Infinitely Many Nodal Solutions for Kirchhoff-Type Equations with Non-odd Nonlinearity

Abstract

In this study, we investigate the existence of infinitely many radial sign-changing solutions with nodal properties for a class of Kirchhoff-type equations possessing non-odd nonlinearity. By combining variational methods and analysis techniques, we prove that for any positive integer k, the equation has a radial solution that changes signs exactly k times. Furthermore, we demonstrate that the energy of such solutions is an increasing function of k. Owing to the inherent characteristics of these equations, the methods used herein significantly differ from those used in the existing literature. Particularly, we discover a unified method to obtain infinitely many radial sign-changing solutions with nodal properties for local and nonlocal problems.

Appendix A

Lemma A.1

Let $g_i\in C(\mathbb {R}_+)$ for $i=1,2,\dots ,k+1$. If

$$\begin{aligned} \lim _{r\rightarrow \infty }\frac{g_i(r)}{r}=\infty ,\ \ i=1,2,\dots ,k+1, \end{aligned}$$

(6.1)

then

$$\begin{aligned} \lim _{t\in \mathbb {R}_+^{k+1},|t|\rightarrow \infty }\frac{\sum _{i=1}^{k+1} g_i(t_i)}{\sum _{i=1}^{k+1}t_i}=\infty . \end{aligned}$$

Proof

According to the condition (6.1), $g_i$ is bounded below, there exists $C>0$ such that $g_i(r)\geqslant -C$ for $r\in \mathbb {R}_+$ and $i=1,2,\dots ,k+1$. For any given $M>0$, again according to (6.1), there exists $R>0$ such that

$$\begin{aligned} \frac{g_i(r)-kC}{r}>(k+1)M,\ \ r\geqslant R,i=1,2,\dots ,k+1.\ \ \end{aligned}$$

Thus, when $t\in \mathbb {R}_+^{k+1}$ with $|t|=(\sum _{i=1}^{k+1}t_i^2)^{1/2}\geqslant \sqrt{k+1}R$, from the inequality $|t|\leqslant \sqrt{k+1}\max \{t_1,t_2,\dots ,t_{k+1}\}$, it follows that $\max \{t_1,t_2,\dots ,t_{k+1}\}\geqslant R$. Without loss of generality, we may assume that $t_1=\max \{t_1,t_2,\dots ,t_{k+1}\}$. This implies that

$$\begin{aligned} \frac{\sum _{i=1}^{k+1}g_i(t_i)}{\sum _{i=1}^{k+1}t_i} \geqslant \frac{g_1(t_1)-kC}{(k+1)t_1}>M. \end{aligned}$$

This completes the proof. $\square$

Now, we present a simple implicit function theorem. Let us denote $D=[c_1,d_1]\times [c_2,d_2]\times \dots \times [c_k,d_k]$.

Lemma A.2

Let $F\in C([a,b]\times D)$. Assume that for each $y\in D$, $F(\cdot ,y)$ is decreasing on [a, b] and there exists $x_0\in [a,b]$ such that $F(x_0,y)=0$. Then there exists an implicit function $f\in C(D)$ such that $F(f(y),y)=0$ for $y\in D$.

Proof

According to the condition of Lemma A.2, there exists an implicit function $f:D\rightarrow [a,b]$ such that $F(f(y),y)=0$ for $y\in D$. Now, we prove the continuity of f. In fact, suppose that f is discontinuous at some $y_0\in D$. Then there exists $\{y_n\}\subset D$ satisfying $y_n\rightarrow y_0,x_n:=f(y_n)\rightarrow z_0$, but $z_0\ne f(y_0)$. However, according to the continuity of F, we have that

$$\begin{aligned} F(z_0,y_0)=\lim _{n\rightarrow \infty }F(f(y_n),y_n)=0. \end{aligned}$$

Thus $z_0=f(y_0)$. This leads to a contradiction, which completes the proof. $\square$