ^{\prime }(u_n)) \longrightarrow 0\) as \(n \rightarrow +\infty\), has a subsequence which converges in E.

Proof

According to the hypothesis \((H_3)\), for any \(\varepsilon >0\) there exists \(R_0>0\) such that

$$\frac{ F(r, x)}{\vert x\vert ^{p^+}} \ge \kappa -\varepsilon ~ \text{ for } \text{ all } (r,\vert x\vert ) \in [1, N]_{\mathbb {N}} \times ] R_0,+\infty [,$$

so,

$$F(r, x) \ge (\kappa -\varepsilon )\vert x\vert ^{p^+} ~ \text{ for } (r,\vert x\vert ) \in [1, N]_{\mathbb {N}} \times ] R_0,+\infty [.$$

By continuity of the function \(s\rightarrow F(t, s)\), there exists \(K>0\) such that

$$F(r, s) \ge (\kappa -\varepsilon )\vert s\vert ^{p^+}-K, \quad \forall (r, s) \in [1, N]_{\mathbb {N}} \times \mathbb {R}.$$

Then,

$$F(r, u(r)) \ge (\kappa -\varepsilon )\vert u(r)\vert ^{p^+}-K, \quad \forall r\in [1, N]_{\mathbb {N}},$$

and by relation (e), we get

$$\begin{aligned} B(u)&\ge (\kappa -\varepsilon ) \sum _{r=1}^N\vert u(r)\vert ^{p^+}-KN \\&\ge (\kappa -\varepsilon ) 2^{-p^+}\left( N+1\right) ^{\frac{2-p^+}{2}}\Vert u\Vert ^{p^+}-KN, \quad \forall u\in E. \end{aligned}$$

On the other hand, for \(u \in E\) with \(\Vert u \Vert > 1\) such that \(\vert \Delta u(r-1) \vert > 1\) for all \(r\in [1, N]_{\mathbb {N}}\), by (d), we have

$$\begin{aligned} A(u)&\le \frac{2}{p^-}\sum _{r=1}^{N+1} \vert \Delta u(r-1) \vert ^{p (r-1)}\\&\le \frac{2}{p^-} \sum _{r=1}^{N+1} \vert \Delta u(r-1) \vert ^{p^+} \\&\le \frac{2}{p^-} (N+1)\Vert u \Vert ^{p^+}, \end{aligned}$$

it follows that,

$$\begin{aligned} T_\lambda (u)&\le \frac{2}{p^-} (N+1) \Vert u \Vert ^{p^+} - 2^{-p^+}\left( N+1\right) ^{\frac{2-p^+}{2}}(\kappa -\varepsilon )\Vert u\Vert ^{p^+}+K N\\&\le \left[ \frac{2}{p^-} (N+1) - 2^{-p^+}\left( N+1\right) ^{\frac{2-p^+}{2}}(\kappa -\varepsilon )\right] \Vert u \Vert ^{p^+}+K N. \end{aligned}$$

Consequently, for \(\varepsilon <\kappa - \frac{2^{p^+ +1}}{p^-} \left( N+1\right) ^{\frac{p^+}{2}}\), we have

$$T_\lambda (u) \rightarrow -\infty , \text{ as } \Vert u\Vert \rightarrow \infty .$$

Hence \(T_\lambda\) is anti-coercive on E.

Therefore, the functional \(T_\lambda\) is unbounded from below. Furthermore, any (PS) sequence \(\left( u_n\right)\) associated with \(T_\lambda\) will be bounded in E, which is a finite-dimensional space. Thus, the functional \(T_\lambda\) satisfies the (PS)-condition. \(\square\)

Now, we state our main result as follows.

Theorem 4.1

Assume that there exist two real constants \(\xi\) and \(\delta \ge 1\), with

$$\begin{aligned} \xi \ge \sqrt{N+1}\left( N^{\frac{p^+ -p^-}{2}} + N^{\frac{p^{-}}{2}}\right) ^{1 / p^{-}} \end{aligned}$$
(4)

and

$$\begin{aligned} \frac{4 \delta ^{p^{+}} }{ p^{-}} < R(\xi ), \end{aligned}$$
(5)

such that

$$\begin{aligned} (F_1):&~~~~\frac{\sum _{t=1}^N \max _{\vert s\vert \le \xi } F(t, s)}{R(\xi )}<\frac{p^{-} \sum _{t=1}^N F(t, \delta )}{4 \delta ^{p^{+}}},\\ (F_2):&~~~~ F(t, \delta ) \ge 0 \text { for any } t \in [1, N]_\mathbb {N}. \end{aligned}$$

Therefore, for every

$$\begin{aligned} \lambda \in \Lambda _1:=\left] \frac{4 \delta ^{p^{+}}}{p^{-} \sum _{t=1}^N F(t, \delta )}, \frac{R(\xi )}{\sum _{t=1}^N \max _{\vert s\vert \le \xi } F(t, s)}\right[, \end{aligned}$$
(6)

problem \((P_\lambda )\) has at least one nontrivial solution \(u_0 \in E\).

Proof

In this proof, we will apply Theorem 2.1. By Lemma 4.1, the functional A is coercive, it remains to verify that there exist \(r \in \mathbb {R}\) and \(w \in E\), with \(0<A(w)<r\), such that

$$\begin{aligned} \frac{\sup _{A^{-1}([0, r])} B}{r}<\frac{B(w)}{A(w)} . \end{aligned}$$
(7)

For this reason, we put \(r:= R (\xi )\) and choosing \(w\in E\) such that

$$\begin{aligned} w(t)=\delta ,\text { for all } t\in [1,N]_\mathbb {N}. \end{aligned}$$
(8)

Obviously with \(\delta \ge 1\), we obtain

$$\begin{aligned} A(w)&= \sum _{t=1}^{N+1} \frac{1}{p (t-1)}\left( \vert \Delta w(t-1) \vert ^{p (t-1)} + \sqrt{1+ \vert \Delta w(t-1) \vert ^{2p (t-1)} } -1 \right) \nonumber \\&= \frac{1}{p (0)}\left( \delta ^{p (0)} + \sqrt{1+ \delta ^{2p (0)} } -1 \right) + \frac{1}{p (N)}\left( \delta ^{p (N)} + \sqrt{1+ \delta ^{2p (N)} } -1 \right) \nonumber \\&\le \frac{1}{p^-}\left( 2\delta ^{p^+} + 2\sqrt{1+ \delta ^{2p^+} } -2 \right) \nonumber \\&\le \frac{ 4\delta ^{p^+}}{p^-}. \end{aligned}$$
(9)

So, from (5) it follows that \(0<A(w)<r\).

Now, let \(u \in E\) such that \(u \in A^{-1}([0, r])\) and \(\Vert u\Vert <1\), by using inequality (b), we get

$$\begin{aligned} r\ge A(u)&=\sum _{t=1}^{N+1} \frac{1}{p (t-1)}\left( \vert \Delta u(t-1) \vert ^{p (t-1)} + \sqrt{1+ \vert \Delta u(t-1) \vert ^{2p (t-1)} } -1\right) \\&\ge \sum _{t=1}^{N+1} \frac{1}{p (t-1)} \vert \Delta u(t-1) \vert ^{p (t-1)} \\&\ge \frac{1}{p^{+}(\sqrt{N})^{2-p^{+}}}\Vert u\Vert ^{p^{+}} . \end{aligned}$$

Then as well in (3), from (a) we have

$$\begin{aligned} r\ge A(u)&= \sum _{t=1}^{N+1} \frac{1}{p (t-1)}\left( \vert \Delta u(t-1) \vert ^{p (t-1)} + \sqrt{1+ \vert \Delta u(t-1) \vert ^{2p (t-1)} }-1 \right) \nonumber \\&\ge \frac{2}{p^{+}(\sqrt{N})^{p^{-}-2}}\Vert u\Vert ^{p^{-}}-2N - \frac{N+1}{p^-}, \end{aligned}$$
(10)

for all \(u \in E\) with \(\Vert u\Vert >1\).

Then

$$\Vert u\Vert \le \max \Biggl \{ 1, ~ \left( \frac{rp^+}{2N^{\frac{p^+-2}{2}}} \right) ^{\frac{1}{p^+}},~ \left( \left( r+ 2N+ \frac{N+1}{p^-}\right) \frac{p^+N^{\frac{p^- -2}{2}}}{2}\right) ^{\frac{1}{p^-}} \Biggr \}.$$

By virtue of (4), we obtain

$$rp^+ \ge 2N^{\frac{p^+-2}{2}},$$

this together with (c), gives

$$\vert u(t) \vert \le \sqrt{N+1} \Vert u\Vert \le \sqrt{N+1} \left( \left( r+ 2N+ \frac{N+1}{p^-}\right) \frac{p^+N^{\frac{p^- -2}{2}}}{2}\right) ^{\frac{1}{p^-}} = \xi ,$$

for all \(t \in [1, N]_\mathbb {N}\). Therefore, we have

$$\begin{aligned} \sup _{u \in A^{-1}([0, r])} B(u)=\sup _{u \in A^{-1}([0, r])} \sum _{t=1}^N F(t, u(t)) \le \sum _{t=1}^N \max _{\vert s\vert \le \xi } F(t, s) . \end{aligned}$$
(11)

In view of (9) and (11), taking in consideration \((F_1)\) and \((F_2)\), we have

$$\begin{aligned} \frac{\sup _{A^{-1}([0, r])} B(u)}{r}&\le \frac{\sum _{t=1}^N \max _{\vert s\vert \le \xi } F(t, s)}{R(\xi )} \nonumber \\&<\frac{p^{-} \sum _{t=1}^N F(t, \delta )}{4 \delta ^{p^{+}}} \le \frac{B(w)}{A(w)}. \end{aligned}$$
(12)

Therefore, according to Theorem 2.1, the functional \(T_\lambda\) has at least one critical point \(u_0\) with \(0< A(u_0) < r\), and thus, \(u_0\) is a nontrivial solution of problem \((P_\lambda )\). \(\square\)

Theorem 4.2

We suppose that hypotheses \((H_2)\) and \((H_3)\) hold. Moreover, suppose that inequalities (4) and (5) of Theorem 4.1 are satisfied. Then, for any

$$\lambda \in \Lambda _2:=\bigg ] 0, \frac{R(\xi )}{\sum _{t=1}^N \max _{\vert s\vert \le \xi } F(t, s)}\bigg [,$$

the problem \((P_\lambda )\) admits at least two nontrivial solutions.

Proof

Fix \(\lambda \in \Lambda _2\). The functional A and B defined in (1) verify all regularity assumptions requested in Theorem 2.2. Choosing \(r=R(\xi )\) and w(t) as in (8), we have

$$\frac{\sup _{A^{-1}([0, r])} B(u)}{r} \le \frac{\sum _{t=1}^N \max _{\vert s\vert \le \xi } F(t, s)}{R(\xi )}<\frac{1}{\lambda }.$$

Moreover, by Lemma 4.2, the function \(T_\lambda\) satisfies the (PS) condition and is unbounded from below. Therefore, in view of Theorem 2.2, for any \(\lambda \in \Lambda _2\), the problem \((P_\lambda )\) admits at least two distinct solutions. \(\square\)

Theorem 4.3

Assume that there exist two real numbers \(\xi\) and \(\delta \ge 1\) satisfy the first inequality in (4) and

$$\begin{aligned} \frac{ 4 \delta ^{p^-}}{p^+} > R(\xi ), \end{aligned}$$
(13)

such that the assertions \((F_1)\) and \((F_2)\) in Theorem 4.1 hold. Moreover, if \((H_1)\) is verified, then, for any \(\lambda \in \Lambda _1\), where \(\Lambda _1\) is defined in (6), the problem \((P_\lambda )\) has at last three solutions.

Proof

Similar to the proof of Theorem 4.1, we set w(k) as in (8) and \(r=R(\xi )\). Taking into account (13), we have \(A(w)>r>0\). Then, (12) is satisfied.

Now, in order to apply Theorem 2.3, it remains to establish that the functional \(T_\lambda\) is coercive. Let \(u \in E\) such that \(\Vert u \Vert >1\). We point out that

$$\begin{aligned} \vert u(t)\vert ^{\alpha (t)} \le \vert u(t)\vert ^{\alpha ^{-}}+\vert u(t)\vert ^{\alpha ^{+}}, ~ \forall t \in [1, N]_\mathbb {N},~ u \in E. \end{aligned}$$
(14)

On the other hand, the relation (c) implies that for \(m\ge 2\)

$$\vert u(t) \vert ^m \le (N+1)^{\frac{m}{2}} \Vert u \Vert ^m, \text { for all } t \in [1, N]_\mathbb {N},$$

and thus

$$\sum _{t=1}^N \vert u(t) \vert ^m \le N(N+1)^{\frac{m}{2}} \Vert u \Vert ^m$$

this combined with (14), gives

$$\begin{aligned} \sum _{t=1}^{N}\vert u(t)\vert ^{\alpha (t)}&\le N(N+1)^{\frac{\alpha ^+}{2}} \Vert u \Vert ^{\alpha ^+} + N(N+1)^{\frac{\alpha ^-}{2}} \Vert u \Vert ^{\alpha ^-} \\&\le 2N(N+1)^{\frac{\alpha ^+}{2}} \Vert u \Vert ^{\alpha ^+}. \end{aligned}$$

Now, from \((H_1)\) there exists \(C_1>0\) such that

$$\vert F(t, s)\vert \le C\frac{\vert s \vert ^{\alpha (t)}}{\alpha (t)} + C_1, \text { for all } t \in [1, N]_\mathbb {N},~s\in \mathbb {R},$$

which implies that

$$\begin{aligned} B(u)&=\sum _{t=1}^{N} F(t, u(t)) \le \sum _{t=1}^{N} C\frac{\vert u(t)\vert ^{\alpha (t)}}{\alpha (t)} + NC_1 \nonumber \\&\le \frac{2NC(N+1)^{\frac{\alpha ^+}{2}}}{ \alpha ^-} \Vert u\Vert ^{\alpha ^{+}} +NC_1. \end{aligned}$$
(15)

In addition, the combination of (3) and (15) leads us to

$$\begin{aligned} T_{\lambda }(u) \ge \frac{2}{p^{+}(\sqrt{N})^{p^{-}-2}}\Vert u\Vert ^{p^{-}}-2N - \frac{N+1}{p^-} -\lambda \frac{2NC(N+1)^{\frac{\alpha ^+}{2}}}{ \alpha ^-} \Vert u\Vert ^{\alpha ^{+}} -\lambda NC_1. \end{aligned}$$

Since \(p ^{-}>\alpha ^{+}\), the above inequality means that \(T_{\lambda }\) is coercive.

Therefore, for any \(\lambda \in \Lambda _1\), the functional \(T_{\lambda }\) possesses at least three critical points which are exactly solutions for the problem \((P_\lambda )\). \(\square\)

Now, we provide two examples illustrating Theorems 4.1 and 4.2.

Example 1

Let \(N=3\) and \(p(r)= 3+\frac{r}{4}\) for \(r\in [0, 4]_{\mathbb {N}}\). Then, \(p^- = 3 \text { and } p^+ = 4\).

We consider the function

$$f(r,x) = r-2x, ~~ x \in \mathbb {R},~ r\in [1, 3]_{\mathbb {N}}$$

with

$$F(r,x) = rx-x^2, ~~ x \in \mathbb {R}, ~ r\in [1, 3]_{\mathbb {N}}.$$

In addition, if we choose \(\delta =1\) and \(\xi = (10\sqrt{3})^{\frac{1}{3}}\), we obtain \(R(\xi )=\frac{8}{3}\), and inequalities (4)–(5) are verified. Moreover, \(F(r,1)\ge 0\) for \(r=1, 2, 3\) and

$$\frac{\sum _{t=1}^3\max _{\vert s\vert \le (10\sqrt{3})^{\frac{1}{3}}} F(t, s)}{8/3}= \frac{21}{16}<\frac{3 \sum _{t=1}^N F(t, 1)}{4} = \frac{9}{4}.$$

Then, in view of Theorem 4.1, for all \(\lambda \in \left]\frac{4}{9}, \frac{16}{21} \right[\), the problem \((P_\lambda )\) has at least one nontrivial solution.

Example 2

We keep as in Example 1, \(N=3\), \(p(r)= \frac{r}{4} +3\) for \(r\in [0, 4]_{\mathbb {N}}\), \(\delta =1\) and \(\xi = (10\sqrt{3})^{\frac{1}{3}}\). We get \(p^- = 3\), \(p^+ = 4\) and \(R(\xi )=\frac{8}{3}\).

Let f be the function defined as:

$$f(r,x) = 20e^{4r} x^{3} +1, ~~ x \in \mathbb {R},~ r\in [1, 3]_{\mathbb {N}}$$

with

$$F(r,x) = 5e^{4r} x^{4 } + x, ~~ x \in \mathbb {R}, ~ r\in [1, 3]_{\mathbb {N}}.$$

We have

$$\liminf _{\vert x\vert \rightarrow \infty } \frac{F(r, x)}{\vert x\vert ^{p^+}} \ge 5e^{4} \ge \kappa ,$$

where \(\kappa\) is a constant such that \(\kappa > \frac{2^{p^+ +1}}{p^-} (N+1)^{\frac{p^+}{2}} = \frac{2^{9}}{3}.\)

Then, conditions \((H_2)\) and \((H_3)\) hold. So, by virtue of Theorem 4.2, for any parameter \(\lambda \in \biggl ]0, \frac{8/3}{\sum _{t=1}^3 \max _{\vert s\vert \le (10\sqrt{3})^{\frac{1}{3}}} F(t, s)}\biggr [\), the problem \((P_\lambda )\) has at least two nontrivial solutions.