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.

1 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, 33–40] 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].

2 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 P̊ be the interior set of P. The elements of P̊ are called positive and the elements of −P̊ 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. □