Existence of nodal solutions of nonlinear Lidstone boundary value problems

1.   Introduction

We investigate the existence of nodal solutions for nonlinear Lidstone boundary value problem

where $\lambda > 0$ is a parameter, $c$ is a constant, $m\geq1$ is an integer, $a :[0, r]\rightarrow [0, \infty)$ is continuous with $a\not\equiv0$ on the subinterval within $[0, r]$, and $f: \mathbb{R}\rightarrow \mathbb{R}$ is a continuous function.

Problem (1.1) is called a $Lidstone$ boundary value problem. Lidstone boundary value problems arise in a lot of various fields of applied mathematics and physics. If $m = 1$, problem (1.1) are Newton's equations of motion under Dirichlet boundary conditions. If $m = 2$, problem (1.1) is the elastic beam equation with simple support at both ends. If $m = 3$, problem (1.1) can be used to describe the speed of a flagellate protozoan ^[1]. From this point of view, it is not only of great theoretical significance to discuss such problems, but also of practical application value.

Lidstone boundary value problems in a comparable form to (1.1) have been taken into account in many papers, particularly in second and fourth-order cases, see for example ^[2,3,4,5,6] and ^[7,8,9,10] and references therein. Specifically, Lazer and McKenna ^[10] considered the existence for nodal solutions of (1.1) with $m = 2$, $a(t)\equiv1$, and a "jumping" nonlinearity, i.e.,

The eigenvalues of the linear eigenvalue problem corresponding to (1.2) can be obtained directly by ordinary differential equation calculation, but for the linear eigenvalue problem with weighted function $a(t)$, the eigenvalues cannot be calculated. In addition, it is worth noting that a complicated method was used in ^[10] (see Lemma 2.2) to certify that all zeros of the solutions are simple, moreover, this method does not seem to apply in the case of $m > 2$.

Existence and multiplicity of positive solutions for 2$m$th-order Lidstone boundary value problems have been extensively studied by several authors, see ^{[11,12,13,14,15,16,17]}. For example, Yuan et al. ^[17] considered the existence of a positive solution for the 2$m$th-order Lidstone boundary value problem

by the fixed point theorem of mixed monotone operators. However, relatively little is known about the existence of nodal solutions for 2$m$th-order Lidstone boundary value problems, see ^[18,19]. In ^[19], Xu and Han dealt with the existence of nodal solutions of Lidstone boundary value problems with the assumption that the nonlinearity $f$ is asymptotically linear

under these suppositions:

(A1) $a \in C([0, 1], [0, \infty))$, $a\not\equiv 0$ on subinterval within $[0, 1]$;

(A2) $f \in C(\mathbb{R}, \mathbb{R})$, $f(s)s > 0$ for $s\neq 0$;

(A3) there exist $f_0, f_{\infty} \in (0, \infty)$ such that

Let $\mu_k$ be the $k$th eigenvalue of the linear eigenvalue problem associated with (1.4). By using bifurcation techniques, they obtained

Theorem A. [19,Theorem 1.1.] Let (A1)–(A3) hold. Assume that for some $k \in \mathbb{N}$, either

or

Then, (1.4) possesses two solutions, namely $u^+_k$ and $u^-_k$. The solution $u^+_k$ has precisely $k-1$ simple zeros within the interval $(0, 1)$ and is positive in the vicinity of 0. Similarly, $u^-_k$ also has exactly $k-1$ simple zeros in $(0, 1)$ and is negative near 0.

However, the more general operator $(-1)^m (u^{(2m)}(t)+c u^{(2m-2)}(t))$ and "jumping" nonlinearity are not considered in ^[19].

Inspired by ^[10] and ^[19], the main purpose of this paper is to analyze the existence of nodal solutions for problem (1.1). One of the contributions of this paper is to employ the disconjugacy theory to examine a sufficient condition that guarantees the disconjugacy of $(-1)^{m}(u^{(2m)}(t)+cu^{(2m-2)}(t)) = 0$ on the interval $[0, r]$. In addition, it is probably the first time that we utilize Elias's spectrum theory to explore the spectrum structure of the linear operator $(-1)^{m}(u^{(2m)}(t)+cu^{(2m-2)}(t)) = \lambda a(t)u, \ t\in(0, r)$ coupled with the boundary conditions $u^{(2i)}(0) = u^{(2i)}(r) = 0, \ \ i = 0, 1, \cdots, m-1$. Moreover, we use a bifurcation technique to obtain the existence of nodal solutions for problem (1.1). It is worth noting that a novel method is employed to prove that all zeros of the solution of problem (1.1) are simple, which is based on the well-known Uri Elias formula, see ^[20,21].

The assumptions of this paper are as follows:

(H1) $c$ is a constant with $c < \dfrac{\pi^2}{r^2}$;

(H2) $a \in C([0, r], [0, \infty))$ and $a\not\equiv 0$ on subinterval within $[0, r]$;

(H3) $f \in C(\mathbb{R}, \mathbb{R})$, $f(s)s > 0$ for $s\neq 0$,

for some $f_{+\infty}, f_0 \in (0, \infty).$ Using the disconjugacy theory and Elias's spectrum theory, we first consider the spectrum structure of the linear eigenvalue problem

Theorem 1.1. Assume that (H1) is satisfied, and $a$ satisfies (H2). Then,

(i) There is an infinite series of positive eigenvalues for problem (1.5)

(ii) $\lambda_k[c]\rightarrow \infty$ as $k\rightarrow \infty$.

(iii) For each eigenvalue $\lambda_k[c]$, there is an essentially unique eigenfunction $\varphi_k$. This eigenfunction $\varphi_k$ has exactly $k-1$ simple zeros within the interval $(0, r)$ and is positive in the vicinity of 0.

(iv) For an arbitrary subinterval within $[0, r]$, an eigenfunction corresponding to a sufficiently large eigenvalue will undergo a sign change in that subinterval.

(v) For every $k\in \mathbb{N}$, the algebraic multiplicity for $\lambda_k[c]$ is $1$.

According to Theorem 1.1, we employ bifurcation theory to acquire our main results.

Theorem 1.2. Let (H1), (H2), (A2), and (A3) hold. Suppose that for $k \in \mathbb{N}$, either

or

Then, (1.1) possesses two solutions, namely $u^+_k$ and $u^-_k$. The solution $u^+_k$ has precisely $k-1$ simple zeros within the interval $(0, r)$ and is positive in the vicinity of 0. Similarly, $u^-_k$ also has exactly $k-1$ simple zeros in $(0, r)$ and is negative near 0.

Remark 1.1. If $c = 0$ and $r = 1$, then Theorem 1.2 is transformed into Theorem 1.1 in ^[19].

Furthermore, we consider the case of the "jumping" nonlinearity.

Theorem 1.3. Assume (H1)–(H3) hold. When $\lambda > \dfrac{\lambda_k[c]}{f_0}$, there exist at least $2k-1$ non-trivial solutions of the boundary value problem (1.1). Indeed, there are solutions $w_1^-, \cdots, w_k^-$, such that, for each $1\leq j\leq k$, $w_j^-$ has exactly $j-1$ simple zeros on the open interval $(0, r)$ and is negative close to 0. Also, there are solutions $z_2^+, \cdots, z_k^+$ such that for each $2\leq j\leq k$, $z_j^+$ has precisely $j-1$ simple zeros on the open interval $(0, r)$ and is positive close to 0.

Remark 1.2. If $m = 2$, $a(t)\equiv1$, $f(u) = (u+1)^+-1$, then Theorem 1.3 is transformed into Theorem 1 in ^[10].

Example 1.1. Let $a(t) = \sin\dfrac{\pi}{r}t$ and $c < \dfrac{\pi^2}{r^2}$. We consider the existence of nodal solutions for sixth order boundary value problem

where

It is easy to verify that

Then, the conditions of Theorem 1.3 are fulfilled. Therefore, when $\lambda > \dfrac{\lambda_k[c]}{f_0}$, there exist at least $2k-1$ non-trivial solutions of the boundary value problem (1.6). Indeed, there are solutions $w_1^-, \cdots, w_k^-$, such that, for each $1\leq j\leq k$, $w_j^-$ has exactly $j-1$ simple zeros on the open interval $(0, r)$ and is negative close to 0. Also, there are solutions $z_2^+, \cdots, z_k^+$ such that for each $2\leq j\leq k$, $z_j^+$ has precisely $j-1$ simple zeros on the open interval $(0, r)$ and is positive close to 0.

The remainder of this paper is structured as follows. Section 2 is dedicated to demonstrating the spectrum results of the linear eigenvalue problem (1.5). In Section 3, we study the existence of nodal solutions for the nonlinear problem (1.1) under some suitable conditions via bifurcation theory.

2.   Spectrum results and maximum principle

Definition 2.1. ^[22] Let $p_k \in C[a, b]$ for $k = 1, ..., n.$ A linear differential equation of order $n$

is regarded as disconjugate on an interval $[a, b]$ if every non-trivial solution possesses less than $n$ zeros on $[a, b]$, where multiple zeros are accounted for in accordance with their multiplicity.

Definition 2.2. ^[22] The functions $y_1, \cdots, y_n \in C^n[a, b]$ are claimed to form a Markov system when the $n$ Wronskians

are positive on $[a, b]$.

Lemma 2.1. ^[22] Equation (2.1) possesses a Markov fundamental system of solutions on $[a, b]$ when and only when it is disconjugate on $[a, b]$.

Lemma 2.2. ^[22] Equation (2.1) possesses a Markov fundamental system of solutions when and only when $L$ has a representation

where $D = d/dt$, and

Example 2.1. For every $M\in(-m_0^4, m_1^4)$, the equation $u''''(t)+Mu(t) = 0$ is disconjugate on the interval $[0, 1]$, where $m_0, m_1$ are the first positive solution of the equations

respectively. See ^[23] for details.

Example 2.2. For every $M \in (-\infty, 2\pi)$, $u''' + Mu' = 0$ is disconjugate on $[0, 1]$. See ^[24] for details.

2.1.   Spectrum results

Elias ^[21] developed a theory on the eigenvalue problem

where

and $\rho_i\in C^{n-i}[a, b]$ with $\rho_i > 0$ on $[a, b]$. $\mathcal{L}_0 y, \cdots, \mathcal{L}_{n-1}y$ are called the quasi-derivatives of $y(x)$. To apply Elias's spectrum theory, we have to prove that problem (1.1) can be rewritten in the form of (2.2), i.e., the linear operator

has a factorization of the form

on $[0, r]$, where $v_i\in C^{2m-i}[0, r]$ with $v_i > 0$ on $[0, r]$, and

if and only if

This can be achieved under (H1) by using the disconjugacy theory in ^[22].

Define a linear differential equation of order $2m$

Theorem 2.1. Let (H1) hold. Then, $L[u]$ has a factorization

where $v_k\in C^{2m-k+1}[0, r]$ with $v_k > 0$ \ $(k = 1, 2, 3, \cdots, 2m)$.

Proof. Take $\phi(t)$ as the unique solution of the initial value problem

Subsequently, (H1) along with the Sturm Comparison Theorem leads to the conclusion that

and therefore

if $\sigma > 0$ is small enough. Take

One can easily verify that (2.3) holds true. □

At present, we are able to compute

Therefore, it is easy for us to reach the following conclusion.

Theorem 2.2. Let (H1) hold. Then,

is equivalent to

Remark 2.1. In ^[20], Rynne considered the boundary value problem

He assumes that $L$ is a 2$m$th-order, disconjugate, ordinary differential operator on $(0, \pi)$, together with separated boundary conditions at $0$ and $\pi$. However, we give a constructive proof (Theorem 2.1.) to obtain that $u^{(2m)}(t)+c u^{(2m-2)}(t) = 0$ is disconjugate on the interval $[0, r]$ if and only if $c < \frac{\pi^2}{r^2}$ in Theorem 2.1.

Currently, we are capable of applying Elias's spectrum theory to investigate the eigenvalue problem.

The Proofs of Theorem 1.1. Items (i) to (iv) are direct consequences of Elias (as mentioned in Theorems 1–5 in reference ^[21]) as well as Theorems 2.1 and 2.2. We shall only provide a proof for (v). Let $Y = C[0, r]$ be the Banach space which is equipped with the norm

Define $\bar L: D(\bar L)\rightarrow Y$ by setting

where

For simplicity, we define $\lambda_k: = \lambda_k[c]$. For demonstrating (v), it suffices to prove

Clearly,

Assume by contradiction that the algebraic multiplicity of $\lambda_k$ is greater than $1$. In that case, there exists a vector $u$ that belongs to $\ker(\bar L-\lambda_k)^2$ but not to $\ker (\bar L-\lambda_k)$. Subsequently,

for some $\gamma\neq 0$. After multiplying both sides of the aforementioned equation by $\varphi_k$ and integrating from 0 to $r$, we can infer that

which is a contradiction. □

Remark 2.2. Let $a(t) = 1$. We consider the eigenvalue problem

we see that the eigenvalues are $\lambda_k = \underbrace{\eta_k\cdots\eta_k} \limits_{m-1}(\eta_k-c)$ for $k = 1, 2, \cdots$ with corresponding eigenfunctions $\varphi_k(t) = \sin kt, \ k = 1, 2, \cdots$ for $c < \dfrac{\pi^2}{r^2}$, where $\{\eta_k\}_1^{\infty}$ are the eigenvalues of the problem

It is worth noting that the first eigenvalue $\eta_1 = \dfrac{\pi^2}{r^2}$. Therefore, the eigenvalues of problem (2.6) are simple, the eigenfunctions have only simple zeros on $(0, r)$, and the $k$th eigenfunction possesses exactly $k-1$ simple zeros on the open interval $(0, r).$ If $m = 2$, then problem (2.6) reduces to the situation in ^[10].

2.2.   Maximum principle

Theorem 2.3. Suppose that condition (H1) is satisfied. Take $e\in C[0, r]$, $e\geq0$ in $[0, r]$ with $e\not\equiv0$ in $[0, r]$. If $u\in C^{2m}[0, r]$ satisfies

then $u(t) > 0$ in $(0, r)$.

Proof. Let $\mathcal{D}u = -u''.$ Then,

Let $z_1: = \underbrace{\mathcal{D}\cdots \mathcal{D}}\limits_{m-2}(\mathcal{D}-cI)u = (-1)^{m-1} (u^{(2m-2)}(t)+cu^{(2m-4)}(t))$. From (2.7), then we have that

which implies

Repeating this procedure $m-1$ times, we obtain

which implies

where $z_{m-1} = (\mathcal{D}-cI)u$.

According to $c < \dfrac{\pi^2}{r^2}$, we get

This together with $z_{m-1}(t) > 0$ implies that $u(t) > 0$ for $t\in(0, r).$ □

3.   Existence of a nodal solutions

Let $E = \{u\in C^{2m-2}[0, r]: \ u^{(2i)}(0) = u^{(2i)}(r) = 0, \ \ i = 0, 1, \cdots, m-1 \}$ be the Banach space which is equipped with the norm

Then, $\bar L^{-1}: Y\rightarrow E$ is completely continuous. Here, $\bar L$ is presented as in (2.5).

Take $\zeta, \ \xi \in C(\mathbb{R}, \mathbb{R})$ such that

where $u^+ = \max\{u, 0\}$. Clearly,

Let

Then, $\tilde \xi$ is nondecreasing and

Let us consider

as a bifurcation problem originating from the trivial solution $u\equiv 0$.

Equation (3.3) can be equivalently transformed into

Obviously, the compactness of $\bar L^{-1}$ combined with (3.1) indicates that

In the following content, we adopt the terminology of Rabinowitz ^[25].

Let $S^+_k$ represent the set of functions in $E$ that possess exactly $k-1$ interior nodal (i.e., nondegenerate) zeros within the interval $(0, r)$ and are positive in the vicinity of $t = 0$. Set $S^-_k = -S^+_k$, and $S_k = S^+_k\cup S^-_k$. These sets are disjoint and open in $E$. Let $\Sigma$ denote the closure of the set consisting of nontrivial solutions of (3.4) in $\mathbb{R} \times E$.

Lemma 3.1. Suppose that (H1)–(H3) (or (A2) and (A3)) are satisfied. If $u \in D(\bar L)$ is a nontrivial solution of

then $u$ has only simple zeros in $(0, r)$. Thus, by definition, $u\in S_k$.

Proof. In fact, (3.5) can be rewritten as

where

Obviously, $\hat a(t)$ meets (H2). Therefore, Lemma 2.2 of ^[20] implies that all zeros of $u$ on the interval $(0, r)$ are simple. □

Remark 3.1. We say that $u$ is a $nodal solution$ if all of zeros of the solution are simple. It is a challenging problem to prove that all zeros of the solution are simple, see [10,Lemma 2.2]. In this paper, a novel method is employed to prove that all zeros of the solution of problem (1.1) are simple, which is based on the well-known Uri Elias formula ^[20,21], see Lemma 3.1.

By the Rabinowitz global bifurcation theorem ^[25], there is a continuum $\mathcal{C}_k \subset \Sigma$ of solutions for (3.4) bifurcating from $(\dfrac{\lambda_k}{f_0}, 0)$ which is either unbounded or contains a pair $(\dfrac{\lambda_j}{f_0}, 0)$ for $j\neq k$. By Lemma 1.24 of ^[25], it implies that if $(\lambda, u) \in \mathcal{C}_k$ and is near $(\dfrac{\lambda_k}{f_0}, 0)$, $u = \alpha\varphi_k + \omega$ with $\omega = o(|\alpha|)$. Since $S^{\pm}_k$ is open and $\varphi_k \in S_k$, then

for all positive $\varepsilon$ small enough, where

Define $\hat{\mathcal{C}_k} = \mathcal{C}_k-\mathbb{B}_{\varepsilon}(\dfrac{\lambda_k}{f_0}, 0)$. Then, Lemma 3.1 implies that

Otherwise, there exists $(\bar{\lambda}, \bar{u})\in \hat{C}_k$ such that $\bar{\lambda} > 0$ and $\bar{u}$ is in the boundary of $S_k$. If $\bar{u} = 0$, then $\bar{\lambda} = \lambda_j$ for some $j\neq k$, and so all points in $\hat{C}_k^+$ near $(\lambda_j, 0)$ are in $S_j$, a contradiction. Hence, $\bar{u}\neq0$. Since all the sets $S_j, j = 1, 2, \cdots$, are open, it follows that there is $t_0\in [0, r]$ such that $\bar{u}(t_0) = \bar{u}'(t_0) = 0.$ But, this contradicts Lemma 3.1. Consequently, one has that

It follows that $\mathcal{C}_k$ returning to the set of trivial solution axis is impossible. So, $\mathcal{C}_k$ is unbounded.

Furthermore, by Theorem 2 of ^[26], there exist two continua $\mathcal{C}_k^+$ and $\mathcal{C}_k^-$ composed of the bifurcation branch $\mathcal{C}_k$ which satisfy that either $\mathcal{C}_k^+$ and $\mathcal{C}_k^-$ are both unbounded or $\mathcal{C}_k^+ \cap \mathcal{C}_k^-\neq\ \{(\dfrac{\lambda_k}{f_0}, 0)\}.$ We know that $u = \alpha\varphi_k + \omega$ for $(\lambda, u)\in \mathcal{C}_k \ \backslash \ \{(\dfrac{\lambda_k}{f_0}, 0)\}$ near $(\dfrac{\lambda_k}{f_0}, 0)$. Since $\alpha\varphi_k \in S^{\pm}_k$ if $0\neq \alpha \in \mathbb{R}^{\pm}$ or $\mathbb{R}^{\mp}$, we have that

for all positive $\varepsilon$ small enough. Similar to the above argument, we are able to demonstrate that $\mathcal{C}^{\pm}_k \ \backslash \ \{(\dfrac{\lambda_k}{f_0}, 0)\}$ cannot depart from $\mathbb{R} \times S^{\pm}_k$ outside a neighborhood of $(\dfrac{\lambda_k}{f_0}, 0)$. Therefore, we have that $\mathcal{C}^\pm_k \subset (\mathbb{R} \times S^{\pm}_k\cup \{(\dfrac{\lambda_k}{f_0}, 0)\})$. It follows that both $\mathcal{C}^+_k$ and $\mathcal{C}^-_k$ are unbounded. Otherwise, at the expense of generality, we suppose that $\mathcal{C}^-_k$ is bounded. Then, there is $(\lambda_*, u_*)\in \mathcal{C}^+_k \cap \mathcal{C}^-_k$ such that $(\lambda_*, u_*)\neq (\dfrac{\lambda_k}{f_0}, 0)$ and $u_*\in S^{+}_k\cap S^{-}_k$. This contradicts the definitions of $S^{+}_k$ and $S^{-}_k$.

The Proofs of Theorem 1.2. By applying a similar method as used to prove [19,Theorem 1.1], with appropriate and obvious modifications, we can obtain the desired result. □

The Proofs of Theorem 1.3. We only need to show that

and

Suppose on the contrary that

where

As $C_j^{\nu}$ joins $(\dfrac{\lambda_k}{f_0}, 0)$ to infinity in $\Sigma$, and since $(\lambda, u) = (0, 0)$ is the sole solution of (3.3) in $E$, there exists a sequence $\{(\mu_m, u_m)\} \subset C_j^{\nu}$ such that $\mu_m \in (0, \lambda)$ and $\Vert u_m\Vert_E\rightarrow \infty$ as $m \rightarrow \infty$. We may assume that $\mu_m\rightarrow \bar{\mu} \in [0, \lambda]$ as $m \rightarrow \infty$. Let $v_m = \dfrac{u_m}{\Vert u_m\Vert_E}, m\geq1$. From the fact

we have that

Therefore, since $\bar{L}^{-1}:E\rightarrow E$ is completely continuous, it can be assumed that there is $v\in E$ together with $\Vert v\Vert_E = 1$ such that $\Vert v_m\rightarrow v\Vert_E\rightarrow0$ as $m\rightarrow \infty$. According to

and (3.2), we have that

i.e.,

By (H1), (H2), (3.6), and the fact that $\Vert v\Vert_{E} = 1$, we conclude that $\bar{\mu}a(t)f_{+\infty} v^+\not\equiv 0$, and consequently

According to Theorem 2.3, it is known that $v(t) > 0$, $t\in(0, r)$.

By Theorem 2.3, we know that $v(t) > 0$ in $(0, r)$. This implies that $\bar{\mu}f_{+\infty}$ is the first eigenvalue of $\bar{L}u = \lambda a(t)u$ and $v$ is the corresponding eigenfunction of $\lambda_1$. Hence, $v\in S_1^+$, and therefore, since $S_1^+$ is open and $\Vert v-v_m\Vert_E\rightarrow 0$ as $m\rightarrow \infty,$ we have that $v_m \in S_1^+$ for $m$ large. But, this contradicts the assumption that $(\lambda_m, v_m)\in C^{\nu}_j$ and $(j, \nu)\in\Gamma$. This proves Theorem 1.3. □

Remark 3.2. In this paper, we consider the "jumping" nonlinearity, i.e.,

for some $f_{+\infty}, f_0 \in (0, \infty)$, which complements the main result in ^[19].

Remark 3.3. For the existence result of solutions for (1.1) with $m = 2$, $n\geq1$ (PDE case), see ^{[27,28,29,30]}. For other results on the Lidstone BVPs, see ^[31,32,33].

Use of AI tools declaration

The authors declare they have not used Artificial Intelligence (AI) tools in the creation of this article.

Acknowledgments

This paper was supported by National Natural Science Foundation of China (No.12061064 and No.12361040). The authors would like to thank the editor and the anonymous reviewers for their helpful suggestions.

Conflict of interest

The authors declare there is no conflicts of interest.