Positive Solutions for Dirichlet BVP of PDE Involving \({\varphi_{p}}\)-Laplacian

Abstract

In this paper, we investigate the existence of infinitely many small solutions for problem $(f_{{\phi }_p})$ involving ${\phi }_p$-Laplacian by exploiting critical point theory. Moreover, the present study first attempts to address discrete Dirichlet problems with ${\phi }_p$-Laplacian in relation to some relative existing references. As far as we know, this research of the partial discrete bvp involves ${\phi }_p$-Laplacian for the first time. Our results are illustrated with three examples.

1. Introduction

We focus on the following problem, noted as $(f_{{\phi }_p})$

$$\begin{array}{cc}-[{\Delta }_1\left({\phi }_p\left({\Delta }_{1}s(\chi -1,\varrho )\right)\right)+{\Delta }_2\left({\phi }_p\left({\Delta }_{2}s(\chi ,\varrho -1)\right)\right)]+h(\chi ,\varrho ){\phi }_p\left(s(\chi ,\varrho )\right)\hfill & =\lambda f\left(\right(\chi ,\varrho ),s(\chi ,\varrho \left)\right),\hfill \\ & (\chi ,\varrho )\in \mathbb{Z}(1,c)\times \mathbb{Z}(1,d),\hfill \end{array}$$

with boundary conditions

$$\begin{array}{c}\hfill s(\chi ,0)=s(\chi ,d+1)=0,\chi \in \mathbb{Z}(0,c+1),\\ \hfill s(0,\varrho )=s(c+1,\varrho )=0,\varrho \in \mathbb{Z}(0,d+1),\end{array}$$

(1)

where c and d refer to positive constants, parameter $\lambda$ denotes a positive real number, and ${\Delta }_1$, ${\Delta }_2$ represent the forward difference operator defined by ${\Delta }_{1}s(\chi ,\varrho )=s(\chi +1,\varrho )-s(\chi ,\varrho )$ and ${\Delta }_{2}s(\chi ,\varrho )=s(\chi ,\varrho +1)-s(\chi ,\varrho )$, ${\Delta }_1^{2}s(\chi ,\varrho )={\Delta }_1\left({\Delta }_{1}s(\chi ,\varrho )\right)$ and ${\Delta }_2^{2}s(\chi ,\varrho )={\Delta }_2\left({\Delta }_{2}s(\chi ,\varrho )\right)$, $h(\chi ,\varrho )\ge 0$, ${\phi }_p\left(u\right)=\frac{{p\left|u\right|}^{p-2}u}{2\sqrt{1+{\left|u\right|}^p}}$, $p\ge 2$ and $f\left(\right(\chi ,\varrho ),·)\in C(\mathbb{R},\mathbb{R})$ for each $(\chi ,\varrho )\in \mathbb{Z}(1,c)\times \mathbb{Z}(1,d)$.

When $p=2$, problem $(f_{{\phi }_p})$ becomes the following problem, noted as $(f_{{\varphi }_c})$

$$\begin{array}{cc}-[{\Delta }_1\left({\varphi }_c\left({\Delta }_{1}s(\chi -1,\varrho )\right)\right)+{\Delta }_2\left({\varphi }_c\left({\Delta }_{2}s(\chi ,\varrho -1)\right)\right)]+h(\chi ,\varrho ){\varphi }_c\left(s(\chi ,\varrho )\right)\hfill & =\lambda f\left(\right(\chi ,\varrho ),s(\chi ,\varrho \left)\right),\hfill \\ \hfill & (\chi ,\varrho )\in \mathbb{Z}(1,c)\times \mathbb{Z}(1,d),\hfill \end{array}$$

with (1).

Difference equations have many applications; see [1,2,3,4,5,6,7,8,9]. Important tools for the study of difference equations can be found in [10,11]. In 2003, Yu and Guo [12] investigated a class of second-order difference equations. Since then, numerous researchers have presented a lot of findings, including those of periodic solutions [12,13,14], homoclinic solutions [15,16,17,18,19,20], and BVP [21,22,23,24,25,26,27,28].

We note that the aforementioned difference equations merely involve a single variable. However, difference equations with two variables are rarely studied and are known as pde. Lately, such equations have been widely used in many fields. The study of PDE is a challenging problem, gaining the attention of many researchers who have obtained some results [29,30,31,32,33,34,35,36,37,38].

In 2015, the authors [29] considered the following problem:

$${\Delta }_1^{2}s(\chi -1,\varrho )+{\Delta }_2^{2}s(\chi ,\varrho -1)+\lambda f((\chi ,\varrho ),s(\chi ,\varrho ))=0,(\chi ,\varrho )\in \mathbb{Z}(1,c)\times \mathbb{Z}(1,d),$$

(2)

with (1), they obtained the existence of three solutions of (2) at minimum.

Inspired by these results, the main purpose of this paper is to investigate the existence of infinitely many solutions of the PDE with ${\phi }_p$-Laplacian. The estimation of variational functional is the difficulty presented in this paper. It can be seen that the key role in this paper is the appropriate oscillating behavior of nonlinear term f at the origin. The problem we are considering has strong practical value [39,40]. In [31], the authors considered infinitely many solutions of the perturbed pde with $(p,q)$-Laplacian. In [28], the authors studied infinitely many solutions for the discrete BVP of the Kirchhoff type, and the problem only contains one variable. In [32], the authors considered the three solutions of the pde. In [41], the authors considered a class of fractional q-difference equations. In [42], the authors studied nonlinear fractional $(p,q)$-difference equations. However, in the present work, we consider infinitely many small solutions of the pde with ${\phi }_p$-Laplacian, there is no perturbation term, and it is different from the main tool used in [31]. Different from [28], the problems we are considering have two variables. This is different from the main methods used in [41,42].

The innovations of this paper are listed below.

(1)

We consider the positive solutions of the pde with ${\phi }_p$-Laplacian.

(2)

The difficulty to be overcome in this paper is the estimation of $\parallel s\parallel$ in Theorem 1.

We arrange the remainder of this article as follows. In Section 2, we construct the variational framework associated with $(f_{{\phi }_p})$. In Section 3, we present the major research results. In Section 4, our results are illustrated with three examples. Finally, in Section 5, conclusions are presented.

2. Preliminaries

We consider the following $cd$-dimensional Banach space:

$S=\{s:\mathbb{Z}(0,c+1)\times \mathbb{Z}(0,d+1)\to \mathbb{R}:s(\chi ,0)=s(\chi ,d+1)=0,\chi \in \mathbb{Z}(0,c+1)$ and $s(0,\varrho )=s(c+1,\varrho )=0,\varrho \in \mathbb{Z}(0,d+1)\}$,

$$\parallel s\parallel ={\left(\sum _{\varrho =1}^d\sum _{\chi =1}^{c+1}|{\Delta }_1{s(\chi -1,\varrho )|}^p+\sum _{\chi =1}^c\sum _{\varrho =1}^{d+1}{\left|{\Delta }_{2}s(\chi ,\varrho -1)\right|}^p\right)}^{\frac{1}{p}},s\in S.$$

We define

$$\Phi \left(s\right)={\Phi }_1\left(s\right)+{\Phi }_2\left(s\right),$$

$$\Psi \left(s\right)=\sum _{\varrho =1}^d\sum _{\chi =1}^{c}F((\chi ,\varrho ),s(\chi ,\varrho )),$$

for every $s\in S$, where

${\Phi }_1\left(s\right)={\displaystyle \sum _{\varrho =1}^d}{\displaystyle \sum _{\chi =1}^{c+1}}(\sqrt{1+|{\Delta }_1{s(\chi -1,\varrho )|}^p}-1)+{\displaystyle \sum _{\chi =1}^c}{\displaystyle \sum _{\varrho =1}^{d+1}}(\sqrt{1+|{\Delta }_2{s(\chi ,\varrho -1)|}^p}-1)$,

${\Phi }_2\left(s\right)={\displaystyle \sum _{\varrho =1}^d}{\displaystyle \sum _{\chi =1}^c}h(\chi ,\varrho )(\sqrt{1+{\left|s(\chi ,\varrho )\right|}^p}-1)$,

$F((\chi ,\varrho ),s)={\int }_0^{s}f((\chi ,\varrho ),\tau )d\tau$, for each $\left(\right(\chi ,\varrho ),s)\in \mathbb{Z}(1,c)\times \mathbb{Z}(1,d)\times \mathbb{R}$.

We let

$$I_{\lambda }\left(s\right)=\Phi \left(s\right)-\lambda \Psi \left(s\right),$$

for any $s\in S$. Obviously, $\Phi ,\Psi \in C^1(S,\mathbb{R})$. By careful calculation, we have

$$\begin{array}{cc}\hfill {\Phi }_1^{\prime }\left(s\right)\left(v\right)& =\underset{t\to 0}{lim}\frac{{\Phi }_1(s+tv)-{\Phi }_1\left(s\right)}{t}\hfill \\ \hfill & =\sum _{\varrho =1}^d\sum _{\chi =1}^{c+1}{\phi }_p\left({\Delta }_{1}s(\chi -1,\varrho )\right){\Delta }_{1}v(\chi -1,\varrho )+\sum _{\chi =1}^c\sum _{\varrho =1}^{d+1}{\phi }_p\left({\Delta }_{2}s(\chi ,\varrho -1)\right){\Delta }_{2}v(\chi ,\varrho -1)\hfill \\ \hfill & =-\sum _{\varrho =1}^d\sum _{\chi =1}^c{\Delta }_1{\phi }_p\left({\Delta }_{1}s(\chi -1,\varrho )\right)v(\chi ,\varrho )-\sum _{\chi =1}^c\sum _{\varrho =1}^d{\Delta }_2{\phi }_p\left({\Delta }_{2}s(\chi ,\varrho -1)\right)v(\chi ,\varrho ),\hfill \end{array}$$

$$\begin{array}{c}\hfill {\Phi }_2^{\prime }\left(s\right)\left(v\right)=\underset{t\to 0}{lim}\frac{{\Phi }_2(s+tv)-{\Phi }_2\left(s\right)}{t}=\sum _{\varrho =1}^d\sum _{\chi =1}^{c}h(\chi ,\varrho ){\phi }_p\left(s(\chi ,\varrho )\right)v(\chi ,\varrho ),\end{array}$$

and

$$\begin{array}{c}\hfill {\Psi }^{\prime }\left(s\right)\left(v\right)=\underset{t\to 0}{lim}\frac{\Psi (s+tv)-\Psi \left(s\right)}{t}=\sum _{\varrho =1}^d\sum _{\chi =1}^{c}f((\chi ,\varrho ),s(\chi ,\varrho ))v(\chi ,\varrho ),\end{array}$$

for $s,v\in S$.

Obviously, for any $s,v\in S$,

$$\begin{array}{cc}\hfill {(\Phi -\lambda \Psi )}^{\prime }\left(s\right)\left(v\right)& =-\sum _{\varrho =1}^d\sum _{\chi =1}^c[{\Delta }_1{\phi }_p\left({\Delta }_{1}s(\chi -1,\varrho )\right)+{\Delta }_2{\phi }_p\left({\Delta }_{2}s(\chi ,\varrho -1)\right)\hfill \\ \hfill & -h(\chi ,\varrho ){\phi }_p\left(s(\chi ,\varrho )\right)+\lambda f((\chi ,\varrho ),s(\chi ,\varrho ))]v(\chi ,\varrho ).\hfill \end{array}$$

(3)

As a result, we change the existence of a solution for the problem $(f_{{\phi }_p})$ into the existence of a critical point of $\Phi -\lambda \Psi$ on S.

To attain the positive solution for the problem $(f_{{\phi }_p})$, we offer the following Lemma.

Lemma 1.

Assume that there exists s: $\mathbb{Z}(0,c+1)\times \mathbb{Z}(0,d+1)\to \mathbb{R}$ such that

$$s(\chi ,\varrho )>0\mathit{or}-\left[{\Delta }_1\left({\phi }_p\left({\Delta }_{1}s(\chi -1,\varrho )\right)\right)+{\Delta }_2\left({\phi }_p\left({\Delta }_{2}s(\chi ,\varrho -1)\right)\right)\right]+h(\chi ,\varrho ){\phi }_p\left(s(\chi ,\varrho )\right)\ge 0,$$

(4)

for all $(\chi ,\varrho )\in \mathbb{Z}(1,c)\times \mathbb{Z}(1,d)$.

Then, either $s(\chi ,\varrho )>0$ for all $(\chi ,\varrho )\in \mathbb{Z}(1,c)\times \mathbb{Z}(1,d)$ or $s\equiv 0$.

Proof. 

Let $w\in \mathbb{Z}(1,c),v\in \mathbb{Z}(1,d)$ and

$$s(w,v)=min\left\{s\right(\chi ,\varrho ):\chi \in \mathbb{Z}(1,c),\varrho \in \mathbb{Z}(1,d\left)\right\}.$$

If $s(w,v)>0$, then it is obvious that $s(\chi ,\varrho )>0$ for all $\chi \in \mathbb{Z}(1,c),\varrho \in \mathbb{Z}(1,d)$.

If $s(w,v)\le 0$, then $s(w,v)=min\left\{s(\chi ,\varrho ):\chi \in \mathbb{Z}(1,c+1),\varrho \in \mathbb{Z}(1,d+1)\right\}$, since ${\Delta }_{1}s(w-1,v)=s(w,v)-s(w-1,v)\le 0$, ${\Delta }_{2}s(w,v-1)=s(w,v)-s(w,v-1)\le 0$, and ${\Delta }_{1}s(w,v)=s(w+1,v)-s(w,v)\ge 0$, ${\Delta }_{2}s(w,v)=s(w,v+1)-s(w,v)\ge 0$, ${\phi }_p\left(s\right)$ is increasing in s, and ${\phi }_p\left(0\right)=0$; thus,

$${\phi }_p\left({\Delta }_{1}s(w,v)\right)\ge 0\ge {\phi }_p\left({\Delta }_{1}s(w-1,v)\right),{\phi }_p\left({\Delta }_{2}s(w,v)\right)\ge 0\ge {\phi }_p\left({\Delta }_{2}s(w,v-1)\right).$$

According to ${\Delta }_1\left({\phi }_p\left({\Delta }_{1}s(w-1,v)\right)\right)={\phi }_p\left({\Delta }_{1}s(w,v)\right)-{\phi }_p\left({\Delta }_{1}s(w-1,v)\right)\ge 0$,

${\Delta }_2\left({\phi }_p\left({\Delta }_{2}s(w,v-1)\right)\right)={\phi }_p\left({\Delta }_{2}s(w,v)\right)-{\phi }_p\left({\Delta }_{2}s(w,v-1)\right)\ge 0$.

Thus,

$${\Delta }_1\left({\phi }_p\left({\Delta }_{1}s(w-1,v)\right)\right)+{\Delta }_2\left({\phi }_p\left({\Delta }_{2}s(w,v-1)\right)\right)\ge 0.$$

(5)

According to (4),

$${\Delta }_1\left({\phi }_p\left({\Delta }_{1}s(w-1,v)\right)\right)+{\Delta }_2\left({\phi }_p\left({\Delta }_{2}s(w,v-1)\right)\right)\le h(w,v){\phi }_p\left(s(w,v)\right)\le 0.$$

(6)

Combination of (5) with (6) produces

$${\Delta }_1\left({\phi }_p\left({\Delta }_{1}s(w-1,v)\right)\right)+{\Delta }_2\left({\phi }_p\left({\Delta }_{2}s(w,v-1)\right)\right)=0.$$

So

$${\phi }_p\left({\Delta }_{1}s(w,v)\right)={\phi }_p\left({\Delta }_{1}s(w-1,v)\right)=0.$$

That is, $s(w+1,v)=s(w-1,v)=s(w,v)$. If $w+1=c+1$, there is $s(w,v)=0$. Otherwise, $(w+1)\in \mathbb{Z}(1,c)$. Replacing w with $w+1$ results in $s(w+2,v)=s(w+1,v)$. Continuing this process $(c+1-w)$ times produces $s(w,v)=s(w+1,v)=s(w+2,v)=\dots =s(c+1,v)=0$. Similarly, there is $s(w,v)=s(w-1,v)=s(w-2,v)=\dots =s(0,v)=0$. Thus, $s(\chi ,v)=0$ for every $\chi \in \mathbb{Z}(1,c)$. It can be proven that $s\equiv 0$ in the same manner, and the proof is completed. □

3. Main Results

Put

$$\begin{array}{c}\hfill B=\underset{\xi \to 0^{+}}{lim\; sup}\frac{{\displaystyle \sum _{\varrho =1}^d}{\displaystyle \sum _{\chi =1}^c}F((\chi ,\varrho ),\xi )}{{\xi }^p},H=\sum _{\varrho =1}^d\sum _{\chi =1}^{c}h(\chi ,\varrho ),\\ \hfill h_{*}=min\left\{h(\chi ,\varrho ):\chi \in \mathbb{Z}(1,c),\varrho \in \mathbb{Z}(1,d)\right\}.\end{array}$$

Theorem 1.

Suppose two real sequences $\left\{a_t\right\}$ and $\left\{b_t\right\}$ exist, with $b_t>0$ and $\underset{t\to +\infty }{lim}{b}_t=0$, such that

$$(2c+2d+H)\left(\sqrt{1+a_t^p}-1\right)<(1+h_{*})\left(\sqrt{1+min\left\{\frac{4^p}{{(c+d+2)}^{p-1}},1\right\}{b}_t^p}-1\right),$$

(7)

for $t\in \mathbb{Z}\left(1\right)$, and

$$A=\underset{t\to \infty }{lim\; inf}\frac{{\displaystyle \sum _{\varrho =1}^d}{\displaystyle \sum _{\chi =1}^c}\underset{\left|\xi \right|\le b_t}{max}F((\chi ,\varrho ),\xi )-{\displaystyle \sum _{\varrho =1}^d}{\displaystyle \sum _{\chi =1}^c}F((\chi ,\varrho ),a_t)}{(1+h_{*})\left(\sqrt{1+min\left\{\frac{4^p}{{(c+d+2)}^{p-1}},1\right\}{b}_t^p}-1\right)-(2c+2d+H)\left(\sqrt{1+a_t^p}-1\right)}<\frac{2B}{2c+2d+H}.$$

(8)

Then, for each $\lambda \in \left(\frac{2c+2d+H}{2B},\frac{1}{A}\right)$, problem $(f_{{\phi }_p})$ admits a nontrivial solution sequence which converges to zero.

Proof. 

We prove the theorem using Theorem 2.1 of [43] as an initial step, which is obviously satisfied.

We let

$$r_t=(1+h_{*})\left(\sqrt{1+min\left\{\frac{4^p}{{(c+d+2)}^{p-1}},1\right\}{b}_t^p}-1\right).$$

We let $\Phi \left(s\right)={\Phi }_1\left(s\right)+{\Phi }_2\left(s\right)<r_t$, $s\in S$, where

$${\Phi }_1\left(s\right)=\sum _{\varrho =1}^d\sum _{\chi =1}^{c+1}\left(\sqrt{1+|{\Delta }_1{s(\chi -1,\varrho )|}^p}-1\right)+\sum _{\chi =1}^c\sum _{\varrho =1}^{d+1}\left(\sqrt{1+|{\Delta }_2{s(\chi ,\varrho -1)|}^p}-1\right),$$

$${\Phi }_2\left(s\right)=\sum _{\varrho =1}^d\sum _{\chi =1}^{c}h(\chi ,\varrho )\left(\sqrt{1+{\left|s(\chi ,\varrho )\right|}^p}-1\right).$$

We put

$$v_1(\chi ,\varrho )=\sqrt{1+|{\Delta }_1{s(\chi -1,\varrho )|}^p}-1,v_2(\chi ,\varrho )=\sqrt{1+|{\Delta }_2{s(\chi ,\varrho -1)|}^p}-1,$$

for $(\chi ,\varrho )\in \mathbb{Z}(1,c+1)\times \mathbb{Z}(1,d+1)$. And

$$\begin{array}{cc}\hfill \sum _{\varrho =1}^d\sum _{\chi =1}^{c+1}{\left|{\Delta }_{1}s(\chi -1,\varrho )\right|}^p& \le {\left(\sum _{\varrho =1}^d\sum _{\chi =1}^{c+1}{v}_1(\chi ,\varrho )\right)}^2+2\sum _{\varrho =1}^d\sum _{\chi =1}^{c+1}{v}_1(\chi ,\varrho ),\hfill \end{array}$$

(9)

$$\begin{array}{cc}\hfill \sum _{\chi =1}^c\sum _{\varrho =1}^{d+1}{\left|{\Delta }_{2}s(\chi ,\varrho -1)\right|}^p& \le {\left(\sum _{\chi =1}^c\sum _{\varrho =1}^{d+1}{v}_2(\chi ,\varrho )\right)}^2+2\sum _{\chi =1}^c\sum _{\varrho =1}^{d+1}{v}_2(\chi ,\varrho ).\hfill \end{array}$$

(10)

According to (9) and (10), there is

$$\begin{array}{cc}\hfill {\parallel s\parallel }^p& \le {\left[\sum _{\varrho =1}^d\sum _{\chi =1}^{c+1}{v}_1(\chi ,\varrho )+\sum _{\chi =1}^c\sum _{\varrho =1}^{d+1}{v}_2(\chi ,\varrho )\right]}^2+2\left[\sum _{\varrho =1}^d\sum _{\chi =1}^{c+1}{v}_1(\chi ,\varrho )+\sum _{\chi =1}^c\sum _{\varrho =1}^{d+1}{v}_2(\chi ,\varrho )\right]\hfill \\ \hfill & ={\left({\Phi }_1\left(s\right)\right)}^2+2{\Phi }_1\left(s\right),\hfill \end{array}$$

(11)

which implies that

$${\Phi }_1\left(s\right)\ge \sqrt{{\parallel s\parallel }^p+1}-1.$$

According to (Ref. [30], Proposition 1), there is

$${\parallel s\parallel }_{\infty }^p\le \frac{{(c+d+2)}^{p-1}}{4^p}{\parallel s\parallel }^p,$$

where

$${\parallel s\parallel }_{\infty }=max\left\{\left|s\right(\chi ,\varrho \left)\right|:(\chi ,\varrho )\in \mathbb{Z}(1,c)\times \mathbb{Z}(1,d)\right\},$$

for $s\in S$.

Thus, there is

$${\Phi }_1\left(s\right)\ge \sqrt{\frac{4^p}{{(c+d+2)}^{p-1}}{\parallel s\parallel }_{\infty }^p+1}-1.$$

Obviously,

$${\Phi }_2\left(s\right)\ge h_{*}\left(\sqrt{{\parallel s\parallel }_{\infty }^p+1}-1\right).$$

Therefore,

$$(1+h_{*})\left(\sqrt{1+min\left\{\frac{4^p}{{(c+d+2)}^{p-1}},1\right\}{\parallel s\parallel }_{\infty }^p}-1\right)\le {\Phi }_1\left(s\right)+{\Phi }_2\left(s\right)<r_t,$$

which implies that

$${\parallel s\parallel }_{\infty }^p<max\left\{\frac{{(c+d+2)}^{p-1}}{4^p},1\right\}\left[{\left(\frac{r_t}{1+h_{*}}\right)}^2+\frac{2r_t}{1+h_{*}}\right]=b_t^p.$$

By the definition of $\psi$, there is

$$\psi \left(r_t\right)\le \underset{s\in {\Phi }^{-1}(-\infty ,r_t)}{inf}\frac{{\displaystyle \sum _{\varrho =1}^d}{\displaystyle \sum _{\chi =1}^c}\underset{\left|\xi \right|\le b_t}{max}F((\chi ,\varrho ),\xi )-{\displaystyle \sum _{\varrho =1}^d}{\displaystyle \sum _{\chi =1}^c}F((\chi ,\varrho ),s(\chi ,\varrho ))}{(1+h_{*})\left(\sqrt{1+min\left\{\frac{4^p}{{(c+d+2)}^{p-1}},1\right\}{b}_t^p}-1\right)-\Phi \left(s\right)},$$

for each $t\in \mathbb{Z}\left(1\right)$.

Let $w_t\in S$; define

$$w_t(\chi ,\varrho )=\left\{\begin{array}{cc}{a}_t,\hfill & \mathrm{if}(\chi ,\varrho )\in \mathbb{Z}(1,c)\times \mathbb{Z}(1,d),\hfill \\ 0,\hfill & \mathrm{if}\chi =0,\varrho \in \mathbb{Z}(0,d+1)\mathrm{or}\chi =c+1,\varrho \in \mathbb{Z}(0,d+1),\hfill \\ 0,\hfill & \mathrm{if}\varrho =0,\chi \in \mathbb{Z}(0,c+1)\mathrm{or}\varrho =d+1,\chi \in \mathbb{Z}(0,c+1).\hfill \end{array}\right.$$

Then, using (7),

$$\Phi \left(w_t\right)=(2c+2d+H)\left(\sqrt{1+a_t^p}-1\right)<r_t.$$

Thus,

$$\begin{array}{cc}\hfill \psi \left(r_t\right)& \le \frac{{\displaystyle \sum _{\varrho =1}^d}{\displaystyle \sum _{\chi =1}^c}\underset{\left|\xi \right|\le b_t}{max}F((\chi ,\varrho ),\xi )-{\displaystyle \sum _{\varrho =1}^d}{\displaystyle \sum _{\chi =1}^c}F((\chi ,\varrho ),\left(w_t(\chi ,\varrho )\right))}{(1+h_{*})\left(\sqrt{1+min\left\{\frac{4^p}{{(c+d+2)}^{p-1}},1\right\}{b}_t^p}-1\right)-\Phi \left(w_t\right)}\hfill \\ \hfill & =\frac{{\displaystyle \sum _{\varrho =1}^d}{\displaystyle \sum _{\chi =1}^c}\underset{\left|\xi \right|\le b_t}{max}F((\chi ,\varrho ),\xi )-{\displaystyle \sum _{\varrho =1}^d}{\displaystyle \sum _{\chi =1}^c}F((\chi ,\varrho ),a_t)}{(1+h_{*})\left(\sqrt{1+min\left\{\frac{4^p}{{(c+d+2)}^{p-1}},1\right\}{b}_t^p}-1\right)-(2c+2d+H)\left(\sqrt{1+a_t^p}-1\right)}.\hfill \end{array}$$

(12)

Therefore, by (8), it is known that

$$\gamma \le \underset{t\to +\infty }{lim\; inf}\psi \left(r_t\right)\le A<+\infty .$$

Apparently, $s\equiv 0$ is a global minimum of $\Phi$. With the aim of acquiring the conclusion $\left(a_2\right)$ Theorem 2.1 of [43], it is necessary to demonstrate that $s\equiv 0$ is not a local minimum of $I_{\lambda }$.

In the case of $B=+\infty$, suppose $\left\{c_t\right\}$ is a positive real sequence, and $\underset{t\to +\infty }{lim}{c}_t=0$, such that

$$\sum _{\varrho =1}^d\sum _{\chi =1}^{c}F((\chi ,\varrho ),c_t)\ge \frac{(2c+2d+H)c_t^p}{\lambda },\mathrm{for}t\in \mathbb{Z}\left(1\right).$$

Define sequence $\left\{{\eta }_t\right\}$ in S with

$${\eta }_t(\chi ,\varrho )=\left\{\begin{array}{cc}{c}_t,\hfill & (\chi ,\varrho )\in \mathbb{Z}(1,c)\times \mathbb{Z}(1,d),\hfill \\ 0,\hfill & \mathrm{if}\chi =0,\varrho \in \mathbb{Z}(0,d+1)\mathrm{or}\chi =c+1,\varrho \in \mathbb{Z}(0,d+1),\hfill \\ 0,\hfill & \mathrm{if}\varrho =0,\chi \in \mathbb{Z}(0,c+1)\mathrm{or}\varrho =d+1,\chi \in \mathbb{Z}(0,c+1).\hfill \end{array}\right.$$

Obviously, there is

$$\begin{array}{cc}\hfill I_{\lambda }\left({\eta }_t\right)& =(2c+2d+H)\left(\sqrt{1+c_t^p}-1\right)-\lambda \left(\sum _{\varrho =1}^d\sum _{\chi =1}^{c}F((\chi ,\varrho ),c_t)\right)\hfill \\ \hfill & \le \frac{2c+2d+H}{2}{c}_t^p-(2c+2d+H)c_t^p\hfill \\ \hfill & =-\frac{2c+2d+H}{2}{c}_t^p\hfill \\ \hfill & <0.\hfill \end{array}$$

(13)

If $B<+\infty$, since $\lambda >\frac{2c+2d+H}{2B}$, ${\xi }_0>0$ can be selected. Thus, there is

$$2c+2d+H-2\lambda (B-{\xi }_0)<0.$$

Accordingly, a positive real sequence $\left\{c_t\right\}$ can be found, which satisfies $\underset{t\to +\infty }{lim}{c}_t=0$, and

$$(B-{\xi }_0)c_t^p\le \sum _{\varrho =1}^d\sum _{\chi =1}^{c}F((\chi ,\varrho ),c_t)\le (B+{\xi }_0)c_t^p.$$

Suppose that the sequence $\left\{{\eta }_t\right\}$ in S is identical to the case in which $B=+\infty$. Thus, there is

$$\begin{array}{cc}\hfill I_{\lambda }\left({\eta }_t\right)& =(2c+2d+H)\left(\sqrt{1+c_t^p}-1\right)-\lambda \left(\sum _{\varrho =1}^d\sum _{\chi =1}^{c}F((\chi ,\varrho ),c_t)\right)\hfill \\ \hfill & \le \frac{2c+2d+H-2\lambda (B-{\xi }_0)}{2}{c}_t^p\hfill \\ \hfill & <0.\hfill \end{array}$$

(14)

Since $I_{\lambda }\left(0\right)=0$, through the combination of the above two cases, it can be seen that $s\equiv 0$ is not a local minimum of $I_{\lambda }$ and using Theorem 2.1 of [43], it can be determined that Theorem 1 holds.

Remark 1.

When $B=+\infty$, for each $\lambda \in (0,\frac{1}{A})$, we obtain the same result as Theorem 1.

Now, we let

$$A^{*}=\underset{\xi \to 0^{+}}{lim\; inf}\frac{2max\left\{\frac{{(c+d+2)}^{p-1}}{4^p},1\right\}{\displaystyle \sum _{\varrho =1}^d}{\displaystyle \sum _{\chi =1}^c}\underset{\left|\eta \right|\le \xi }{max}F((\chi ,\varrho ),\eta )}{(1+h_{*}){\xi }^p}.$$

When $A^{*}=0$, we agree to read $\frac{1}{A^{*}}=+\infty$.

Corollary 1.

We assume that

$$A^{*}<\frac{2B}{2c+2d+H},$$

(15)

accordingly, for every $\lambda \in \left(\frac{2c+2d+H}{2B},\frac{1}{A^{*}}\right)$, the same result as Theorem 1 is obtained.

Now, taking into account that positive solutions for the problem $(f_{{\phi }_p})$, we have

Corollary 2.

We suppose that $f\left(\right(\chi ,\varrho ),0)\ge 0$ for each $(\chi ,\varrho )\in \mathbb{Z}(1,c)\times \mathbb{Z}(1,d)$, and

$$\widehat{A}=\underset{\xi \to 0^{+}}{lim\; inf}\frac{2max\left\{\frac{{(c+d+2)}^{p-1}}{4^p},1\right\}{\displaystyle \sum _{\varrho =1}^d}{\displaystyle \sum _{\chi =1}^c}\underset{0\le s\le \xi }{max}{\int }_0^{s}f((\chi ,\varrho ),t)dt}{(1+h_{*}){\xi }^p}<\frac{2B}{2c+2d+H},$$

(16)

then, for every $\lambda \in \left(\frac{2c+2d+H}{2B},\frac{1}{\widehat{A}}\right)$, problem $(f_{{\phi }_p})$ admits a positive solution sequence which converges to zero.

Proof. 

We let

$$f^{*}((\chi ,\varrho ),\xi )=\left\{\begin{array}{cc}f\left(\right(\chi ,\varrho ),\xi ),\hfill & \mathrm{if}\xi >0,\hfill \\ f\left(\right(\chi ,\varrho ),0),\hfill & \mathrm{if}\xi \le 0,\hfill \end{array}\right.$$

(17)

for $(\chi ,\varrho )\in \mathbb{Z}(1,c)\times \mathbb{Z}(1,d)$; since $f\left(\right(\chi ,\varrho ),0)\ge 0$, we see that

$$\underset{0\le \left|s\right|\le \xi }{max}{\int }_0^s{f}^{*}((\chi ,\varrho ),t)dt=\underset{0\le s\le \xi }{max}{\int }_0^{s}f((\chi ,\varrho ),t)dt,$$

for all $\xi \ge 0$. It is clear from Corollary 1 that problem $\left(f_{{\phi }_p}\right)$ with f replaced by $f^{*}$ admits the same result as Theorem 1 for every $\lambda \in \left(\frac{2c+2d+H}{2B},\frac{1}{\widehat{A}}\right)$. Furthermore, the foregoing solutions are positive based on Lemma 1. □

In particular, when p = 2, according to Theorem 1, we can obtain the result for the multiplicity of the Dirichlet problems with ${\varphi }_c$-Laplacian.

Theorem 2.

Suppose two real sequences $\left\{a_t\right\}$ and $\left\{b_t\right\}$ exist, with $b_t>0$ and $\underset{t\to +\infty }{lim}{b}_t=0$, such that

$$(2c+2d+H)\left(\sqrt{1+a_t^2}-1\right)<(1+h_{*})\left(\sqrt{1+min\left\{\frac{4^2}{c+d+2},1\right\}{b}_t^2}-1\right),$$

(18)

for $t\in \mathbb{Z}\left(1\right)$, and

$$C<\frac{2E}{2c+2d+H}.$$

(19)

Then, for each $\lambda \in \left(\frac{2c+2d+H}{2E},\frac{1}{C}\right)$, problem $(f_{{\varphi }_c})$ admits a nontrival solution sequence which converges to zero, where

$$C=\underset{t\to \infty }{lim\; inf}\frac{{\displaystyle \sum _{\varrho =1}^d}{\displaystyle \sum _{\chi =1}^c}\underset{\left|\xi \right|\le b_t}{max}F((\chi ,\varrho ),\xi )-{\displaystyle \sum _{\varrho =1}^d}{\displaystyle \sum _{\chi =1}^c}F((\chi ,\varrho ),a_t)}{(1+h_{*})\left(\sqrt{1+min\left\{\frac{4^2}{c+d+2},1\right\}{b}_t^2}-1\right)-(2c+2d+H)\left(\sqrt{1+a_t^2}-1\right)},$$

$$E=\underset{\xi \to 0^{+}}{lim\; sup}\frac{{\displaystyle \sum _{\varrho =1}^d}{\displaystyle \sum _{\chi =1}^c}F((\chi ,\varrho ),\xi )}{{\xi }^2}.$$

Remark 2.

With $E=+\infty$, it can be concluded from Theorem 2 that each $\lambda \in (0,\frac{1}{C})$, the same result as Theorem 2.

Corollary 3.

Assume that

$$C_{*}=\underset{\xi \to 0^{+}}{lim\; inf}\frac{2max\left\{\frac{c+d+2}{4^2},1\right\}{\displaystyle \sum _{\varrho =1}^d}{\displaystyle \sum _{\chi =1}^c}\underset{\left|\eta \right|\le \xi }{max}F((\chi ,\varrho ),\eta )}{(1+h_{*}){\xi }^2}<\frac{2E}{2c+2d+H}.$$

(20)

Then, for each $\lambda \in \left(\frac{2c+2d+H}{2E},\frac{1}{C_{*}}\right)$, the same result as Theorem 2 is obtained.

Remark 3.

With $E=+\infty$, it can be concluded from Corollary 3 that each $\lambda \in (0,\frac{1}{C_{*}})$, the same result as Corollary 3. With $C_{*}=0$, it can be concluded from Corollary 3 that each $\lambda \in \left(\frac{2c+2d+H}{2E},+\infty \right)$, the same result as Corollary 3. With $E=+\infty$ and $C_{*}=0$, it can be concluded from Corollary 3 that each $\lambda \in (0,+\infty )$, the same result as Corollary 3.

Corollary 4.

Suppose $f\left(\right(\chi ,\varrho ),0)\ge 0$ for every $(\chi ,\varrho )\in \mathbb{Z}(1,c)\times \mathbb{Z}(1,d)$, and

$$\widehat{C}=\underset{\xi \to 0^{+}}{lim\; inf}\frac{2max\left\{\frac{c+d+2}{4^2},1\right\}{\displaystyle \sum _{\varrho =1}^d}{\displaystyle \sum _{\chi =1}^c}\underset{0\le s\le \xi }{max}{\int }_0^{s}f((\chi ,\varrho ),t)dt}{(1+h_{*}){\xi }^2}<\frac{2E}{2c+2d+H}.$$

(21)

Then, for each $\lambda \in \left(\frac{2c+2d+H}{2E},\frac{1}{\widehat{C}}\right)$, problem $(f_{{\varphi }_c})$ admits a sequence of positive solutions which converges to zero.

Remark 4.

With $E=+\infty$, it can be concluded from Corollary 4 that each $\lambda \in (0,\frac{1}{\widehat{C}})$, the same result as Corollary 4. With $\widehat{C}=0$, it can be concluded from Corollary 4 that each $\lambda \in \left(\frac{2c+2d+H}{2E},+\infty \right)$, the same result as Corollary 4. With $E=+\infty$ and $\widehat{C}=0$, it can be concluded from Corollary 4 that each $\lambda \in (0,+\infty )$, the same result as Corollary 4.

4. Examples

Example 1.

Consider problem $(f_{{\phi }_3})$ with

$$f((\chi ,\varrho ),s)=f\left(s\right)=\left\{\begin{array}{cc}{\left|s\right|}^{p-2}s(2p+p\epsilon +2psin(\epsilon lns))+2\epsilon cos(\epsilon lns)),\hfill & s\ne 0,\hfill \\ 0,\hfill & s=0,\hfill \end{array}\right.$$

(22)

for every $(\chi ,\varrho )\in \mathbb{Z}(1,2)\times \mathbb{Z}(1,2)$. Then,

$$F((\chi ,\varrho ),s)=F\left(s\right)={\int }_0^{s}f\left(\xi \right)d\xi =s^p(2+\epsilon +2sin(\epsilon lns)),\mathit{for}s>0.$$

Since $f\left(s\right)\ge 0$ for $s\ge 0$, it can be seen that $F\left(s\right)$ is increasing in $s\in [0,+\infty )$.

Suppose $h(\chi ,\varrho )=\frac{1}{\chi \times \varrho }$, so $H={\displaystyle \sum _{\chi =1}^2}{\displaystyle \sum _{\varrho =1}^2}h(\chi ,\varrho )=\frac{9}{4}$.

Let

$${\xi }_t=exp(-\frac{2t\pi +\frac{3}{2}\pi }{\epsilon }),{\tau }_t=exp(-\frac{2t\pi +\frac{\pi }{2}}{\epsilon }).$$

Then, $\underset{t\to +\infty }{lim}{\xi }_t=0=\underset{t\to +\infty }{lim}{\tau }_t$, and

$$\frac{F\left({\xi }_t\right)}{{\xi }_t^p}=4+\epsilon ,\frac{{max}_{0\le s\le {\tau }_t}F\left(s\right)}{{\tau }_t^p}=\frac{F\left({\tau }_t\right)}{{\tau }_t^p}=\epsilon ,$$

which implies that

$$\widehat{A}\le \frac{2cdmax\left\{\frac{{(c+d+2)}^{p-1}}{4^p},1\right\}\epsilon }{1+h_{*}},B\ge (4+\epsilon )cd.$$

Let $\epsilon =\frac{1}{10,000}$, such that

$$\frac{2cdmax\left\{\frac{{(c+d+2)}^{p-1}}{4^p},1\right\}\epsilon }{1+h_{*}}=0.00064<\frac{2(4+\epsilon )cd}{2c+2d+H}\approx 3.12203,$$

namely,

$$\widehat{A}\le \frac{2cdmax\left\{\frac{{(c+d+2)}^{p-1}}{4^p},1\right\}\epsilon }{1+h_{*}}<\frac{2(4+\epsilon )cd}{2c+2d+H}\le \frac{2B}{2c+2d+H}.$$

Then, (16) holds.

In accordance with Corollary 2, for every $\lambda \in (0.32030,1562.5)$, the problem considered leads to the same conclusion as Corollary 2.

Next, we provide another example.

Example 2.

Consider problem $(f_{{\varphi }_c})$ with

$$f((\chi ,\varrho ),s)=f\left(s\right)=\left\{\begin{array}{cc}s(6+8\epsilon +6cos(\epsilon ln\left|s\right|)-3\epsilon sin(\epsilon ln\left|s\right|\left)\right),\hfill & s\ne 0,\hfill \\ 0,\hfill & s=0,\hfill \end{array}\right.$$

(23)

for every $(\chi ,\varrho )\in \mathbb{Z}(1,2)\times \mathbb{Z}(1,2)$. Then,

$$F((\chi ,\varrho ),s)=F\left(s\right)={\int }_0^{s}f\left(\xi \right)d\xi =s^2(3+4\epsilon +3cos(\epsilon lns)),\mathit{for}s>0.$$

Since $f\left(s\right)\ge 0$ for $s\ge 0$, it can be seen that $F\left(s\right)$ is increasing in $s\in [0,+\infty )$.

Suppose that $h(\chi ,\varrho )=\frac{1}{\chi +\varrho }$, so $H={\displaystyle \sum _{\chi =1}^2}{\displaystyle \sum _{\varrho =1}^2}h(\chi ,\varrho )=\frac{17}{12}$.

Let

$${\omega }_t=exp(-\frac{2t\pi }{\epsilon }),{\nu }_t=exp(-\frac{2t\pi +\pi }{\epsilon }).$$

Then, $\underset{t\to +\infty }{lim}{\omega }_t=0=\underset{t\to +\infty }{lim}{\nu }_t$, and

$$\frac{F\left({\omega }_t\right)}{{\omega }_t^2}=6+4\epsilon ,\frac{{max}_{0\le s\le {\nu }_t}F\left(s\right)}{{\nu }_t^2}=\frac{F\left({\nu }_t\right)}{{\nu }_t^2}=4\epsilon ,$$

which implies that

$$\widehat{C}\le \frac{8cdmax\left\{\frac{c+d+2}{4^2},1\right\}\epsilon }{1+h_{*}},E\ge (6+4\epsilon )cd.$$

Let $\epsilon =\frac{1}{10,000}$, such that

$$\frac{8cdmax\left\{\frac{c+d+2}{4^2},1\right\}\epsilon }{1+h_{*}}=0.00256<\frac{2(6+4\epsilon )cd}{2c+2d+H}\approx 5.09750,$$

namely,

$$\widehat{C}\le \frac{8cdmax\left\{\frac{c+d+2}{4^2},1\right\}\epsilon }{1+h_{*}}<\frac{2(6+4\epsilon )cd}{2c+2d+H}\le \frac{2E}{2c+2d+H}.$$

Then, (21) holds.

In accordance with Corollary 4, for every $\lambda \in (0.19617,390.625)$, the problem considered leads to the same conclusion as Corollary 4.

To exemplify Corollary 1, we offer the following example.

Example 3.

Consider problem $(f_{{\phi }_3})$ with

$$f((\chi ,\varrho ),s)=f\left(s\right)=\left\{\begin{array}{cc}{s}^{p-1}p(p+4p\epsilon +pcos(\epsilon lns))-\epsilon sin(\epsilon lns)),\hfill & s\ne 0,\hfill \\ 0,\hfill & s=0,\hfill \end{array}\right.$$

(24)

for every $(\chi ,\varrho )\in \mathbb{Z}(1,2)\times \mathbb{Z}(1,2)$. Then

$$F((\chi ,\varrho ),s)=F\left(s\right)={\int }_0^{s}f\left(\xi \right)d\xi =s^p(1+4\epsilon +cos(\epsilon lns)),\mathit{for}s>0.$$

Since $f\left(s\right)\ge 0$ for $s\ge 0$, it can be seen that $F\left(s\right)$ is increasing in $s\in [0,+\infty )$.

Suppose $h(\chi ,\varrho )=\frac{1}{\chi \times \varrho }$, so $H={\displaystyle \sum _{\chi =1}^2}{\displaystyle \sum _{\varrho =1}^2}h(\chi ,\varrho )=\frac{9}{4}$, $h_{*}=min\{h(\chi ,\varrho ):\chi \in \mathbb{Z},\varrho \in \mathbb{Z}\}=\frac{1}{4}$.

Let $\epsilon =\frac{1}{10,000}$, such that

$$A^{*}=\underset{\xi \to 0^{+}}{lim\; inf}\frac{2max\left\{\frac{{(c+d+2)}^{p-1}}{4^p},1\right\}{\displaystyle \sum _{\varrho =1}^d}{\displaystyle \sum _{\chi =1}^c}\underset{\left|\eta \right|\le \xi }{max}F((\chi ,\varrho ),\eta )}{(1+h_{*}){\xi }^p}=\underset{\xi \to 0^{+}}{lim\; inf}\frac{8\times 4\epsilon }{1.25}=0.00256.$$

$$B=\underset{\xi \to 0^{+}}{lim\; sup}\frac{{\displaystyle \sum _{\varrho =1}^d}{\displaystyle \sum _{\chi =1}^c}F((\chi ,\varrho ),\xi )}{{\xi }^p}=4\times (2+4\epsilon )=8+16\epsilon .$$

$$\frac{2B}{2c+2d+H}=\frac{2\times (8+16\epsilon )}{2\times 2+2\times 2+\frac{9}{4}}=1.561,$$

namely,

$$A^{*}<\frac{2B}{2c+2d+H}.$$

Then, (15) holds.

In accordance with Corollary 1, for every $\lambda \in (0.640,390.625)$, the problem we consider leads to the same conclusion as Corollary 1.

5. Conclusions

In this paper, the existence of infinitely many solutions is acquired for the PDE involving ${\phi }_p$-Laplacian. In [27], the difference equations are different from the equation we study. However, the equations in our study contain two variables and are more complex in computational processing than that in [27]. Theorems 1 and 2 show that the existence of infinitely many solutions is obtained. First, from Theorem 2.1 of [43], we acquire a nontrivial solution sequence which converges to zero in Theorem 1. When $a_t=0$ for $t\in \mathbb{Z}\left(1\right)$, the same result as the conclusion of Theorem 1 is obtained in Corollary 1. Furthermore, by Lemma 1, we obtain a positive solution sequence which converges to zero in Corollary 2. When $p=2$, according to Theorem 1, we can obtain the result for the multiplicity of the Dirichlet problems with ${\varphi }_c$-Laplacian, and the result is the same as above.