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Infinite number of solutions for some elliptic eigenvalue problems of Kirchhoff-type with non-homogeneous material

Abstract

In this paper, using variational method, we study the existence of an infinite number of solutions (some are positive, some are negative, and others are sign-changing) for a non-homogeneous elliptic Kirchhoff equation with a nonlinear reaction term.

Introduction

In this paper, we consider the following nonlocal equation:

$$ \textstyle\begin{cases} -M(x, \Vert u \Vert ^{2})\triangle u=\lambda f(x,u),\quad x\in \Omega , \\ u|_{x\in \partial \Omega }=0, \end{cases} $$
(1.1)

where Ω is a bounded open domain of \(\mathbb{R}^{N}\) with smooth boundary and

$$ \textstyle\begin{cases} f\in C(\overline{\Omega }\times \mathbb{R},\mathbb{R}), \\ M(x,t)=a(x)+b(x)t,\quad \Vert u \Vert =\int _{\Omega } \vert \nabla u \vert ^{2}\,dx, \end{cases} $$

with a, \(b\in C^{\gamma }(\overline{\Omega })\), \(\gamma \in (0,1)\), \(a(x)\geq a_{0}>0\), \(b(x)\geq 0\). Problem (1.1) is the steady-state problem associated with

$$ \textstyle\begin{cases} u_{tt}-M(x, \Vert u \Vert ^{2})\triangle u=f,&(x,t)\in (0,+\infty )\times \Omega , \\ u=0,&(x,t)\in \partial \Omega \times (0,+\infty ), \\ u(x,0)=u_{0}(x), \qquad u_{t}(x,0)=u_{1}(x),&x\in \Omega , \end{cases} $$
(1.2)

which is an open problem proposed by Lions [25] as a generalization of

$$ \textstyle\begin{cases} u_{tt}-M( \Vert u \Vert ^{2})\triangle u=f,&(x,t)\in (0,+\infty )\times \Omega , \\ u=0,&(x,t)\in \partial \Omega \times (0,+\infty ), \\ u(x,0)=u_{0}(x), \qquad u_{t}(x,0)=u_{1}(x),&x\in \Omega , \end{cases} $$
(1.3)

where \(M(t)=a+bt\) with \(a>0\) and \(b>0\). In [15, 24], the authors noted that Problem (1.2) models small vertical vibrations of an elastic string with fixed ends when the density of the material is not constant. When \(M(x,t)\) is independent of x, Problem (1.1) can be simplified to

$$ \textstyle\begin{cases} -M( \Vert u \Vert ^{2})\triangle u=\lambda f(x,u),\quad x\in \Omega , \\ u|_{x\in \partial \Omega }=0. \end{cases} $$
(1.4)

The steady-state problem (1.4) associated with Problem (1.3) has received a lot of attention in the literature (usually using variational methods); see [2, 3, 10, 14, 16, 20, 22, 23, 29, 3340] and the references therein.

There are many papers in the literature on sign-changing solutions for Dirichlet problems; see [4, 5, 7, 12, 19, 21] and their references. In [43], Zhang and Perera obtained sign-changing solutions for a class of Problem (1.4) using variational methods and invariant sets of descent flow; in [31] using minimax methods and invariant sets of descent flow, Mao and Zhang established the existence of sign-changing solutions; and in [35] combining the constraint variational method and the quantitative deformation lemma, Shuai proved that Problem (1.4) possesses one least energy sign-changing solution. Other results on the existence of sign-changing solutions for Kirchhoff equations can be found in [5, 9, 28, 30, 37] and their references.

Since \(M(x,t)\) is dependent on x in Problem (1.1), the variational approach cannot be used to discuss it in a direct way, and fixed point theory and the Galerkin method were used to establish existence in [33] and [38]. In [15], Figueiredo et al. established the existence and uniqueness of a positive solution of Problem (1.1) via bifurcation theory, and in [17], Huy and Quan considered a generalization of Problem (1.1)

$$ \textstyle\begin{cases} -M(x, \Vert u \Vert ^{2})\triangle u=\lambda f(x,u,\nabla u)-g(x,u, \nabla u),\quad x\in \Omega , \\ u|_{x\in \partial \Omega }=0, \end{cases} $$

and established existence results for both non-degenerate and degenerate cases of the function M using the fixed point index theory. We note, to the best of our knowledge, that there are no results in the literature on the existence of a sign-changing solution for Problem (1.1). In this paper (motivated by [21]) using the steepest descent method for gradient mappings of the isoperimetric variational problem (see [6]) and the method of invariant sets of descending flow in critical point theory (see [27]), we establish the existence of an infinite number of solutions (some are positive, some are negative, and others are sign-changing). Some ideas come from [18] and [42].

Main result

In this section, we suppose that f satisfies the following conditions:

(1) \(f:\overline{\Omega }\times \mathbb{R}\to \mathbb{R}\) is locally Lipschitz continuous;

(2) \(f(x,t)t\geq 0\) and \(f(x,t)\not \equiv 0\) in \(\Omega \times (-\delta ,0)\cup \Omega \times (0,\delta )\);

(3) \(|f(x,t)|\leq c_{1}|t|^{p}+c_{2}\), where \(c_{1}\), \(c_{2}\in \mathbb{R}^{+}\), \(1\leq p<\frac{N+2}{N-2}\) if \(N\geq 3\) and \(1\leq p<+\infty \) if \(N=1\) and \(N=2\).

Let \(A:= \mathbb{N}\) in our main result. The main theorem is as follows.

Theorem 2.1

Suppose that f satisfies (1), (2), and (3). Then Problem (1.1) has an infinite number of positive solutions \(\{u_{1,\alpha }\}_{\alpha \in A}\), an infinite number of negative solutions \(\{u_{2,\alpha }\}_{\alpha \in A}\), and an infinite number of sign-changing solutions \(\{u_{3,\alpha }\}_{\alpha \in A}\).

First we establish the following lemma for Problem (1.1).

Lemma 2.1

Problem (1.1) has a nontrivial solution if and only if there exists \(r>0\) such that the following problem

$$ \textstyle\begin{cases} - \triangle u=\lambda \frac{1}{M(x,r^{2})}f(x,u),& x \in \Omega , \\ u=0,& x\in \partial \Omega \end{cases} $$
(2.1)

has a nontrivial solution u with \(\|u\|=r\).

Proof

Sufficiency. There exists \(r>0\) such that Problem (2.1) has a nontrivial solution u with \(\|u\|=r\), and so u satisfies

$$ \textstyle\begin{cases} - \triangle u=\lambda \frac{1}{M(x,r^{2})}f(x,u)= \lambda \frac{1}{M(x, \Vert u \Vert ^{2})}f(x,u),& x\in \Omega , \\ u=0,& x\in \partial \Omega . \end{cases} $$

Clearly, u is a nontrivial solution of Problem (1.1).

Necessity. Problem (1.1) has a nontrivial solution u. Let \(r=\|u\|>0\). Then u satisfies

$$ \textstyle\begin{cases} - \triangle u=\lambda \frac{1}{M(x, \Vert u \Vert ^{2})}f(x,u)= \lambda \frac{1}{M(x,r^{2})}f(x,u),& x\in \Omega , \\ u=0,& x\in \partial u, \end{cases} $$

that is, u is a nontrivial solution of Problem (2.1) with \(\|u\|=r\).

The proof is completed. □

For given \(r>0\), set

$$ S_{r}= \biggl\{ u\in H_{0}^{1}(\Omega ): \int _{\Omega } \vert \nabla u \vert ^{2}\,dx=r^{2} \biggr\} \quad \text{ and }\quad \overline{S}_{r}=S_{r} \cap C_{0}^{1}( \overline{\Omega }). $$

From Lemma 2.1, we only consider the existence of a nontrivial solution of Problem (2.1) in \(S_{r}\).

Set

$$\begin{aligned}& g(x,u)=\frac{1}{M(x,r)}f(x,u),\qquad G(x,u)= \int _{0}^{u}g(x,t)\,dt, \quad \forall u\in \mathbb{R}, \\& \Psi (u)=- \int _{\Omega }G\bigl(x,u(x)\bigr)\,dx,\quad \forall u\in H_{0}^{1}( \Omega ), \end{aligned}$$

and

$$ F=\Psi |_{S_{r}},\quad \overline{F}=F|_{\overline{S}_{r}}. $$

Note that

$$ F'(u)= \Psi '(u)- \frac{(\Psi '(u),u)}{ \Vert u \Vert ^{2}}u=-T(u)u-K\mathbb{G}(u), $$

where \((\cdot ,\cdot )\) is the inner product in \(H_{0}^{1}(\Omega )\) given by \((u,v)=\int _{\Omega }\nabla u\cdot \nabla v\,dx\), \(K=(-\triangle )^{-1}\) with the Dirichlet boundary condition, \(\mathbb{G}\) is the Nemitskii operator induced by g and

$$ T(u)=\frac{(\Psi '(u),u)}{r^{2}}. $$

From condition (2), we have \((\Psi '(u),u)<0\) for all \(u\in S_{r}\) (see Lemma 1.0 in [19]), and we know that the solutions of Problem (2.1) correspond to the critical points of f.

In order to discuss Problem (2.1), for \(r>0\), we let (here \(n\in \mathbb{N}\))

$$\begin{aligned}& f_{n}(x,t)= \textstyle\begin{cases} f(x,t),&\text{if } \vert t \vert \leq n, \\ f(x,n)+t-n, & \text{if } t>n, \\ f(x,-n)+t+n, &\text{if } t< -n, \end{cases}\displaystyle \end{aligned}$$
(2.2)
$$\begin{aligned}& g_{n}(x,t)=\frac{1}{M(x,r^{2})}f_{n}(x,t), \end{aligned}$$
(2.3)
$$\begin{aligned}& G_{n}(x,u)= \int _{0}^{u}g_{n}(x,t)\,dt, \quad \forall u\in \mathbb{R} \end{aligned}$$
(2.4)

and consider

$$ \textstyle\begin{cases} -\triangle u=\lambda g_{n}(x,u),\quad x\in \Omega , \\ u|_{x\in \partial \Omega }=0. \end{cases} $$
(2.5)

Let

$$ \Psi _{n}(u)=- \int _{\Omega }G_{n}\bigl(x,u(x)\bigr)\,dx,\quad \forall u\in H_{0}^{1}( \Omega ), $$

and

$$ F_{n}=\Psi _{n}|_{S_{r}}, \quad \overline{F}_{n}=F_{n}|_{\overline{S}_{r}}. $$
(2.6)

We obtain that

$$ F_{n}'(u)=\Psi _{n}'(u)- \frac{(\Psi _{n}'(u),u)}{ \Vert u \Vert ^{2}}u=-T_{n}(u)u-K \mathbb{G}_{n}(u), $$
(2.7)

where \(\mathbb{G}_{n}\) is the corresponding Nemitskii operator to \(g_{n}\) and

$$ T_{n}(u)=\frac{(\Psi '_{n}(u),u)}{r^{2}}. $$

From the definition of \(F_{n}\) in (2.6), we know that the solutions of Problem (2.5) correspond to the critical points of \(F_{n}\).

From the definition of \(g_{n}\) and conditions (1), (2), and (3), it is easy to see that \(g_{n}\) also satisfies (1), (2), and (3) uniformly with respect to n and

(1)′ there exists \(L_{n}>0\) such that

$$ \bigl\vert g_{n}(x,t_{1})-g_{n}(x,t_{2}) \bigr\vert \leq L_{n} \vert t_{1}-t_{2} \vert ,\quad \forall x \in \overline{\Omega }, t_{1},t_{2} \in \mathbb{R}. $$
(2.8)

We shall need the following results later.

Lemma 2.2

(see [1])

Let Ω be a bounded, open subset of \(\mathbb{R}^{N}\), and suppose that Ω is \(C^{1}\). Assume that \(N< p\leq +\infty \) and \(u\in W^{k+1,p}(\Omega )\). Then there is \(u^{*}\in C^{k,\gamma }(\overline{\Omega })\) with \(u(x)=u^{*}(x)\) a.e. \(x\in \Omega \) such that

$$ \bigl\Vert u^{*} \bigr\Vert _{C^{k,\gamma }}\leq C \Vert u \Vert _{W^{k+1,p}}; $$

here the constant C depends only on p, N, and Ω.

Lemma 2.3

(see [13])

Let Ω be a bounded open subset of \(\mathbb{R}^{N}\) with a \(C^{1}\) boundary. Assume that \(u\in W^{k,p}(\Omega )\).

(1) If

$$ k< \frac{n}{p}, $$

then \(u\in L^{q}(\Omega )\), where

$$ \frac{1}{q}=\frac{1}{p}-\frac{k}{N}. $$

Also

$$ \Vert u \Vert _{L^{q}(\Omega )}\leq C \Vert u \Vert _{W^{k,p}(\Omega )}; $$

here the constant C depends only on k, p, N, and Ω.

(2) If

$$ k>\frac{n}{p}, $$

then \(u\in C^{k-[\frac{n}{p}]-1,\gamma }(\overline{\Omega })\), where

$$ \gamma =\textstyle\begin{cases} [\frac{n}{p}]+1-\frac{n}{p}, & \textit{if } \frac{n}{p} \textit{ is not an integer}, \\ \textit{any positive number }< 1, & \textit{if } \frac{n}{p} \textit{ is an integer}. \end{cases} $$

Also

$$ \Vert u \Vert _{C^{k-[\frac{n}{p}]-1,\gamma }(\overline{\Omega })}\leq C \Vert u \Vert _{W^{k,p}( \Omega )}; $$

here the constant C depends only on k, p, N, and Ω.

Lemma 2.4

(see [11])

Let p, \(1\leq p\leq p_{0}=(N + 2)/(N - 2)\) (so that \(2\leq p+1\leq 2^{*}\)), and let \(\beta =(2^{*}/N)(2^{*}-(p+1))\). Then, for each γ, \(0\leq \gamma \leq \beta \), there exists \(c > 0\) such that

$$ \Vert u \Vert _{p+1}^{p+1}\leq c \Vert \nabla u \Vert _{2}^{p+1-\gamma } \Vert u \Vert _{2}^{\gamma } $$

for all \(u\in W_{0}^{1,2}(\Omega )\). (Here and henceforth \(\|u\|_{p}\) denotes the norm of u in \(L^{p}(\Omega )\).)

Lemma 2.5

(see [8])

Let X be a Banach space and F be a closed subset in X. Assume that \(V:X\to Y\) is locally continuous and

$$ \lim_{h\downarrow 0}\frac{d(u+hV(u),F)}{h}=0 $$
(2.9)

for all \(u\in \partial F\), where \(d(\cdot ,\cdot )\) is the distance on X. If \(u_{0}\in F\) and \(\sigma (t)\)(\(0\leq t<\omega _{+}(u_{0})\)) is the solution of the initial value problem

$$ \textstyle\begin{cases} \frac{d\sigma }{\,dt}=V(\sigma ), \\ \sigma (0,u_{0})=u_{0}, \end{cases} $$

then \(\sigma (t)\in F\) for all \(t\in [0,\omega _{+}(u_{0}))\).

For each n, we consider

$$ \textstyle\begin{cases} \frac{d\sigma }{dt}=-F_{n}'(\sigma )=T_{n}(\sigma ) \sigma +K\mathbb{G}_{n}(\sigma ), \quad t\geq 0, \\ \sigma (0,u_{0})=u_{0} \end{cases} $$
(2.10)

in \(H_{0}^{1}(\Omega )\) for \(u_{0}\in S_{r}\), where \(F'_{n}\) is defined in (2.7). Since (1)′, (2), and (3) hold, we have the following.

Lemma 2.6

(see [32])

Let \(c< b<0\). For every \(u\in F_{n}^{-1}([c,b])\), if \(\sigma _{n}(t,u)\) is a solution of Problem (2.10) in \([0,+\infty )\) (see step 3 in Lemma 2.7), then either there is a unique \(t(u)\in [0,+\infty )\) such that \(F_{n}(\sigma _{n}(t(u),u))=c\) or there is a critical point v of \(F_{n}\) in \(F_{n}^{-1}([c,b])\) such that \(\sigma _{n}(t,u)\to v\) as \(t\to +\infty \).

Lemma 2.7

Under conditions (1), (2), and (3), Problem (2.10) has a unique solution \(\sigma _{n}(t,u_{0})\) on \([0,+\infty )\), which satisfies:

(i) \(\sigma _{n}(t,u_{0})\in S_{r}\) for all \(u_{0}\in S_{r}\); \(\sigma _{n}(t,u_{0})\in \overline{S_{r}}\) for all \(u_{0}\in \overline{S_{r}}\);

(ii) there exists \(u_{n}\in S_{r}\) such that \(\lim_{t\to +\infty }\sigma _{n}(t,u_{0})) \stackrel{H_{0}^{1}}{=}u_{n}\) for \(u_{0}\in S_{r}\);

(iii) if \(u_{0}\in \overline{S}_{r}\), then \(u_{n}\in \overline{S}_{r}\) and \(\lim_{t\to +\infty }\sigma _{n}(t,u_{0}) \stackrel{C_{0}^{1}}{=}u_{n}\).

Proof

The proof is divided into six steps.

Step 1. We show that \(F_{n}'(u)=-T_{n}(u)u-K\mathbb{G}_{n}(u)\) is globally Lipschitz continuous with respect to \(H_{0}^{1}(\Omega )\), that is, there is \(M>0\) such that

$$ \bigl\Vert F_{n}'(u_{1})-F_{n}'(u_{2}) \bigr\Vert _{H_{0}^{1}}\leq M \Vert u_{1}-u_{2} \Vert _{H_{0}^{1}}, \quad \forall u_{1}, u_{2}\in S_{r}. $$

Let \(2^{*}=\frac{2N}{N-2}\). From (2.8), we have

$$\begin{aligned} \bigl\Vert \mathbb{G}_{n}(u_{1})- \mathbb{G}_{n}(u_{2}) \bigr\Vert _{L^{2^{*}}} =& \biggl( \int _{\Omega } \bigl\vert g_{n}(x,u_{1})-g_{n}(x,u_{2}) \bigr\vert ^{2^{*}}\,dx \biggr)^{1/2^{*}} \\ \leq& \biggl( \int _{\Omega }L_{n}^{2^{*}} \bigl\vert u_{2}(x)-u_{1}(x) \bigr\vert ^{2^{*}}\,dx \biggr)^{1/2^{*}} \\ =& L_{n} \Vert u_{1}-u_{2} \Vert _{L^{2^{*}}}, \end{aligned}$$

i.e., \(\mathbb{G}_{n}\) is globally Lipschitz in the \(L^{2^{*}}\) topology. Note that

$$ H_{0}^{1}(\Omega )\hookrightarrow L^{2^{*}}( \Omega ) \stackrel{\mathbb{G}}{\hookrightarrow }L^{2^{*}}(\Omega ) \stackrel{K}{\rightarrow } H_{0}^{1}(\Omega ), $$

K is a bounded linear operator, and so

$$ \bigl\Vert K\mathbb{G}(u_{1})-K\mathbb{G}(u_{2}) \bigr\Vert \leq \overline{L}_{n} \Vert u_{1}-u_{2} \Vert $$

for some positive constant \(\overline{L}_{n}\), where \(\|\cdot \|\) denotes the norm in \(H_{0}^{1}(\Omega )\). Note

$$\begin{aligned} \bigl\vert T_{n}(u_{1})-T_{n}(u_{2}) \bigr\vert =&\frac{1}{r^{2}}|\bigl(K \mathbb{G}_{n}(u_{1}),u_{1} \bigr)-(K\mathbb{G}_{n}) (u_{2}),u_{2})| \\ \leq &\frac{1}{r}\overline{L}_{n} \Vert u_{1}-u_{2} \Vert + \frac{1}{r^{2}} \bigl\Vert K\mathbb{G}(u_{2}) \bigr\Vert \Vert u_{1}-u_{2} \Vert , \end{aligned}$$

and

$$ \bigl\Vert T_{n}(u_{1})u_{1}-T_{n}(u_{2})u_{2} \bigr\Vert \leq \bigl\vert T_{n}(u_{1})-T_{n}(u_{2}) \bigr\vert \Vert u_{1} \Vert + \bigl\vert T_{n}(u_{2}) \bigr\vert \Vert u_{1}-u_{2} \Vert . $$

Since \(\|K\mathbb{G}_{n}(u)\|\) is bounded in \(S_{r}\), so \(T_{n}(u)\) is bounded also. Thus \(F'_{n}(u)\) is globally Lipschitz continuous.

Step 2. We show that \(F_{n}'(u)=-T_{n}(u)u-K\mathbb{G}_{n}(u)\) is globally Lipschitz continuous with respect to \(C_{0}^{1}(\overline{\Omega })\), that is, there is \(\overline{M}>0\) such that

$$ \bigl\Vert F_{n}'(u_{1})-F_{n}'(u_{2}) \bigr\Vert _{C_{0}^{1}}\leq \overline{M} \Vert u_{1}-u_{2} \Vert _{C_{0}^{1}},\quad \forall u_{1}, u_{2}\in \overline{S_{r}}. $$

Let \(l>N\). From (2.8), we have

$$ \bigl\Vert G_{n}(u_{1})-G_{n}(u_{2}) \bigr\Vert _{L^{l}}= \biggl( \int _{\Omega } \bigl\vert g_{n}(x,u_{1})-g_{n}(x,u_{2}) \bigr\vert ^{l}\,dx \biggr)^{1/l} \leq L_{n} \Vert u_{1}-u_{2} \Vert _{L^{l}}, $$

i.e., \(\mathbb{G}_{n}\) is globally Lipschitz in the \(L^{l}(\Omega )\) topology. Note that

$$ C_{0}^{1}(\overline{\Omega })\hookrightarrow L^{l}(\Omega ) \stackrel{\mathbb{G}}{\rightarrow }L^{l}(\Omega ) \stackrel{K}{\rightarrow }W^{2,l}(\Omega ) \cap W_{0}^{2,l}(\Omega ) \hookrightarrow C_{0}^{1}(\overline{\Omega }), $$

K is a bounded linear operator, so there exists \(\overline{L}'_{n}>0\) such that

$$ \bigl\Vert K\mathbb{G}_{n}(u_{1})-K \mathbb{G}_{n}(u_{2}) \bigr\Vert _{C_{0}^{1}}\leq \overline{L}'_{n} \Vert u_{1}-u_{2} \Vert _{C_{0}^{1}}. $$

Note

$$\begin{aligned}& \bigl\vert T_{n}(u_{1})-T_{n}(u_{2}) \bigr\vert \\& \quad =\frac{1}{r^{2}} \bigl\vert \bigl(K \mathbb{G}_{n}(u_{1}),u_{1} \bigr)-(K\mathbb{G}_{n}) (u_{2}),u_{2}) \bigr\vert \\& \quad =\frac{1}{r^{2}} \bigl\vert \bigl(K\mathbb{G}_{n}(u_{1}),u_{1} \bigr)-\bigl(K \mathbb{G}_{n}(u_{2}),u_{1} \bigr) + \bigl(K\mathbb{G}_{n}(u_{2}),u_{1} \bigr)-(K\mathbb{G}_{n}) (u_{2}),u_{2}) \bigr\vert \\& \quad =\frac{1}{r^{2}} \bigl\vert \bigl(K\mathbb{G}_{n}(u_{1})-K \mathbb{G}_{n}(u_{2}),u_{1}\bigr)+ \bigl(K \mathbb{G}_{n}(u_{2}),u_{1}-u_{2} \bigr) \bigr\vert \\& \quad \leq \frac{1}{r}\overline{L}'_{n} \Vert u_{1}-u_{2} \Vert _{C_{0}^{1}}+ \frac{1}{r^{2}} \bigl\Vert K\mathbb{G}(u_{2}) \bigr\Vert _{C_{0}^{1}} \Vert u_{1}-u_{2} \Vert _{C_{0}^{1}}, \end{aligned}$$

and

$$ \bigl\Vert T_{n}(u_{1})u_{1}-T_{n}(u_{2})u_{2} \bigr\Vert _{C_{0}^{1}}\leq \bigl\vert T_{n}(u_{1})-T_{n}(u_{2}) \bigr\vert \Vert u_{1} \Vert _{C_{0}^{1}}+ \bigl\vert T_{n}(u_{2}) \bigr\vert \Vert u_{1}-u_{2} \Vert _{C_{0}^{1}}. $$

Since \(\|K\mathbb{G}_{n}(u)\|_{C_{0}^{1}}\) is bounded in \(\overline{S}_{r}\), so \(T_{n}(u)\) is bounded also. Thus \(F'_{n}(u)\) is globally Lipschitz continuous.

Step 3. We show that Problem (2.10) has a unique solution \(\sigma _{n}(t,u_{0})\) with maximal interval \([0,+\infty )\) for \(u_{0}\in S_{r}\) and \(\sigma _{n}(t,u_{0})\in S_{r}\) for all \(t\in [0,+\infty )\).

The theory of Cauchy problems of ordinary differential equations together with step 1 implies that (2.10) has a unique solution \(\sigma _{n}(t,u_{0})\) with maximal interval \([0,\omega _{+}(u_{0}))\) for \(u_{0}\in S_{r}\). Note

$$\begin{aligned}& \sigma _{n}(t,u_{0})=e^{-w(t)} \biggl\{ u_{0}+ \int _{0}^{t}e^{w(s)}K \mathbb{G} \bigl(\sigma _{n}(s,u_{0})\bigr)\,ds \biggr\} \\& \quad \text{where } w(t)=- \int _{0}^{t}T_{n}\bigl(\sigma _{n}(s,u_{0})\bigr)\,ds. \end{aligned}$$

Since \(d\|\sigma _{n}(t,u_{0})\|^{2}/dt\equiv 0\) for all \(t\in [0,\omega _{+}(u_{0}))\), we have \(\sigma _{n}(t,u_{0})\in S_{r}\) for \(t\in [0,\omega _{+}(u_{0}))\) if \(u_{0}\in S_{r}\).

Also, since \(g_{n}(\sigma _{n}(t,u_{0}))\) is bounded in \(H_{0}^{1}\) if \(u_{0}\in S_{r}\), then \(\omega _{+}(u_{0})=+\infty \) (see [32]).

Step 4. We show that Problem (2.10) has a unique solution \(\sigma _{n}(t,u_{0})\) with maximal interval \([0,+\infty )\) for \(u_{0}\in \overline{S}_{r}\) and \(\sigma _{n}(t,u_{0})\in \overline{S}_{r}\) for all \(t\in [0,+\infty )\).

Since step 2 holds, the proof of step 4 is similar to that of step 3, so we omit it.

Step 5. For \(u_{0}\in S_{r}\), we show that there exists \(u_{n}\in S_{r}\) such that

$$ \lim_{t\to +\infty }\sigma _{n}(t,u_{0})=u_{n} \quad \text{in } H_{0}^{1}. $$

First, since \(F_{n}(u)<0\) for \(u\in S_{r}\), choose \(b=F_{n}(u_{0})<0\). Since \(S_{r}\) is bounded and weakly convergent closed and \(F_{n}\) is weakly semi-continuous from below, we have \(\inf_{u\in S_{r}}F_{n}(u)>-\infty \). Let \(c<\inf_{u\in S_{r}}F_{n}(u)\). Then \(u_{0}\in F_{n}^{-1}([c,b])\). From Lemma 2.6, there exists \(u_{n}\in S_{r}\) such that

$$ \lim_{t\to +\infty }\sigma _{n}(t,u_{0})=u_{n} \quad \text{in } H_{0}^{1}. $$

Step 6. For \(u_{0}\in \overline{S}_{r}\), there exists \(u_{n}\in \overline{S}_{r}\) such that

$$ \lim_{t\to +\infty }\sigma _{n}(t,u_{0})=u_{n} \quad \text{in } C_{0}^{1}. $$

Using the proof of step 5, step 2 guarantees the conclusion is true. □

Let P be the positive cone in \(C_{0}^{1}(\overline{\Omega })\) and be the interior set of P. The elements of are called positive and the elements of − are called negative.

Lemma 2.8

Under condition (1) and (2), the flow in Lemma 2.7has the following properties:

$$ \sigma _{n}(t,u_{0})\in \pm \mathring{P} \quad \textit{for }u_{0}\in \pm \mathring{P}\cap \overline{S}_{r} \textit{ and } t\in [0,+\infty ). $$

Proof

The proof follows the ideas in Lemma 1 and 6 in [26].

(1) We show that \(K\mathbb{G}_{n}(u_{0})\in \mathring{P}\) for \(u_{0}\in P-\{\theta \}\).

Let \(v=K\mathbb{G}_{n}(u_{0})\), and we have

$$ -\triangle v=g_{n}(x,u_{0})\geq \not \equiv 0,\quad \forall x\in \Omega , v|_{\partial \Omega }=0. $$

The strong maximum principle implies that \(v\in \mathring{P}\).

(2) We show that

$$ K\mathbb{G}_{n}\bigl(\sigma _{n}(t,u_{0}) \bigr)\in \mathring{P} \quad \text{ for } u_{0} \in P\cap \overline{S}_{r}, \text{ and } t>0. $$
(2.11)

Now \(\forall u\in P\), choose \(\delta >0\) small such that, for all \(\delta >h>0\), we have

$$ u+h(\bigl(T_{n}(u)u+K\mathbb{G}_{n}(u)\bigr)= \bigl(1+hT_{n}(u)\bigr)u+hK\mathbb{G}_{n}(u) \in P, $$

i.e., (2.9) is satisfied. Now Lemma 2.5 guarantees that the solution \(\sigma _{n}(t,u_{0})\) of the initial value problem (2.10) satisfies \(\sigma _{n}(t,u_{0})\in P\) for all \(t\in [0,+\infty )\) (in fact \(\sigma _{n}(t,u_{0})\in P\cap \overline{S_{r}}\) since \(u_{0} \in P\cap \overline{S_{r}}\)). Hence (as in (1)) (2.11) holds.

(3) We show that

$$ \sigma _{n}(t,u_{0})\in \mathring{P} \quad \text{for }u_{0}\in \mathring{P} \cap \overline{S}_{r} \text{ and } t\in [0,+\infty ). $$

Let \(w(t)=-\int _{0}^{t}T_{n}(\sigma _{n}(s,u_{0}))\,ds\). We have \(w'(t)>0\), \(w(t)>0\), and \(w(t)\) is strictly increasing. Let \(w^{-1}(t)\) be the inverse function of \(w(t)\). It follows from (2.11), for \(u_{0}\in P\cap \overline{S}_{r}\), that

$$ \bigl(1/w'(t)\bigr)K\mathbb{G}_{n} \bigl(\sigma _{n}(t,u_{0})\bigr)\in \mathring{P}. $$
(2.12)

Let \(A(t)=(1/w'(t))K\mathbb{G}_{n}(\sigma (t,u_{0}))\) and \(E_{t}=\{A(s):0\leq s\leq t\}\). Note that \(E_{t}\) is a compact set in \(C_{0}^{1}(\overline{\Omega })\) and (2.12) implies that \(E_{t}\subseteq \mathring{P}\) and hence \(\overline{co}E_{t}\subseteq \mathring{P}\), where \(\overline{co}E_{t}\) is the closed convex set hull of \(E_{t}\) in \(C_{0}^{1}(\overline{\Omega })\). Note

$$\begin{aligned} \frac{1}{e^{w(t)}-1} \int _{0}^{t}e^{w(s)}K \mathbb{G}_{n}\bigl(\sigma _{n}(s,u_{0}) \bigr)\,ds&= \frac{1}{e^{w(t)}-1} \int _{1}^{e^{w(t)}} \frac{K\mathbb{G}_{n}(\sigma _{n}(w^{-1}(\ln (s)),u_{0}))}{w'(w^{-1}(\ln (s)))}\,ds \\ &= \lim_{m\to +\infty }\frac{1}{m}\sum _{i=1}^{m}A(w^{-1}\biggl( \ln \biggl(1+\frac{i}{m}\bigl(e^{w(t)}-1\bigr)\biggr)\biggr). \end{aligned}$$

Therefore

$$ \frac{1}{e^{w(t)}-1} \int _{0}^{t}e^{w(t)}K\mathbb{G} \bigl(\sigma _{n}(s,u_{0})\bigr)\,ds \in \overline{co}F_{t}\in \mathring{P}, $$

and this together with

$$ \sigma _{n}(t,u_{0})=e^{-w(t)} \biggl\{ u_{0}+ \int _{0}^{t}e^{w(s)}K \mathbb{G} \bigl(\sigma _{n}(s,u_{0})\bigr)\,ds \biggr\} ,\quad t\in [0,+\infty ) $$

and

$$ e^{-w(t)}= \bigl(1- e^{-w(t)} \bigr)\frac{1}{e^{w(t)}-1} $$

yields

$$ \sigma _{n}(t,u_{0})\in \mathring{P} \quad \text{for }u_{0}\in \mathring{P} \cap \overline{S}_{r} \text{ and } t\in [0,+\infty ). $$

For the case \(u_{0}\in (-\mathring{P})\), the proof is similar, so we omit it.

The proof is completed. □

Lemma 2.9

Under conditions (1), (2), and (3), Problem (2.5) has at least one positive solution \(u_{1,n}\in \overline{S}_{r}\cap P\), one negative solution \(u_{2,n}\in \overline{S}\cap (-P)\), and one sign-changing solution \(u_{3,n}\in \overline{S}_{r}\cap (C_{0}^{1}-(-P\cup P))\).

Proof

Let \(e_{1}\) be an eigenfunction corresponding to the first eigenvalue of the Dirichlet eigenvalue problem: \(-\triangle u=\lambda u\) in Ω, \(u|_{\partial \Omega }=0\), \(e_{2}\) be an eigenfunction corresponding to the second one with \(\|e_{1}\|=\|e_{2}\|=r\). Let \(\Lambda ={\mathrm{span}}\{e_{1},e_{2}\}\cap S_{r}\). Note that \(\Psi _{n}(u)<0\) for each \(n>0\) if \(u\not \equiv 0\) and

$$ \Lambda =\{\cos \theta e_{1}+\sin \theta e_{2}:0\leq \theta \leq 2 \pi \} $$

is a compact set in \(S_{r}\). Then there exists \(\alpha _{n}>0\) such that

$$ \max \bigl\{ \Psi _{n}(u):u\in \Lambda \bigr\} =\max \bigl\{ \Psi (u):u\in \Lambda \bigr\} < - \alpha _{n} . $$
(2.13)

Set

$$ \Lambda ^{\pm }=\bigl\{ u\in \Lambda :\sigma _{n}(t,u)\in \pm \mathring{P} \text{ for some } t>0\bigr\} . $$

(1) We show that \(\Lambda ^{\pm }\neq\emptyset \).

Since \(e_{1}\in \overline{S}_{r}\cap \mathring{P}\), \(-e_{1}\in \overline{S}_{r}\cap (-\mathring{P})\), Lemma 2.8 guarantees that \(\sigma _{n}(t,e_{1})\in \mathring{P}\) and \(\sigma _{n}(t,-e_{1})\in (-\mathring{P})\) for \(t\in [0,+\infty )\). Therefore, \(\Lambda ^{\pm }\neq\emptyset \).

(2) We show that Problem (2.5) has at least one positive solution \(u_{1,n}\) and one negative solution \(u_{2,n}\).

Consider \(\sigma _{n}(t,e_{1})\), \(t\in [0,+\infty )\). Lemma 2.7 guarantees that there exists \(u_{1,n}\in \overline{S}_{r}\cap P\) such that

$$ \lim_{t\to +\infty }\sigma _{n}(t,e_{1}) \stackrel{C_{0}^{1}}{=}u_{1,n}, $$

and \(u_{1,n}\) is a critical point of \(F_{n}\) in \(\overline{S}_{r}\cap P\). Then \(u_{1,n}\) is a solution of Problem (2.5) and \(u_{1,n}\in \overline{S}_{r}\). By using the strong maximum principle, we have \(u_{1,n}\in \mathring{P}\).

For \(\sigma _{n}(t,-e_{1})\), \(t\in [0,+\infty )\), a similar argument to that of \(\sigma _{n}(t,e_{1})\) shows that there exists \(u_{2,n}\in \overline{S}_{r}\cap (-\mathring{P})\) such that \(u_{2,n}\) is a solution of Problem (2.5).

(3) We show that Problem (2.5) has at least one sign-changing solution \(u_{3,n}\in \overline{S}_{r}\cap (C_{0}^{1}-(P\cup (-P)))\).

From the proof of step 2, \(e_{1}\in \Lambda ^{+}\), \(-e_{1}\in \Lambda ^{-}\). Note that both \(\Lambda ^{+}\) and \(\lambda ^{-}\) are open sets of Λ since \(\sigma _{n}(t,u)\) depends continuously on u (see [32]). From Lemma 2.8, we have \(\Lambda ^{+}\cap \Lambda ^{-}=\emptyset \), and the connectedness of Λ implies that there is \(u_{0}\in \Lambda -(\Lambda ^{+}\cup \Lambda ^{-})\). By Lemma 2.7, \(\sigma _{n}(t,u_{0})\to u_{3,n}\), a critical point of \(F_{n}\), in \(H_{0}^{1}(\Omega )\) as \(t\to +\infty \). Then \(u_{3,n}\) is a solution of Problem (2.5) and \(u_{3,n}\in \overline{S}_{r}\). From (iii) of Lemma 2.7, we have that \(\lim_{t\to +\infty }\sigma _{n}(t,u_{0})=u_{3,n}\) in the \(C_{0}^{1}(\overline{\Omega })\)-topology. Therefore \(u_{3,n}\notin \mathring{P}\cap (-\mathring{P})\) since \(\sigma _{n}(t,u_{0})\notin \mathring{P}\cap (-\mathring{P})\). Then \(u_{3,n}\notin {P}\cap (-P)\) by using the strong maximum principle. Hence \(u_{3,n}\) changes its sign in Ω. Also \(\Psi _{n}(u_{3,n})<\Psi _{n}(\sigma _{n}(t,u_{0}))<-\alpha _{n}\), \(\forall t\in [0,+\infty )\) since \(\sigma _{n}\) is a negative descent flow. □

Proof

(Theorem 2.1) We only prove the existence of sign-changing solutions of Problem (1.1) since the proofs of the existence of positive solutions and negative solutions are similar, so we omit them.

From Lemma 2.9, for given \(r>0\), Problem (2.5) has at least one sign-changing solution \(u_{3,n}\) with \(u_{3,n}\in \overline{S}_{r}\cap (C_{0}^{1}-(P\cup (-P)))\), where \(\lambda ^{-1}_{3,n}=\int _{\Omega }g_{n}(x,u_{3,n})u_{n}\,dx/r^{2}>0\) and \(g_{n}\) is defined in (2.3) for each \(n\in \mathbb{N}\).

(1) We first prove that \(\{\lambda _{3,n}\}\) is bounded.

Since \(\{u_{3,n}\}\) is bounded in the \(H_{0}^{1}(\Omega )\) topology, we may assume that it converges weakly to \(u^{*}\) in \(H_{0}^{1}(\Omega )\). Then \(u_{3,n}\to u^{*}\) in \(L^{p+1}(\Omega )\) since \(1\leq p<\frac{N+2}{N-2}\). There exists a number \(c>0\) such that, for all \(t\in \mathbb{R}\) and for all \(n=1, 2, \ldots \) ,

$$ \bigl\vert g_{n}(x,t) \bigr\vert \leq c \bigl(1+ \vert t \vert ^{p}\bigr) \quad \text{and}\quad \bigl\vert G_{n}(x,t) \bigr\vert \leq c\bigl(1+ \vert t \vert ^{p+1}\bigr), $$
(2.14)

where \(G_{n}\) is defined in (2.4).

From Lemma A.1 in [41], there exists a subsequence of \(\{u_{3,n}\}\), denoted also by \(\{u_{3,n}\}\), and there exists \(h\in L^{p+1}(\Omega )\) such that \(u_{n}\to u^{*}\) a.e. in Ω, \(|u^{*}(x)|\leq h(x)\), \(|u_{3,n}(x)|\leq h(x)\) a.e. in Ω. From the Lebesgue dominated convergence theorem, we have

$$ \int _{\Omega }G_{n}(x,u_{3,n})\,dx\to \int _{\Omega }G\bigl(x, u^{*}\bigr)\,dx \quad \text{as }n\to +\infty . $$
(2.15)

Let \(n>k_{0}=\max \{\|u\|_{C(\overline{\Omega })}:u\in \Lambda \}\). By the definitions of \(\Psi _{n}\) and Ψ, if \(n>k_{0}\), we have \(\Psi _{n}(u)=\Psi (u)\) for \(u\in \Lambda \).

From (2.2), (2.13), there exists a positive constant \(\alpha >0\) such that

$$ -\Psi _{n}(u)= \int _{\Omega }G_{n}(x,u)\,dx=-\Psi (u)= \int _{\Omega }G(x,u)\,dx> \alpha >0 \quad \text{for }u\in \Lambda , n>k_{0}. $$

Since \(u_{3,n}\) is obtained along a descending flow, it follows that

$$ \int _{\Omega }G_{n}(x,u_{3,n})\,dx\geq \int _{\Omega }G_{n}\bigl(x,\sigma (0,u_{0,n})\bigr)\,dx= \int _{\Omega }G_{n}(x,u_{0,n})\,dx>\alpha >0,\quad \forall n>k_{0}, $$

for some \(u_{0,n}\in \Lambda \), where \(\sigma (t,u_{0,n})\) is a solution of Problem (2.5). From (2.15), we have \(\int _{\Omega }G(x,u^{*})\,dx>0\). Hence \(u^{*}\not \equiv 0\). Similar to (2.15), we have

$$ \int _{\Omega }g_{n}(x,u_{3,n})u_{3,n}\,dx \to \int _{\Omega }g\bigl(x,u^{*}\bigr)u^{*}\,dx \stackrel{\bigtriangleup }{=} 2\beta >0. $$

Thus \(0<\lambda _{3,n}= \frac{r^{2}}{\int _{\Omega }g_{n}(x,u_{3,n})u_{n}\,dx/r^{2}}<r^{2}/ \beta \) for n large enough.

(2) We prove that \(\{u_{3,n}\}\) is bounded in the \(C_{0}^{1}\)-topology.

Choose a sequence of numbers \(\{q_{i}\}\) satisfying

$$ q_{1}=\frac{2N}{N-2}< q_{2}< \cdots < q_{m-1}< q_{m}=2Np $$

and

$$ \frac{1}{q_{i+1}}\geq \frac{p}{q_{i}}-\frac{2}{N}, \quad i=1,2,\ldots , m-1. $$

Let \(p_{i}=q_{i}/p\) (\(i=1, 2, \ldots , m\)). From Lemma 2.4, we have

$$ H_{0}^{1}\hookrightarrow L^{q_{1}}(\Omega ). $$
(2.16)

From (2.14), one has (note \(p_{1}\) \(p=q_{1}\))

$$ \bigl\Vert \lambda _{3,n}\mathbb{G}_{n}(u_{3,n}) \bigr\Vert _{L^{p_{1}}}\leq \frac{1}{\beta }\biggl( \int _{\Omega } \bigl\vert g_{n}(x,u_{3,n}) \bigr\vert ^{p_{1}}\,dx\biggr)^{ \frac{1}{p_{1}}}\leq \frac{2c}{\beta } \bigl( \vert \Omega \vert ^{\frac{1}{p_{1}}}+ \Vert u_{3,n} \Vert _{L^{q_{1}}}^{p}\bigr)$$
(2.17)

for large n. Since K is a bounded linear operator, one has together with Lemma 2.3

$$ L^{p_{1}}(\Omega ) \stackrel{K}{\rightarrow }W^{2,p_{1}}(\Omega )\cap W_{0}^{1,p_{1}}( \Omega ) \hookrightarrow L^{q_{2}}(\Omega ). $$
(2.18)

Combining (2.16), (2.17), and (2.18), we have

$$ \{u_{3,n}\}_{n\geq k}\subseteq L^{q_{2}}(\Omega ) \text{ is bounded}, \quad \text{ for large } k. $$
(2.19)

Repeating the progress of (2.17), (2.18), and (2.19) for \(i=2, 3, \ldots , m\), we have

$$ \{u_{3,n}\}_{n\geq k}\subseteq L^{q_{m}}(\Omega ) \text{ is bounded}, \quad \text{for large } k. $$

We have (note \(p_{m}\) \(p=q_{m}\))

$$ \bigl\Vert \lambda _{3,n}\mathbb{G}_{n}(u_{3,n}) \bigr\Vert _{L^{p_{m}}}\leq \frac{1}{\beta }\biggl( \int _{\Omega } \bigl\vert g_{n}(x,u_{3,n}) \bigr\vert ^{p_{m}}\,dx\biggr)^{ \frac{1}{p_{m}}}\leq \frac{2c}{\beta } \bigl( \vert \Omega \vert ^{\frac{1}{p_{m}}}+ \Vert u_{3,n} \Vert _{L^{q_{m}}}^{p}\bigr)$$

for large n, which together with boundedness of the linear operator K guarantees that

$$ \{u_{3,n}\}_{n\geq k}\subseteq W^{2,p_{m}}\cap W_{0}^{1,p_{m}} \text{ is bounded}, \quad \text{for large } k.$$

Now Lemma 2.2 implies (note \(p_{m}=2N>N\)) that

$$ \{u_{3,n}\}_{n\geq k}\subseteq W^{2,p_{m}}\cap W_{0}^{1,p_{m}} \hookrightarrow C_{0}^{1}(\overline{\Omega }) \text{ is bounded},\quad \text{for large } k.$$
(2.20)

(3) We consider sign-changing solutions of Problem (1.1) in \(S_{r}\).

From (2.20), set \(L>0\) such that

$$ \Vert u_{3,n} \Vert _{C_{0}^{1}(\overline{\Omega })}\leq L, \quad n\in \{1,2, \ldots \}. $$
(2.21)

Choose \(n_{0}>L\). From the definitions of \(f_{n_{0}}\) and \(g_{n_{0}}\) in (2.2) and (2.3), we have together with (2.21) that

$$ f_{n_{0}}\bigl(x,u_{3,n_{0}}(x)\bigr)=f\bigl(x,u_{3,n_{0}}(x) \bigr),\quad x\in \overline{\Omega } $$

and

$$ g_{n_{0}}\bigl(x,u_{3,n_{0}}(x)\bigr)=\frac{1}{M(x,r^{2})}f \bigl(x,u_{3,n_{0}}(x)\bigr), \quad x\in \overline{\Omega }, $$

which implies that \(u_{3,n_{0}}\) is a sign-changing solution of Problem (2.1) with

$$ \lambda =r^{2}\Bigm/ \int _{\Omega }g\bigl(x,u_{3,n_{0}}(x)\bigr)u_{3,n_{0}}(x)\,dx. $$

For \(r>0\) given above, write \(u_{3,r}(x)=u_{3,n_{0}}(x)\) for \(x\in \overline{\Omega }\). Lemma 2.1 guarantees that \(u_{3,r}\) is a sign-changing solution of Problem (1.1) with

$$ \lambda =r^{2}\Bigm/ \int _{\Omega }g\bigl(x,u_{3,r}(x)\bigr)u_{3,r}(x)\,dx. $$

Similarly, we obtain a set \(\{u_{1,r}\}_{r\in A}\) of positive solutions of Problem (1.1) and a set \(\{u_{2,r}\}_{r \in A}\) of negative solutions of Problem (1.1).

The proof is completed. □

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Acknowledgements

We thank the referees for their valuable suggestions. The work is supported by the NSFC of China (62073203) and the Fund of Natural Science of Shandong Province (ZR2018MA022).

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Correspondence to Baoqiang Yan.

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Yan, B., O’Regan, D. & Agarwal, R.P. Infinite number of solutions for some elliptic eigenvalue problems of Kirchhoff-type with non-homogeneous material. Bound Value Probl 2021, 44 (2021). https://doi.org/10.1186/s13661-021-01522-9

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MSC

  • 35J60
  • 35B32
  • 35J25
  • 35J62

Keywords

  • Kirchhoff equations
  • Sign-changing solution
  • Variational method