Nontrivial Solutions of a Class of Fourth-Order Elliptic Problems with Potentials

Abstract

This paper deals with a fourth-order elliptic equation with Dirichlet boundary conditions. Using a variant form of the mountain pass theorem, we prove the existence of nontrivial solutions to this problem. Furthermore, we discuss the fundamental properties of the representation of the solution by considering two cases. Our results not only make previous results more general but also show new insights into fourth-order elliptic problems.

1. Introduction

Since the pioneering studies of Ambrosetti, Brezis, and Cerami (see [1,2]), there has been a significant amount of interest in second-order elliptic problems with mixed nonlinearities. However, as far as the authors are aware, the case of the fourth-order elliptic issue appears to receive very few results. With regard to u, the classical criteria of sublinearity at 0 and superlinearity at ∞ are introduced here as local counterparts.

We now provide a brief summary of the relevant findings. In [3], Ying and Jiang investigated the following nonlinear fourth-order elliptic equations

$$\left\{\begin{array}{c}{\Delta }^{2}u-\Delta u+V\left(x\right)u-\frac{1}{2}u\Delta \left(u^2\right)=g\left(u\right),\mathrm{in}{\mathbb{R}}^N,\hfill \\ u\in H^2\left({\mathbb{R}}^N\right)\hfill \end{array}\right.$$

where ${\Delta }^2=\Delta (\Delta )$ is the biharmonic operator, V is an indefinite potential, and g grows subcritically and satisfies the Ambrosetti–Rabinowitz type condition $g\left(t\right)t\ge$$\mu G\left(t\right)\ge 0$ with $\mu >3$. Using Morse theory, they obtained nontrivial solutions of the above equations.

Moreover, Mao and Wang [4] proved the multiplicity solutions of the following nonlocal fourth-order elliptic problem

$${\Delta }^{2}u-\left(1+\lambda {\int }_{{\mathbb{R}}^3}{|\nabla u|}^{2}dx\right)\Delta u+V\left(x\right)u=f(x,u)\mathrm{in}{\mathbb{R}}^3,$$

where ${\Delta }^2=\Delta (\Delta )$ is the bi-harmonic operator, and $\lambda \ge 0$ is a constant. They focused on the case in which $f(x,u)$ involves a combination of convex and concave terms, and the potential $V\left(x\right)$ is allowed to be sign-changing.

In this work, we study the following fourth-order elliptic equations

$$\left\{\begin{array}{cc}{\Delta }^{2}u+c\Delta u=\lambda V\left(x\right){\left|u\right|}^{p-2}u+g(x,u),\hfill & \mathrm{in}\mathsf{\Omega },\hfill \\ u=\Delta u=0\hfill & \mathrm{on}\partial \mathsf{\Omega },\hfill \end{array}\right.$$

(1)

where $\mathsf{\Omega }\subset {\mathbb{R}}^N$ is a smooth bounded domain, ${\Delta }^2$ denotes a biharmonic operator, $1<p<2$, $\lambda$ is a non-negative parameter, c is a constant, and $V\in L^{\infty }\left(\mathsf{\Omega }\right)$. The potential $V\left(x\right)\ge 0$ and the function $g(x,t):\overline{\mathsf{\Omega }}\times \mathbb{R}\to \mathbb{R}$ are both continuous.

An and Liu used the well-known mountain pass theorem in [5] to demonstrate the existence of nontrivial solutions to (1). They provide the following assumptions for $g(x,t)$:

$\left(A_1\right)g(x,t)\in C(\overline{\mathsf{\Omega }}\times \mathbb{R});f(x,t)\ge 0,\forall x\in \overline{\mathsf{\Omega }},t>0;f(x,t)\equiv 0,\forall x\in \overline{\mathsf{\Omega }},t\le 0$.

$\left(A_2\right)\left|g(x,t)\right|\le a\left(x\right)+{b\left|t\right|}^p$, for a.e. $x\in \mathsf{\Omega }$ and $\forall t\in \mathbb{R}$, where $a\left(x\right)\in L^q\left(\mathsf{\Omega }\right),b\in \mathbb{R}$ and $1<p<\frac{N+4}{N-4},N>4$ and $\frac{1}{p}+\frac{1}{q}=1$.

$\left(A_3\right)$ For a.e. $x\in \mathsf{\Omega },\frac{g(x,t)}{t}$ is nondecreasing with respect to $t>0$.

$\left(A_4\right){lim}_{t\to 0^{+}}\frac{g(x,t)}{t}=\mu ,{lim}_{t\to +\infty }\frac{g(x,t)}{t}=l(0<l<+\infty )$ uniformly a.e. $x\in \mathsf{\Omega }$, where $\mu <{\mathrm{\Lambda }}_1<l,l\ne {\mathrm{\Lambda }}_k=$${\lambda }_k\left({\lambda }_k-c\right)$ are constants, and ${\lambda }_k(k=1,2,\dots )$ are the eigenvalues of $-\Delta$ in $H_0^1\left(\mathsf{\Omega }\right)$.

They demonstrated that Problem (1) has at least one nontrivial solution if $c<{\lambda }_1$ and Conditions $\left(A_1\right)-\left(A_4\right)$ hold. On the one hand, we use a variation of the mountain pass theorem in place of the standard one and discover that Conditions $\left(A_2\right)$ and $\left(A_3\right)$ in [5] can be omitted and that $l\ne {\mathrm{\Lambda }}_k={\lambda }_k\left({\lambda }_k-c\right)$ in Condition $\left(A_4\right)$ in [5] can also be omitted. We demonstrate that the result mentioned above is likewise accurate (see Theorem 1). Additionally, our findings in the manuscript outperform a comparable finding in [5] (see Theorems 1 and 2).

Fourth-order elliptic equations are commonly used to describe a variety of occurrences in physics, engineering, and other disciplines. A great number of authors have focused their attention on fourth-order problems in recent years (see [6,7,8,9,10,11,12,13,14,15] and references therein).

Problem (1) presents a model for studying traveling waves in suspension bridges, according to Lazer and McKenna in [16]. According to the authors of [17,18], Problem (1) also occurs in other types of spacecraft, such as communication satellites, space shuttles, and space stations, all of which are outfitted with huge antennas installed on long flexible masts (beams). A considerable number of studies [5,19,20,21] on Problem (1) have been undertaken in recent years. For example, Yin and Jiang [3] prove the existence of nontrivial solutions of a modified fourth-order elliptic equation by applying the Morse theory. Wu and Chen [22] investigated the multiplicity result for a fourth-order elliptic problem in whole space ${\mathbb{R}}^N$. Moreover, they showed that this problem has a ground state by using variational methods.

We see that the trivial solution $u=0$ is always allowed for Problem (1). Therefore, it makes sense to inquire as to whether there are any other solutions. We have further theorems in this direction.

The present paper is motivated by [23]. The goal of this note is to prove the existence of nontrivial solutions to (1) under suitable conditions on $g(x,t)$ [24,25,26,27,28,29,30].

The following assumptions about $g(x,t)\in C(\overline{\mathsf{\Omega }}\times \mathbb{R})$ will be considered in this paper:

$$\begin{array}{c}\hfill \left(F_1\right):\left\{\begin{array}{c}g(x,t)\ge 0,\mathrm{if}\forall x\in \overline{\mathsf{\Omega }},t>0,\hfill \\ g(x,t)\equiv 0,\mathrm{if}\forall x\in \overline{\mathsf{\Omega }},t\le 0.\hfill \end{array}\right.\end{array}$$

$${\displaystyle \left(F_2\right):\underset{t\to 0^{+}}{lim}\frac{g(x,t)}{t}=q\left(x\right),\underset{t\to +\infty }{lim}\frac{g(x,t)}{t}=l(0<l\le +\infty )\mathrm{uniformly}\mathrm{a}.\mathrm{e}.x\in \mathsf{\Omega },}$$

where $0\le q\left(x\right)\in L^{\infty }\left(\mathsf{\Omega }\right)$, ${\parallel q\left(x\right)\parallel }_{\infty }<{\lambda }_1$, and ${\lambda }_1$ is the first eigenvalue of $({\Delta }^2+c\Delta ,H^2\left(\mathsf{\Omega }\right)\cap$$H_0^1\left(\mathsf{\Omega }\right))$.

$\left(F_3\right)$: For a.e. $x\in \mathsf{\Omega }$, $\frac{g(x,t)}{t}$ is nondecreasing with respect to $t>0$.

Now, we disclose our primary findings, which pertain to the existence results for Problem (1).

Theorem 1.

If $\left(F_1\right)$ and $\left(F_2\right)$ hold, then Equation (1) admits a nontrivial solution when ${\lambda }_1<l<+\infty$.

Theorem 2.

If $\left(F_1\right)$–$\left(F_3\right)$ hold, then Equation (1) admits a nontrivial solution when $l=+\infty$ and $g(x,t)$ is subcritical, i.e.,

$$\underset{t\to +\infty }{lim}\frac{g(x,t)}{t^{r-1}}=0$$

uniformly a.e. $x\in \mathsf{\Omega }$ for some $r\in (2,\frac{2N}{N-4})$.

Remark 1.

If $\lambda =0$ in Problem (1), then we obtain the same results as in [23] (see Theorems 1(ii) and 2).

Remark 2.

If $c=0$ and $\lambda =0$ in Problem (1), then we obtain the same results as in [11] (see Theorems 1.2 and 1.2${}^{\prime }$).

2. Preliminaries

The Hilbert space $H=H^2\left(\mathsf{\Omega }\right)\cap H_0^1\left(\mathsf{\Omega }\right)$ with the inner product

$${\displaystyle {\langle u,v\rangle }_H={\int }_{\mathsf{\Omega }}\left(\Delta u\Delta v+\nabla u.\nabla v\right)dx}$$

as well as the norm ${\displaystyle {\parallel u\parallel }_H^2={\int }_{\mathsf{\Omega }}{|\Delta u|}^{2}dx+{\int }_{\mathsf{\Omega }}{|\nabla u|}^{2}dx.}$ Denote ${\mu }_k(k\in \mathbb{N})$ as the eigenvalue and $u_k(k\in \mathbb{N})$ as the associated eigenfunctions of

$$\left\{\begin{array}{cc}-\Delta u=\mu u,\hfill & \mathrm{in}\mathsf{\Omega },\hfill \\ u=0\hfill & \mathrm{on}\partial \mathsf{\Omega },\hfill \end{array}\right.$$

(2)

where each eigenvalue ${\mu }_k$ appears as many times as the multiplicity. Note that $0<{\mu }_1<{\mu }_2\le {\mu }_3\le \dots \le {\mu }_k\to +\infty$ and $u_1>0$ for $x\in \mathsf{\Omega }$. It is clear that ${\lambda }_k={\mu }_k({\mu }_k-c)$ are the following problem’s eigenvalues

$$\begin{array}{c}\hfill \left\{\begin{array}{cc}{\Delta }^{2}u+c\Delta u=\lambda u,\hfill & \mathrm{in}\mathsf{\Omega },\hfill \\ u=\Delta u=0\hfill & \mathrm{on}\partial \mathsf{\Omega },\hfill \end{array}\right.\end{array}$$

(3)

and the associated eigenfunctions are $u_k$.

Suppose that $c<{\mu }_1$. We introduce a new norm:

$${\displaystyle {\parallel u\parallel }^2={\int }_{\mathsf{\Omega }}{|\Delta u|}^{2}dx-c{\int }_{\mathsf{\Omega }}{|\nabla u|}^{2}dx,\forall u\in H.}$$

Obviously, the norm $\parallel .\parallel$ with a norm on H are equivalent due to the [11]. Furthermore, for all $\forall u\in H$, Poincaré inequality holds (for more details see [9,10]):

$${\parallel u\parallel }^2\ge {\lambda }_1{\parallel u\parallel }_{L^2}^2.$$

(4)

Next, we give a definition of a weak solution of Equation (1), if u satisfies

$${\displaystyle {\int }_{\mathsf{\Omega }}\left(\Delta u\Delta v-c\nabla u.\nabla v-{\lambda V\left(x\right)\left|u\right|}^{p-2}uv-g(x,u)v\right)dx=0,\forall v\in H^{*},}$$

where $H^{*}$ is the dual space of H.

The weak solution of Equation (1) is equivalent to the critical point of the Euler–Lagrange functional

$$\begin{array}{c}\hfill {\displaystyle I\left(u\right)=\frac{1}{2}{\int }_{\mathsf{\Omega }}\left({|\Delta u|}^2-c{|\nabla u|}^2\right)dx-\frac{\lambda }{p}{\int }_{\mathsf{\Omega }}V\left(x\right){\left|u\left(x\right)\right|}^{p}dx-{\int }_{\mathsf{\Omega }}G(x,u)dx,}\end{array}$$

(5)

where $G(x,u)={\int }_0^{u}g(x,t)dt$.

Obviously, $I\in C^1(H,\mathbb{R})$, and

$$\langle \nabla I\left(u\right),v\rangle ={\int }_{\mathsf{\Omega }}\left(\Delta u\Delta v-c\nabla u.\nabla v-{\lambda V\left(x\right)\left|u\right|}^{p-2}uv-g(x,u)v\right)dx,\forall u,v\in H.$$

Lemma 1

(see [23]). If $u\in H^2\left(\mathsf{\Omega }\right)\cap H_0^1\left(\mathsf{\Omega }\right)$, then the two norms $\parallel u\parallel$ and ${\parallel \Delta u\parallel }_2$ are equivalent where ${\parallel u\parallel }_2={\parallel u\parallel }_{L^2}={\left({\int }_{\mathsf{\Omega }}{\left|u\right|}^2\right)}^{\frac{1}{2}}$.

Lemma 2

(see [23]). If $u_n\to u$ in $L^p\left(\mathsf{\Omega }\right)$ for $1\le p<+\infty$, then $u_n^{+}\to u^{+}$ in $L^p\left(\mathsf{\Omega }\right)$.

Lemma 3.

Let $\left(F_1\right)$ and $\left(F_2\right)$ be true.

(i)

There exist ρ, R $>0$ such that $I\left(u\right)\ge R$ for all $u\in H^2\left(\mathsf{\Omega }\right)\cap H_0^1\left(\mathsf{\Omega }\right)$ with $\parallel u\parallel =\rho$.

(ii)

If ${\phi }_1>0$ is the ${\lambda }_1$-eigenfunction of $\left({\Delta }^2+c\Delta ,H^2\left(\mathsf{\Omega }\right)\cap H_0^1\left(\mathsf{\Omega }\right)\right)$, and ${\lambda }_1<l<+\infty$, then $I\left(t{\phi }_1\right)\to -\infty$ as $t\to +\infty$.

Proof. 

(i) In light of the conditions $\left(F_1\right)$ and $\left(F_2\right)$, it follows that for any $\epsilon >0$, there exists $A=A\left(\epsilon \right)>0$, such that, for all $(x,t)\in \mathsf{\Omega }\times \mathbb{R}$ and $\alpha \in (1,\frac{N+4}{N-4})$,

$$G(x,t)\le \frac{1}{2}\left({\parallel q\left(x\right)\parallel }_{\infty }+\epsilon \right)t^2+A{\left|t\right|}^{\alpha +1}.$$

(6)

Let $\epsilon >0$, such that ${\parallel q\left(x\right)\parallel }_{\infty }+\epsilon <{\lambda }_1$. We have ${\parallel u\parallel }_{\alpha +1}^{\alpha +1}\le K{\parallel u\parallel }^{\alpha +1}$. According to Lemma 1 and the Sobolev inequality, we have

$$\begin{array}{ccc}\hfill I\left(u\right)& =& {\displaystyle \frac{1}{2}{\parallel u\parallel }^2-\frac{\lambda }{p}{\int }_{\mathsf{\Omega }}V\left(x\right){\left|u\left(x\right)\right|}^{p}dx-{\int }_{\mathsf{\Omega }}G(x,u)dx}\hfill \\ & \ge & {\displaystyle \frac{1}{2}{\parallel u\parallel }^2-\frac{{\lambda \left|h\right|}_{\infty }}{p}{\parallel u\parallel }_{L^p}^p-\frac{1}{2}\left({\parallel q\left(x\right)\parallel }_{\infty }+\epsilon \right){\int }_{\mathsf{\Omega }}{\left|u\right|}^{2}dx-A{\int }_{\mathsf{\Omega }}{\left|u\right|}^{\alpha +1}dx}\hfill \\ & =& \frac{1}{2}{\parallel u\parallel }^2-\frac{{\lambda \left|h\right|}_{\infty }}{p}{\parallel u\parallel }_{L^p}^p-\frac{1}{2}\left({\parallel q\left(x\right)\parallel }_{\infty }+\epsilon \right){\parallel u\parallel }_{L^2}^{{}^2}-A{\parallel u\parallel }_{L^{\alpha +1}}^{{}^{\alpha +1}}\hfill \\ & \ge & \frac{1}{2}\left(1-\frac{{\parallel q\left(x\right)\parallel }_{\infty }+\epsilon }{{\lambda }_1}\right){\parallel u\parallel }^2-\frac{{\lambda \left|h\right|}_{\infty }}{p}{K\parallel u\parallel }^p-AK{\parallel u\parallel }^{\alpha +1}\hfill \\ & =& \left[\frac{1}{2}\left(1-\frac{{\parallel q\left(x\right)\parallel }_{\infty }+\epsilon }{{\lambda }_1}\right)-{\lambda C\parallel u\parallel }^{p-2}-C_s{\parallel u\parallel }^{\alpha -1}\right]{\parallel u\parallel }^2.\hfill \end{array}$$

(7)

Let

$$G\left(t\right)=\lambda C{t}^{p-2}+C_s{t}^{\alpha -1}.$$

We prove that $t_0$ exists in the sense that

$$G\left(t_0\right)<\frac{1}{2}\left(1-\frac{{\parallel q\left(x\right)\parallel }_{\infty }+\epsilon }{{\lambda }_1}\right).$$

Indeed,

$$G^{{}^{\prime }}\left(t\right)=\lambda C(p-2)t^{p-3}+C_s(\alpha -1)t^{\alpha -2}.$$

Setting

$$G^{{}^{\prime }}\left(t\right)=0,$$

we obtain

$$t_0={\left(\frac{\lambda C(2-p)}{C_s(\alpha -1)}\right)}^{\frac{1}{\alpha -p+1}}.$$

At $t=t_0$, $G\left(t\right)$ reaches its minimum.

Let $\beta =\frac{C(2-p)}{C_s(\alpha -1)}$, $\tilde{p}=\frac{p-2}{\alpha -p+1}$, $\tilde{q}=\frac{\alpha -1}{\alpha -p+1}$, $k=\frac{1}{2}\left(1-\frac{{\parallel q\left(x\right)\parallel }_{\infty }+\epsilon }{{\lambda }_1}\right)$.

Substituting $t_0$ in $Q\left(t\right)$, we have

$$G\left(t_0\right)<\frac{1}{2}\left(1-\frac{{\parallel q\left(x\right)\parallel }_{\infty }+\epsilon }{{\lambda }_1}\right),0<\lambda <{\lambda }^{*},$$

where

$${\lambda }^{*}={\left(\frac{k}{C^{\tilde{p}}+C_s{\beta }^{\tilde{q}}}\right)}^{\frac{1}{\tilde{q}}}.$$

Take $\rho =t_0$, then there exists $R>0$ such that the lemma holds. This completes the proof.

(ii) According to Fatou’s lemma, if we have ${\lambda }_1<l<+\infty$, we obtain

$$\begin{array}{ccc}\hfill {\displaystyle \underset{t\to +\infty }{lim}\frac{I\left(t{\phi }_1\right)}{t^2}}& \le & {\displaystyle \frac{1}{2}\parallel {\phi }_1{\parallel }^2-\underset{t\to +\infty }{lim}\frac{{\lambda \left|t\right|}^{p-2}}{p}{\int }_{\mathsf{\Omega }}V\left(x\right)|{\phi }_1{|}^{p}dx-{\int }_{\mathsf{\Omega }}\underset{t\to +\infty }{lim}\frac{G(x,t{\phi }_1)}{{\left(t{\phi }_1\right)}^2}{\left|{\phi }_1\right|}^{2}dx}\hfill \\ & \le & \frac{{\lambda }_1}{2}\parallel {\phi }_1{\parallel }_2^2-\underset{t\to +\infty }{lim}\frac{{\lambda \left|t\right|}^{p-2}}{p}{\int }_{\mathsf{\Omega }}V\left(x\right)|{\phi }_1{|}^{p}dx-\frac{l}{2}{\parallel {\phi }_1\parallel }_2^2\hfill \\ & \le & \frac{1}{2}({\lambda }_1-l)\parallel {\phi }_1{\parallel }_2^2-\underset{t\to +\infty }{lim}\frac{{\lambda \left|t\right|}^{p-2}}{p}{\int }_{\mathsf{\Omega }}V\left(x\right){\left|{\phi }_1\right|}^{p}dx<0(1<p<2).\hfill \end{array}$$

Then,

$$\underset{t\to +\infty }{lim}I\left(t{\phi }_1\right)=-\infty .$$

The proof is complete. □

Lemma 4.

Assuming $\left(F_1\right)$ and $\left(F_3\right)$ hold, if $\langle I^{{}^{\prime }}\left(u_n\right),u_n\rangle \to 0$ as $n\to \infty$, then there exists a subsequence $\left(u_n\right)$ such that $I\left(t{u}_n\right)\le \frac{1+t^2}{2n}+I\left(u_n\right)$ for all $t>0$ and $n\in \mathbb{N}$.

Proof. 

By $\langle I^{{}^{\prime }}\left(u_n\right),u_n\rangle \to 0$ as $n\to \infty$, we obtain

$$\frac{-1}{n}<\langle I^{{}^{\prime }}\left(u_n\right),u_n\rangle =\parallel u_n{\parallel }^2-{\int }_{\mathsf{\Omega }}g(x,u_n\left(x\right))u_n\left(x\right)dx-\lambda {\int }_{\mathsf{\Omega }}V\left(x\right){\left|u_n\left(x\right)\right|}^{p}dx<\frac{1}{n},\forall n\in \mathbb{N}.$$

We argue that for all $t>0$ and $n\in \mathbb{N}$,

$${\displaystyle I\left(t{u}_n\right)\le \frac{t^2}{2n}+{\int }_{\mathsf{\Omega }}\left(\frac{1}{2}g(x,u_n\left(x\right))u_n\left(x\right)dx-G(x,u_n\left(x\right))\right)dx+\lambda \left({\int }_{\mathsf{\Omega }}V\left(x\right)|u_n{|}^p-\frac{1}{p}V\left(x\right){\left|u_n\right|}^p\right)dx.}$$

In fact, a fixed $n\in \mathbb{N}$ and $x\in \mathsf{\Omega }$, if we let

$$h\left(t\right)=\frac{t^2}{2}g(x,u_n)u_n\left(x\right)-G(x,t{u}_n\left(x\right))+\frac{\lambda t^2}{2}V\left(x\right)|u_n{|}^p-\frac{\lambda t^p}{p}V\left(x\right){\left|u_n\right|}^p\forall t>0,$$

then

$$\begin{array}{ccc}\hfill h^{{}^{\prime }}\left(t\right)& =& tg(x,u_n)u_n\left(x\right)-g(x,t{u}_n)u_n\left(x\right)+\lambda tV\left(x\right)|u_n{|}^p-\lambda t^{p-1}V\left(x\right){\left|u_n\right|}^p\hfill \\ & =& {\displaystyle t\left(\frac{g(x,u_n)}{u_n}|u_n{|}^2-\frac{g(x,t{u}_n)}{t{u}_n}{\left|u_n\right|}^2\right)+t\lambda V\left(x\right){\left|u_n\right|}^p(1-t^{p-1}),}\hfill \end{array}$$

(8)

hence,

$$\begin{array}{c}\hfill \left\{\begin{array}{c}{h}^{{}^{\prime }}\left(t\right)\ge 0\mathrm{if}0<t<1,\hfill \\ h{}^{\prime }\left(t\right)\le 0\mathrm{if}t\ge 1.\hfill \end{array}\right.\end{array}$$

(9)

Thus, we have

$$h\left(t\right)\le h\left(1\right)\forall t>0.$$

Therefore, we obtain

$$\begin{array}{ccc}\hfill I\left(t{u}_n\right)& =& \frac{t^2}{2}\parallel u_n{\parallel }^2-{\int }_{\mathsf{\Omega }}G(x,t{u}_n)dx-\frac{\lambda t^p}{p}{\int }_{\mathsf{\Omega }}V\left(x\right){|u_n|}^{p}dx\hfill \\ & <& \frac{t^2}{2}\left[\frac{1}{n}+{\int }_{\mathsf{\Omega }}g(x,u_n)u_{n}dx+\lambda {\int }_{\mathsf{\Omega }}V\left(x\right){\left|u_n\right|}^{p}dx\right]-{\int }_{\mathsf{\Omega }}G(x,t{u}_n)dx-\frac{\lambda t^p}{p}{\int }_{\mathsf{\Omega }}V\left(x\right){|u_n|}^{p}dx\hfill \\ & \le & {\displaystyle \frac{t^2}{2n}+{\int }_{\mathsf{\Omega }}\left[\frac{t^2}{2}g(x,u_n)u_n-G(x,t{u}_n)\right]dx+{\int }_{\mathsf{\Omega }}\left[\frac{\lambda t^2}{2}{\int }_{\mathsf{\Omega }}V\left(x\right)|u_n{|}^{p}dx-\frac{\lambda t^p}{p}{\int }_{\mathsf{\Omega }}V\left(x\right){|u_n|}^{p}dx\right]}\hfill \\ & \le & {\displaystyle \frac{t^2}{2n}+{\int }_{\mathsf{\Omega }}\left[\frac{1}{2}g(x,u_n)u_n-G(x,u_n)\right]dx+{\int }_{\mathsf{\Omega }}\left[\frac{\lambda }{2}{\int }_{\mathsf{\Omega }}V\left(x\right)|u_n{|}^{p}dx-\frac{\lambda }{p}{\int }_{\mathsf{\Omega }}V\left(x\right){\left|u_n\right|}^{p}dx\right].}\hfill \end{array}$$

Our assertion is proved.

However, one has

$$\begin{array}{ccc}\hfill I\left(u_n\right)& =& \frac{1}{2}\parallel u_n{\parallel }^2-{\int }_{\mathsf{\Omega }}G(x,u_n)dx-\frac{\lambda }{p}{\int }_{\mathsf{\Omega }}V\left(x\right){\left|u_n\right|}^{p}dx\hfill \\ & \ge & \frac{1}{2}\left[-\frac{1}{n}+{\int }_{\mathsf{\Omega }}g(x,u_n)u_{n}dx+\lambda {\int }_{\mathsf{\Omega }}V\left(x\right){\left|u_n\right|}^{p}dx\right]-{\int }_{\mathsf{\Omega }}G(x,u_n)dx-\frac{\lambda }{p}{\int }_{\mathsf{\Omega }}V\left(x\right){\left|u_n\right|}^{p}dx,\hfill \end{array}$$

i.e.,

$${\int }_{\mathsf{\Omega }}\left[\frac{1}{2}g(x,u_n)u_n-G(x,u_n)\right]dx+{\int }_{\mathsf{\Omega }}\left[\frac{\lambda }{2}V\left(x\right)|u_n{|}^p-\frac{\lambda }{p}V\left(x\right){\left|u_n\right|}^p\right]dx\le \frac{1}{2n}+I\left(u_n\right).$$

From the above, we can find that if

$$I\left(t{u}_n\right)\le \frac{t^2}{2n}+\frac{1}{2n}+I\left(u_n\right),$$

then

$$I\left(t{u}_n\right)\le \frac{1+t^2}{2n}+I\left(u_n\right),\forall t>0andn\in \mathbb{N}.$$

□

3. The Proofs of Main Results

Proof of Theorem 1. 

With ${\lambda }_1<l<+\infty$, using Lemma 3 (i) and (ii), we may obtain a $t_0>0$ large enough to satisfy $I\left(t_0{\phi }_1\right)<0$. Define

$$\begin{array}{ccc}& \mathsf{\Gamma }& =\left\{\gamma \in C\left([0,1],H^2\left(\mathsf{\Omega }\right)\cap H_0^1\left(\mathsf{\Omega }\right)\right):\gamma \left(0\right)=0,\gamma \left(1\right)=t_0{\phi }_1\right\}\hfill \\ & c_0& =\underset{\gamma \in \mathsf{\Gamma }}{inf}\underset{0\le t\le 1}{max}I\left(\gamma \left(t\right)\right).\hfill \end{array}$$

Then, $c_0>R>0$ and, according to Proposition 2.1 in [23], there exists a sequence $\left(u_n\right)\subset H^2\left(\mathsf{\Omega }\right)\cap H_0^1\left(\mathsf{\Omega }\right)$ such that

$$\begin{array}{c}\hfill I\left(u_n\right)=\frac{1}{2}\parallel u_n{\parallel }^2-\frac{\lambda }{p}{\int }_{\mathsf{\Omega }}V\left(x\right){\left|u_n\right|}^{p}dx-{\int }_{\mathsf{\Omega }}G(x,u_n\left(x\right))dx=c_0+o\left(1\right),\end{array}$$

(10)

and

$$\begin{array}{c}\hfill (1+\parallel u_n\parallel )\parallel I^{{}^{\prime }}\left(u_n\right){\parallel }_{H^{-2}\left(\mathsf{\Omega }\right)}\underset{n\to \infty }{\to }0,\end{array}$$

(11)

where $H^{-2}\left(\mathsf{\Omega }\right)$ represents the dual space of $H^2\left(\mathsf{\Omega }\right)\cap H_0^1\left(\mathsf{\Omega }\right)$. We denote by $o\left(1\right)$ any quantity that approaches zero as $n\to +\infty$. By (11), it follows that

$$\begin{array}{c}\hfill \langle I^{{}^{\prime }}\left(u_n\right),u_n\rangle =\parallel u_n{\parallel }^2-\lambda {\int }_{\mathsf{\Omega }}V\left(x\right){\left|u_n\right|}^{p}dx-{\int }_{\mathsf{\Omega }}g(x,u_n)u_{n}dx=o\left(1\right).\end{array}$$

(12)

It follows from standard argument and the theory of Sobolev embedding that there exists a subsequence $\left(u_n\right)$ that strongly converges to u, which is a critical point of functional I if $\left(u_n\right)$ is bounded in $H^2\left(\mathsf{\Omega }\right)\cap H_0^1\left(\mathsf{\Omega }\right)$.

In order to prove Theorem 1, we first show that $\left(u_n\right)$ is bounded in $H^2\left(\mathsf{\Omega }\right)\cap H_0^1\left(\mathsf{\Omega }\right)$. In contradiction, we assume $\parallel u_n\parallel \to +\infty$ and take

$$\begin{array}{c}\hfill t_n=\frac{1}{\parallel u_n\parallel },v_n=t_n{u}_n=\frac{u_n}{\parallel u_n\parallel }.\end{array}$$

(13)

Clearly, $\left(v_n\right)$ is bounded in $H^2\left(\mathsf{\Omega }\right)\cap H_0^1\left(\mathsf{\Omega }\right)$. We can deduce that by taking a subsequence from Lemma 2,

$$\begin{array}{cc}\hfill v_n& \underset{n\to \infty }{\rightharpoonup }w\mathrm{in}{H}^2\left(\mathsf{\Omega }\right)\cap H_0^1\left(\mathsf{\Omega }\right),\hfill \\ \hfill v_n& \underset{n\to \infty }{\to }w\mathrm{in}{L}^2\left(\mathsf{\Omega }\right),v_n\underset{n\to \infty }{\to }w\mathrm{a}.\mathrm{e}.\mathrm{in}\mathsf{\Omega },\hfill \\ \hfill v_n^{+}& \underset{n\to \infty }{\to }{v}^{+}\mathrm{in}{L}^2\left(\mathsf{\Omega }\right),v_n^{+}\underset{n\to \infty }{\to }{v}^{+}\mathrm{a}.\mathrm{e}.\mathrm{in}\mathsf{\Omega }.\hfill \end{array}$$

We argue that

$$\begin{array}{c}\hfill w\ne 0.\end{array}$$

(14)

In addition, we can show from Conditions $\left(F_1\right)$ and $\left(F_2\right)$ that there exists $M>0$ such that $\frac{\left|g\right(x,t\left)\right|}{\left|t\right|}\le M$ for all $x\in \mathsf{\Omega }$ and $t\ne 0.$ Then, using (12) and (13), we obtain

$$\begin{array}{c}\hfill {\displaystyle {\int }_{\mathsf{\Omega }}\frac{g(x,u_n)}{u_n}|v_n{|}^{2}dx=\frac{\lambda }{\parallel u_n{\parallel }^2}{\int }_{\mathsf{\Omega }}V\left(x\right){\left|u_n\right|}^{p}dx+\frac{{\displaystyle {\int }_{\mathsf{\Omega }}g(x,u_n)u_{n}dx}}{\parallel u_n{\parallel }^2}=1+o\left(1\right).}\end{array}$$

This means that

$$\begin{array}{cc}\hfill 1+o\left(1\right)& =\frac{\lambda }{\parallel u_n{\parallel }^2}{\int }_{\mathsf{\Omega }}V\left(x\right)|u_n{|}^{p}dx+{\int }_{\mathsf{\Omega }}\frac{g(x,u_n)}{u_n}{\left|v_n\right|}^{2}dx\hfill \\ \hfill & \le \frac{{\lambda \left|h\right|}_{\infty }}{\parallel u_n{\parallel }^2}{\int }_{\mathsf{\Omega }}|u_n{|}^{p}dx+M{\int }_{\mathsf{\Omega }}|v_n{|}^{2}dx\to M{\int }_{\mathsf{\Omega }}{\left|w\right|}^{2}dx,\hfill \end{array}$$

so $w\ne 0$.

By $I^{{}^{\prime }}\left(u_n\right)\underset{n\to \infty }{\to }0$, for all $\phi \in H^2\left(\mathsf{\Omega }\right)\cap H_0^1\left(\mathsf{\Omega }\right)$, we have

$${\displaystyle {\int }_{\mathsf{\Omega }}\left(\Delta u_n\Delta \phi -c\nabla u_n\nabla \phi \right)dx={\int }_{\mathsf{\Omega }}g(x,u_n)\phi dx+\lambda {\int }_{\mathsf{\Omega }}V\left(x\right){\left|u_n\right|}^{p-2}{u}_n\phi dx+o\left(1\right).}$$

Then, by (13) and $\left(F_1\right)$, we have

$$\begin{array}{c}\hfill {\int }_{\mathsf{\Omega }}(\Delta v_n\Delta \phi -c\nabla v_n\nabla \phi )dx={\int }_{\mathsf{\Omega }}\frac{g(x,u_n^{+})}{u_n^{+}}{v}_n^{+}\phi dx+\frac{\lambda }{\parallel u_n\parallel }{\int }_{\mathsf{\Omega }}V\left(x\right){\left|u_n\right|}^{p-2}{u}_n\phi dx+o\left(1\right).\end{array}$$

(15)

Since $\frac{\left|g\right(x,t)}{\left|t\right|}\le M$,

$$\frac{g(x,u_n^{+})}{u_n^{+}}{v}_n^{+}\to 0=l{v}^{+}\mathrm{a.e.}x\in \mathsf{\Omega },\mathrm{if}{v}^{+}\left(x\right)=0.$$

If $v^{+}\left(x\right)>0$, then we obtain that $u_n^{+}\left(x\right)=v_n^{+}\left(x\right)\parallel u_n\parallel \underset{n\to \infty }{\to }+\infty$, a.e. $x\in \mathsf{\Omega }$, so by $\left(F_2\right)$ we have

$$\frac{g(x,u_n^{+})}{u_n^{+}}{v}_n^{+}\to l{v}^{+}\mathrm{a.e.}x\in \mathsf{\Omega }.$$

$\left(\frac{g(x,u_n^{+})}{u_n^{+}}{v}_n^{+}\right)$ is bounded in $L^2\left(\mathsf{\Omega }\right)$ since $\frac{\left|g\right(x,t\left)\right|}{\left|t\right|}\le M$, hence there exists a subsequence such that

$$\frac{g(x,u_n^{+})}{u_n^{+}}{v}_n^{+}\rightharpoonup l{v}^{+}\mathrm{weakly\; in}{L}^2\left(\mathsf{\Omega }\right).$$

Therefore, we have

$$\begin{array}{c}\hfill {\displaystyle {\int }_{\mathsf{\Omega }}\frac{g(x,u_n^{+})}{u_n^{+}}{v}_n^{+}\phi dx\underset{n\to \infty }{\to }l{\int }_{\mathsf{\Omega }}{v}^{+}\phi dx,\forall \phi \in H^2\left(\mathsf{\Omega }\right)\cap H_0^1\left(\mathsf{\Omega }\right).}\end{array}$$

(16)

Since $v_n\underset{n\to \infty }{\rightharpoonup }w$ weakly in $H^2\left(\mathsf{\Omega }\right)\cap H_0^1\left(\mathsf{\Omega }\right)$, by Lemma 1, we obtain

$$\begin{array}{c}\hfill {\int }_{\mathsf{\Omega }}(\Delta v_n\Delta \phi -c\nabla v_n\nabla \phi )dx\to {\int }_{\mathsf{\Omega }}(\Delta w\Delta \phi -c\nabla w\nabla \phi )dx,\forall \phi \in H^2\left(\mathsf{\Omega }\right)\cap H_0^1\left(\mathsf{\Omega }\right).\end{array}$$

(17)

Using (15)–(17), we obtain

$$\begin{array}{c}\hfill {\int }_{\mathsf{\Omega }}(\Delta w\Delta \phi -c\nabla w\nabla \phi )dx=l{\int }_{\mathsf{\Omega }}{v}^{+}\phi dx.\end{array}$$

(18)

Taking $\phi ={\phi }_1\left(x\right)$ in (18), we obtain

$$\begin{array}{c}\hfill {\int }_{\mathsf{\Omega }}(\Delta w\Delta {\phi }_1-c\nabla w\nabla {\phi }_1)dx=l{\int }_{\mathsf{\Omega }}{v}^{+}{\phi }_{1}dx.\end{array}$$

(19)

Observe that ${\phi }_1$ is positive first eigenfunction of

$$\left\{\begin{array}{cc}{\Delta }^{2}u+c\Delta u={\lambda }_{1}u,\hfill & x\in \mathsf{\Omega },\hfill \\ u=\Delta u=0,\hfill & x\in \partial \mathsf{\Omega },\hfill \end{array}\right.$$

(20)

then for each $e\in H^2\left(\mathsf{\Omega }\right)\cap H_0^1\left(\mathsf{\Omega }\right)$, we have

$${\displaystyle {\int }_{\mathsf{\Omega }}\left(\Delta e\Delta {\phi }_1-\nabla e\nabla {\phi }_1\right)dx={\lambda }_1{\int }_{\mathsf{\Omega }}e{\phi }_{1}dx.}$$

Using $e=w\left(x\right)$ , we obtain

$$\begin{array}{c}\hfill {\displaystyle {\int }_{\mathsf{\Omega }}\left(\Delta w\Delta {\phi }_1-\nabla w\nabla {\phi }_1\right)dx={\lambda }_1{\int }_{\mathsf{\Omega }}w{\phi }_{1}dx,}\end{array}$$

(21)

thus combining (19) and (21), we have

$$\begin{array}{c}\hfill l{\int }_{\mathsf{\Omega }}{v}^{+}{\phi }_{1}dx={\lambda }_1{\int }_{\mathsf{\Omega }}w{\phi }_{1}dx.\end{array}$$

(22)

Notice that

$${\displaystyle {\int }_{\mathsf{\Omega }}{v}^{+}{\phi }_{1}dx-{\int }_{\mathsf{\Omega }}w{\phi }_{1}dx={\int }_{{\mathsf{\Omega }}_{-}}-w{\phi }_{1}dx\ge 0,}$$

where ${\mathsf{\Omega }}_{-}=\left\{z\in \mathsf{\Omega }:v\left(z\right)<0\right\}$. □

Now, we assert that the two situations $|{\mathsf{\Omega }}_{-}| =0$ and $|{\mathsf{\Omega }}_{-}| >0$ are incompatible:

Case 1: If $|{\mathsf{\Omega }}_{-}|=0$, then $w\left(x\right)\ge 0$ a.e. x in $\mathsf{\Omega }$. Since $w\left(x\right)\ne 0$, we obtain ${\displaystyle {\int }_{\mathsf{\Omega }}w{\phi }_{1}dx>0}$. Then (22) means that

$${\displaystyle l{\int }_{\mathsf{\Omega }}w{\phi }_{1}dx=l{\int }_{\mathsf{\Omega }}{v}^{+}{\phi }_{1}dx={\lambda }_1{\int }_{\mathsf{\Omega }}w{\phi }_{1}dx,}$$

which is impossible due to the fact that $l>{\lambda }_1$.

Case 2: If $|{\mathsf{\Omega }}_{-}|>0$, then ${\displaystyle {\int }_{{\mathsf{\Omega }}_{-}}-w{\phi }_{1}dx>0}$ and ${\displaystyle {\int }_{\mathsf{\Omega }}{v}^{+}{\phi }_{1}dx>{\int }_{\mathsf{\Omega }}w{\phi }_{1}dx}$.

According to (22), we have

$${\displaystyle l{\int }_{\mathsf{\Omega }}{v}^{+}{\phi }_{1}dx={\lambda }_1{\int }_{\mathsf{\Omega }}w{\phi }_{1}dx<{\lambda }_1{\int }_{\mathsf{\Omega }}{v}^{+}{\phi }_{1}dx,}$$

which is also impossible due to the fact that $l>{\lambda }_1$, if ${\displaystyle {\int }_{\mathsf{\Omega }}{v}^{+}{\phi }_{1}dx>0}$, which also is in contradiction to $0<0$, if ${\displaystyle {\int }_{\mathsf{\Omega }}{v}^{+}{\phi }_{1}dx=0}$.

Thus, we show the boundedness of sequence $\left(u_n\right)$ in $H^2\left(\mathsf{\Omega }\right)\cap H_0^1\left(\mathsf{\Omega }\right)$.

Proof of Theorem 2. 

Similarly to the proof of Theorem 1, we can also identify a subsequence $\left(u_n\right)\subset H^2\left(\mathsf{\Omega }\right)\cap H_0^1\left(\mathsf{\Omega }\right)$ such that (10) and (12) hold for the case of $l=+\infty$. As a result, to establish Theorem 2, we only need to show that $\left(u_n\right)$ is bounded in $H^2\left(\mathsf{\Omega }\right)\cap H_0^1\left(\mathsf{\Omega }\right)$. If not, we can assume that $\parallel u_n\parallel \to +\infty$ and let

$$\begin{array}{c}\hfill t_n=\frac{2\sqrt{c_0}}{\parallel u_n\parallel },v_n=t_n{u}_n=\frac{2\sqrt{c_0}{u}_n}{\parallel u_n\parallel }.\end{array}$$

(23)

Obviously, $\left(v_n\right)$ is bounded in $H^2\left(\mathsf{\Omega }\right)\cap H_0^1\left(\mathsf{\Omega }\right)$. Similarly, we also have

$$\begin{array}{ccc}\hfill v_n& \underset{n\to \infty }{\rightharpoonup }& w\mathrm{in}{H}^2\left(\mathsf{\Omega }\right)\cap H_0^1\left(\mathsf{\Omega }\right),\hfill \\ \hfill v_n& \underset{n\to \infty }{\to }& w\mathrm{in}{L}^2\left(\mathsf{\Omega }\right),v_n\to w\mathrm{a.e.\; in}\mathsf{\Omega },\hfill \\ \hfill v_n^{+}& \underset{n\to \infty }{\to }& v^{+}\mathrm{in}{L}^2\left(\mathsf{\Omega }\right),v_n^{+}\to v^{+}\mathrm{a.e.\; in}\mathsf{\Omega }.\hfill \end{array}$$

Observing that $l\equiv +\infty$ in $\left(F_2\right)$, we have for each given $K>0$,

$$\begin{array}{c}\hfill \underset{n\to +\infty }{lim}\frac{g(x,u_n^{+})}{u_n^{+}}|v_n^{+}{|}^2\ge K{\left|v^{+}\right|}^2\mathrm{almost}\mathrm{everywhere}x\in \mathsf{\Omega }.\end{array}$$

(24)

In fact, for almost everywhere $x\in \mathsf{\Omega }$, if $v^{+}\left(x\right)=0$, then the equality is true. If $v^{+}\left(x\right)>0$, it follows from (23) that $u_n^{+}\left(x\right)\to +\infty$. With $|v_n^{+}{\left(x\right)|}^2\underset{n\to \infty }{\to }{\left|v^{+}\left(x\right)\right|}^2$, let ${ϵ}_0=\frac{1}{2}{\left|v^{+}\left(x\right)\right|}^2,$ thus there exists a positive $N_1$ such that

$$\begin{array}{c}\hfill -{ϵ}_0<|v_n^{+}{\left(x\right)|}^2-{\left|v^{+}\left(x\right)\right|}^2<{ϵ}_0\mathrm{for}n\ge N_1,\end{array}$$

(25)

which means that

$$\begin{array}{c}\hfill |v_n^{+}{\left(x\right)|}^2>|v^{+}{\left(x\right)|}^2-{ϵ}_0=\frac{1}{2}{\left|v^{+}\right|}^2.\end{array}$$

(26)

With $l\equiv +\infty$ in $\left(F_2\right)$, and $u_n^{+}\left(x\right)\underset{n\to \infty }{\to }+\infty$, we obtain

$$\underset{n\to +\infty }{lim}\frac{g(x,u_n^{+}\left(x\right))}{u_n^{+}\left(x\right)}=+\infty .$$

As a result, for any given $K>0$, we can find a positive $N_2(x,K)$ such that

$$\begin{array}{c}\hfill \frac{g(x,u_n^{+}\left(x\right))}{u_n^{+}\left(x\right)}\ge 2K\mathrm{for}n\ge N_2.\end{array}$$

(27)

Set $N(x,K)=max\left\{N_1,N_2\right\}$, then, by (26) and (27), we have

$$\begin{array}{c}\hfill \frac{g(x,u_n^{+})}{u_n^{+}}|v_n^{+}{|}^2\ge K{\left|v^{+}\right|}^2\mathrm{for}n\ge N.\end{array}$$

Therefore, (24) is true.

In view of (12), (23) and (24), we obtain

$$\begin{array}{ccc}\hfill 4c_0& =& \underset{n\to +\infty }{lim}{\parallel v_n\parallel }^2\hfill \\ & =& \underset{n\to +\infty }{lim}4c_0\left[{\displaystyle \frac{{\displaystyle {\int }_{\mathsf{\Omega }}g(x,u_n^{+})u_n^{+}dx}}{\parallel u_n{\parallel }^2}+\frac{\lambda }{\parallel u_n{\parallel }^2}{\int }_{\mathsf{\Omega }}V\left(x\right){\left|u_n^{+}\right|}^{p}dx}\right]\hfill \\ & =& {\displaystyle \underset{n\to +\infty }{lim}\left[{\displaystyle {\int }_{\mathsf{\Omega }}\frac{g(x,u_n^{+})}{u_n^{+}}{\left|v_n^{+}\right|}^{2}dx}\right]\ge K{\int }_{\mathsf{\Omega }}{\left|v^{+}\right|}^{2}dx.}\hfill \end{array}$$

Choose $K>0$ big enough such that $\parallel v^{+}{\parallel }_2=0$, so $v^{+}\left(x\right)\equiv 0.$ Therefore, we have

$$\underset{n\to +\infty }{lim}{\int }_{\mathsf{\Omega }}G(x,v_n\left(x\right))dx=\underset{n\to +\infty }{lim}{\int }_{\mathsf{\Omega }}G(x,v_n^{+}\left(x\right))=0$$

and

$$\begin{array}{c}\hfill I\left(v_n\right)=\frac{1}{2}{\parallel v_n\parallel }^2+o\left(1\right)=2c_0+o\left(1\right).\end{array}$$

(28)

Using $\parallel u_n\parallel \to +\infty$ as $n\to +\infty$ and, with (23), we know that $t_n\to 0$ as $n\to +\infty$, which implies that according to Lemma 4 and (10), we have

$$\begin{array}{c}\hfill I\left(v_n\right)=I\left(t_n{u}_n\right)\le \frac{1+t_n^2}{2n}+I\left(u_n\right)\to c_0>0.\end{array}$$

(29)

It is obvious that (28) and (29) are incongruous, which reveals that $\left(u_n\right)$ is bounded, and the proof of Theorem 2 is complete. □

Remark 3.

If we assume the following conditions:

(F${}_0$)

$g(x,0)=0$ and $g(x,-t)=-g(x,t)$ for all $x\in \mathsf{\Omega },t\in \mathbb{R}$.

(F${}_4$)

${\displaystyle \underset{t\to |\infty |}{lim}(g(x,t)t-2G(x,t))=-\infty }$ uniformly in $\mathsf{\Omega }$, and (F${}_2$) with $c<{\lambda }_1$; ${\lambda }_k({\lambda }_1-c)<l$,

we can prove that there exists $l^{★}>0$ such that for $l\in (0,l^{★})$. Problem (1) has infinitely many solutions by employing the same method used by C. Liu and J. Wang (see [24]).

4. Conclusions

In our work, we showed the existence of nontrivial solutions to Problem (1) through a variation of the mountain pass theorem. We also discussed the main characteristics of the solution representation by looking at two cases. We discovered that Conditions (A2) and (A3) in [5] can be removed, and we established that our results are also accurate by using a variation of the mountain pass theorem rather than the usual one (see Theorem 1). Furthermore, in comparison to the findings that are in [5], our findings in this paper performed significantly better (see Theorems 1 and 2).

This finding supports some previously published research. In Theorem 1, we studied the case of ${\lambda }_1<l<+\infty$, and we found that our conditions are not sufficient to study the cases if $0<l<{\lambda }_1$ and if $l={\lambda }_1$. This issue has been left as an open question for researchers who are interested in the subject.

Moreover, the presence of the term V in our problem makes it more challenging to investigate the uniqueness and convergence of solutions. Therefore, we leave this topic as an unsolved question for experts in this field.