Bifurcation from infinity and multiplicity results for an elliptic system from biology

Abstract

This article is concerned with the bifurcation from infinity of the following elliptic system arising from biology $\begin{array}{l} - κ Δ u = λ u + f (x, u) - v, \\ - Δ v = u - v, \end{array}$ in a bounded domain $Ω \subset R^{N}$ . We regard this problem as a stationary problem of some reaction-diffusion system. By using a method of a pure dynamical nature, we will establish some multiplicity results on bifurcations from infinity for this system under an appropriate Landesman-Lazer type condition.

Keywords

Conley index bifurcation from infinity elliptic equation parabolic equation

1. Introduction

This article is concerned with the following semilinear elliptic equations $\begin{array}{l} (1.1) & \{\begin{matrix} - κ Δ u = λ u + f (x, u) - v, \\ - Δ v = u - v, \end{matrix} \end{array}$ in Ω, associated with the Dirichlet boundary condition $\begin{array}{l} (1.2) & u (x, t) = v (x, t) = 0, x \in \partial Ω, \end{array}$ where $Ω \subset R^{N}$ ( $N ⩾ 1$ ) is a bounded domain with smooth boundary $\partial Ω$ , $λ \in R$ is the bifurcation parameter, $κ > 0$ and f is locally Lipschitz continuous, and satisfies the Landesman-Lazer type condition: $\begin{matrix} (1.3) & \underset{s \to + \infty}{lim inf} f (x, s) ⩾ \overline{f} > 0, \underset{s \to - \infty}{lim sup} f (x, s) ⩽ - \underline{f} < 0 \end{matrix}$ uniformly with respect to $x \in \bar{Ω}$ . We mainly investigate the bifurcation from infinity and multiplicity of solutions of this elliptic system.

It is well-known that the problem (1.1) can be regarded as a stationary one of a system of some reaction-diffusion equation from biology [29]. This reaction-diffusion equation aims to explain the formation of spatial patterns in biological systems; see [24,28,30] for details. In the past decades the investigation of system (1.1) has attracted much attention; see e.g., [3,7,11,12,29] and the references therein. Especially, by using variational methods, Rothe [29] established some global existence results of branches of solutions to (1.1). Later, De Figueiredo and Mitidieri [7] investigated the system (1.1) for $λ = 0$ . Using monotone iteration techniques and variational methods, the authors gave the existence of pairs (one pair or even two pairs) of solutions to (1.1) under different conditions on f. Furthermore, by applying topological degree, Chiappinelli and De Figueiredo [3] studied the multiplicity of solutions for system (1.1). Precisely, let ${\hat{λ}}_{1}$ be the first eigenvalue of the corresponding linear system. They proved that for λ near ${\hat{λ}}_{1}$ , if $λ ⩾ {\hat{λ}}_{1}$ , then (1.1) has at least one pair of solutions, and if $λ < {\hat{λ}}_{1}$ , then (1.1) has at least two distinct pair of solutions, one such pair are positive in Ω.

The bifurcation and multiple solutions for elliptic equations near resonance has been widely studied by researchers. For example, Mawhin and Schmitt [20] first considered the following elliptic system $\begin{matrix} (1.4) & \{\begin{matrix} - Δ u = λ u + f (x, u), & x \in Ω, \\ u (x) = 0, & x \in \partial Ω . \end{matrix} \end{matrix}$ Specifically, assume $λ_{k}$ is an isolated eigenvalue of odd multiplicity. By using the degree theory, the authors verified that if f satisfies some appropriate Landesman-Lazer type condition, there is $ϵ > 0$ such that for $λ \in [λ_{k}, λ_{k} + ϵ]$ , equation (1.4) has at least one solution, and at least two distinct solutions for $λ \in [λ_{k} - ϵ, λ_{k})$ . Motivated by this work, Chiappinelli, Mawhin and Nugari [4] investigated the bifurcation from infinity and multiple solutions for problem (1.4) near the first eigenvalue $λ_{1}$ . We mention that in [4], the nonlinearity f can be unbounded. Later, under some conditions on f, Paiva and Massa [8] further showed that (1.4) has at least two solutions for λ near the eigenvalue $λ_{k}$ ( $k ⩾ 2$ ) without the condition that $λ_{k}$ is of odd multiplicity. Recently, by utilizing some dynamical arguments, Li, Li and Zhang [14] gave some results on the bifurcation and multiplicity of solutions for (1.4). Concerning this topic for elliptic equations under various boundary conditions, the interested reader is referred to [2,9,13,15,18,21,26,27,33–35,38], etc.

Inspired by these works mentioned above, in this article, we further study the bifurcation from infinity and multiplicity of solutions for (1.1) by using a method of a pure dynamical nature. Note that for each fixed u, the second equation in (1.1) can be uniquely solved. Specifically, for each fixed u, the problem $\begin{matrix} (1.5) & \{\begin{matrix} - Δ v + v = u, & x \in Ω, \\ v = 0, & x \in \partial Ω \end{matrix} \end{matrix}$ has a unique solution v in terms of u. Denote by $B u$ the solution of (1.5). Then we can rewrite the system (1.1) as the following equivalent system $\begin{matrix} (1.6) & \{\begin{matrix} - κ Δ u + B u = λ u + f (x, u), & x \in Ω, \\ u = 0, & x \in \partial Ω, \end{matrix} \end{matrix}$ which can be regarded as a stationary one of the following dynamic equation $\begin{matrix} (1.7) & \{\begin{matrix} u_{t} - Δ u + B u = λ u + f (x, u), & x \in Ω, \\ u = 0, & x \in \partial Ω . \end{matrix} \end{matrix}$ By giving some detailed descriptions on the dynamical behaviors of (1.7), we establish some results on bifurcations from infinity and multiplicity of solutions for the elliptic system (1.1). Precisely, let $μ_{k}$ be an isolated eigenvalue of $D = - Δ + B$ . It will be shown that there exists $ε > 0$ such that for every $λ \in I_{-} = [μ_{k} - ε, μ_{k})$ , the elliptic system (1.1) has at least two distinct pairs of solutions $(u_{λ}^{1}, v_{λ}^{1})$ and $(u_{λ}^{\infty}, v_{λ}^{\infty})$ satisfying $\begin{matrix} (1.8) & lim_{λ \to μ_{k}^{-}} {({‖ u_{λ}^{\infty} ‖}^{2} + {‖ v_{λ}^{\infty} ‖}^{2})}^{\frac{1}{2}} = \infty, \end{matrix}$ whereas $(u_{λ}^{1}, v_{λ}^{1})$ remains bounded with respect to $λ \in I_{-}$ ;

Finally, we pay our attention to a particular but important case where $\begin{matrix} f (x, s) = o (| s |) as | s | \to 0 uniformly w.r.t. x \in \overline{Ω} . \end{matrix}$ In this case, we shall present some more detailed information on the bifurcation and multiplicity of solutions for the elliptic system (1.1) near the isolated eigenvalue $μ_{k}$ . Roughly speaking, when f satisfies the Landesman-Lazer type condition (1.3), we prove that there exists at least one-sided neighborhood $I_{1}$ of $μ_{k}$ such that for each $λ \in I_{1} ∖ {μ_{k}}$ , (1.1) has at least three distinct pairs of solutions. Furthermore, there is an open dense subset J of $R$ so that (1.1) has at least four distinct pairs of solutions for every $λ \in I_{-} \cap J$ , where $I_{-} = [μ_{k} - ε, μ_{k})$ , which partially improves the results in [14, Theorem 5.10].

We remark that our approach in studying the problem (1.1) is of pure dynamical systems, which is different from those in the literature. From the point of view of dynamical systems and by developing some techniques in [3,7,14,29], we can obtain some more clear descriptions on the dynamical behaviors of the dynamic equation (1.7) near resonance. We also mention that, if, instead of (1.3), we assume that $\begin{matrix} (1.9) & \underset{s \to + \infty}{lim sup} f (x, s) ⩽ - \overline{f} < 0, \underset{s \to - \infty}{lim inf} f (x, s) ⩾ \underline{f} > 0 \end{matrix}$ uniformly for $x \in Ω$ , then “dual” versions of all our results hold true.

This article is organized as follows. Section 2 is devoted to making some preliminaries. In Section 3 we present some detailed descriptions on the dynamical behavior of (1.7), and establish some theorems on bifurcations from infinity for the dynamic equation (1.7). In Section 4, we discuss the bifurcation from infinity and multiplicity of solutions of the elliptic system (1.1).

2. Preliminaries

In this section we are concerned with some preliminaries.

2.1. Local semiflows

In this subsection we briefly recall some basic concepts on dynamical systems.

A local semiflow Φ on a complete metric space X is a continuous mapping from an open set $D (Φ) \subset R_{+} \times X$ to X and it fulfills the following conditions.

For any $x \in X$ , there exists $0 < T_{x} ⩽ \infty$ with $\begin{matrix} (t, x) \in D (Φ) ⟺ t \in [0, T_{x}); \end{matrix}$

$Φ (0, \cdot) = {id}_{X}$ , and for every $x \in X$ and $t, s \in R_{+}$ with $t + s ⩽ T_{x}$ , then $\begin{matrix} Φ (t + s, x) = Φ (t, Φ (s, x)) . \end{matrix}$

For convenice, we rewrite

Φ (t, x)

Φ (t) x

A continuous mapping γ on an interval $J \subset R$ to X is called a trajectory (or solution) of Φ, if $\begin{matrix} γ (t) = Φ (t - s) γ (s), \forall t, s \in J, t ⩾ s . \end{matrix}$ We call the trajectory γ on $R$ a full trajectory. The orbit of a full trajectory γ is the following set $\begin{matrix} orb (γ) = {γ (t) : t \in R}, \end{matrix}$ which is called a full orbit.

Let γ be a full trajectory. Then the ω-limit set $ω (γ)$ and $ω^{*}$ -limit set $ω^{*} (γ)$ are defined, respectively, by $\begin{array}{l} ω (γ) = {y \in X : there is t_{n} \to \infty such that γ (t_{n}) \to y}, \\ ω^{*} (γ) = {y \in X : there is t_{n} \to - \infty such that γ (t_{n}) \to y} . \end{array}$

We say that Φ does not explode in a set $N \subset X$ , if $Φ ([0, T_{x})) x \subset N$ implies $T_{x} = \infty$ .

Definition 2.1 ([31]).

A set $N \subset X$ is called admissible, if for any sequences $x_{n} \in N$ and $t_{n} \to \infty$ satisfying $Φ ([0, t_{n}]) x_{n} \subset N$ for all n, the sequence $Φ (t_{n}) x_{n}$ has a convergent subsequence. N is called strongly admissible, if it is admissible, and Φ does not explode in N.

Definition 2.2.
Φ is called asymptotically compact on X, if every bounded set $B \subset X$ is strongly admissible.

A set $S \subset X$ is called positively invariant (resp.,invariant), if it satisfies $Φ (t) S \subset S$ (resp., $Φ (t) S = S$ ) for all $t ⩾ 0$ .

A compact invariant set $A \subset X$ is called an attractor of Φ, if it attracts a neighborhood V of itself, that is, $\begin{matrix} lim_{t \to \infty} d_{H} (Φ (t) V, A) = 0 . \end{matrix}$
2.2. Conley index

In this subsection, we collect some basic concepts on the Conley index; see [5,23,31,36], etc., for details.

Now we always assume that the local semiflow Φ on X is asymptotically compact.

A compact invariant set S is called isolated, if there exists a neighborhood N of S so that S is the maximal compact invariant set of Φ in $\overline{N}$ . Correspondingly, N is called an isolating neighborhood of S.

Assume that B is a bounded closed subset of X. $x \in \partial B$ is called a strict egress (resp., strict ingress, bounce-off) point of B, if for each trajectory $γ : [- τ, s] \to X$ satisfying $γ (0) = x$ , where $τ ⩾ 0$ and $s > 0$ , then the following conditions hold:

(1) there is $ε \in (0, s)$ such that $\begin{matrix} γ (t) \notin B (resp., γ (t) \in int B, resp., γ (t) \notin B), \forall t \in (0, ε); \end{matrix}$

(2) if $τ > 0$ , there exists a number $δ \in (0, τ)$ such that $\begin{matrix} γ (t) \in int B (resp., γ (t) \notin B, resp., γ (t) \notin B), \forall t \in (- δ, 0) . \end{matrix}$ Let $B^{i}$ (resp., $B^{e}$ , $B^{b}$ ) denote the set of all strict ingress (resp., strict egress, bounce-off) points of the closed set B, and write $B^{-} = B^{e} \cup B^{b}$ .

$B \subset X$ is called an isolating block [31] if $B^{-}$ is closed and $\partial B = B^{i} \cup B^{-}$ .

Let N, E be two closed subsets of X. We say that E is an exit set of N, if it satisfies the following properties:

E is N-positively invariant, specifically, for each $x \in E$ and $t ⩾ 0$ , $\begin{matrix} Φ ([0, t]) x \subset N ⟹ Φ ([0, t]) x \subset E; \end{matrix}$

for every $x \in N$ , if there is $t_{1} > 0$ satisfying $Φ (t_{1}) x \notin N$ , then there exists $t_{0} \in [0, t_{1}]$ so that $Φ (t_{0}) \in E$ .

Assume that S is a compact isolated invariant set of Φ. A pair of bounded closed subsets $(N, E)$ is said to be an index pair of S, if (1) $N ∖ E$ is an isolating neighborhood of S; (2) E is an exit set of N. By virtue of [31], we deduce that if B is a bounded isolating block, then $(B, B^{-})$ is an index pair of the maximal compact invariant set S of Φ in B.

Definition 2.3.
The homotopy Conley index of S, denoted by $h (Φ, S)$ is defined to be the homotopy type $[(N / E, [E])]$ of the pointed space $(N / E, [E])$ for every index pair $(N, E)$ of S.

Let S be a compact isolated invariant set of Φ. By $H_{}$ and $H^{}$ we denote the singular homology and cohomology theories with the coefficient group $Z$ , respectively. Applying $H_{}$ and $H^{}$ to $h (Φ, S)$ , we can obtain the homology and cohomology Conley indices of S.

The Poincaré polynomial of S, denoted by $p (t, S)$ is defined to be the formal polynomial $\begin{matrix} p (t, S) = \sum_{q = 0}^{\infty} β_{q} t^{q}, \end{matrix}$ where $β_{q} = rank H_{q} (h (Φ, S))$ . If S has a Morse decomposition $M = {M_{1}, \dots, M_{l}}$ , the following Morse equation $\begin{matrix} p (t, M_{1}) + \dots + p (t, M_{l}) = p (t, S) + (1 + t) Q (t) \end{matrix}$ holds for some formal polynomial $Q (t) = \sum_{q = 0}^{\infty} d_{q} t^{q}$ with $d_{q} \in Z_{+}$ .
3. Dynamic bifurcation from infinity for system (1.7)

In this section we first give some results on bifurcations from infinity for the dynamic system (1.7). For this purpose, we always assume that the function f in (1.7) satisfies

(A) f is bounded, and locally Lipschitz continuous in u on $Ω \times H_{0}^{1} (Ω)$ uniformly with respect to $x \in Ω$ .

3.1. Mathematical setting

Set $X = L^{2} (Ω)$ and $Y = H_{0}^{1} (Ω)$ . For each fixed $u \in X$ , let $B u$ be the unique solution of (1.5). It is clear to see that the operator B can be expressed as $\begin{matrix} B = {(- Δ + 1)}^{- 1}, \end{matrix}$ associated with the homogeneous Dirichlet boundary condition. We observe from [3,12] that B is positive, self-adjoint and compact in X. Denote the usual inner product and norm on X by $⟨ \cdot, \cdot ⟩$ and $| \cdot |$ , respectively. Define the norm $‖ \cdot ‖$ on Y by $\begin{matrix} ‖ u ‖ = {(\int_{Ω} | \nabla u |^{2} d x + \int_{Ω} (B u) u d x)}^{1 / 2}, u \in Y . \end{matrix}$ Then one can easily see that the above norm $‖ \cdot ‖$ on Y is well-defined.

Let $D = - κ Δ + B$ associated with the homogeneous Dirichlet boundary condition. Then we see from [7] that the operator D is selfadjoint and has a compact resolvent. Denote by $σ (D)$ and $μ_{k}$ the spectrum and the eigenvalue of D, respectively. Then $\begin{matrix} μ_{k} = κ λ_{k} + \frac{1}{1 + λ_{k}}, k = 1, 2, \dots, \end{matrix}$ where $λ_{k} > 0 (k = 1, 2, \dots)$ is the eigenvalue of $- Δ$ under the homogeneous Dirichlet boundary condition.

Define the Nemitski operator $\tilde{f} (u)$ from Y to X as $\begin{matrix} \tilde{f} (u) (x) = f (x, u), u \in Y . \end{matrix}$ Then the dynamic equation (1.7) can be expressed as the following abstract system on Y: $\begin{matrix} (3.1) & u_{t} + D u = λ u + \tilde{f} (u) . \end{matrix}$ It is trivial to see that D is a sectorial operator. Moreover, by the assumption on f, we deduce that $\tilde{f} (u)$ is locally Lipschitz continuous. Thus by virtue of the basic theory on evolution equations (see e.g., [10]), the Cauchy problem of (3.1) is well-posed on Y. Precisely, for every $u_{0} \in Y$ , the equation (3.1) has a unique (local) solution $u (t)$ in Y. Let $Φ_{λ}$ denote the semiflow generated by (3.1). Then one can deduce from the boundedness of f that $Φ_{λ}$ is actually a global semiflow in Y.

Assume that $μ_{k}$ is an isolated eigenvalue of the operator D. Take two real numbers $α_{1}$ and $α_{2}$ such that the spectrum $σ (D)$ of D has a decomposition $σ (D) = σ^{-} \cup σ^{0} \cup σ^{+}$ with $\begin{matrix} σ^{-} = σ (D) \cap {Re λ ⩽ α_{1}}, σ^{0} = {μ_{k}}, σ^{+} = σ (D) \cap {Re λ ⩾ α_{2}} . \end{matrix}$ Clearly, $Re σ (D) \cap (α_{1}, α_{2}) = {μ_{k}}$ . Thus the space X can be decomposed into the orthogonal direct sum of its subspaces $X^{-}$ , $X^{0}$ , $X^{+}$ . Noth that $X^{-}$ , $X^{0}$ are finite-dimensional. Denote by $P^{i} (i \in {-, 0, +})$ the projection from X to $X^{i}$ . Set $\begin{matrix} Y^{i} = Y \cap X^{i}, i \in {-, 0, +} . \end{matrix}$ It is trivial to see that $Y^{-}$ and $Y^{0}$ coincide with $X^{-}$ and $X^{0}$ , respectively.

Definition 3.1.
We say that the system (3.1) bifurcates from infinity at $λ = μ_{k}$ (or, $(\infty, μ_{k})$ is a bifurcation point), if there exist a sequence $λ_{n}$ with $λ_{n} \to μ_{k}$ ( $n \to \infty$ ) and a bounded full solution $u_{n} = u_{λ_{n}} (t)$ of (3.1) satisfying $\begin{matrix} ‖ u_{n} ‖_{\infty} \to \infty, as n \to \infty, \end{matrix}$ where $‖ u_{λ} ‖_{\infty} = {sup}_{t \in R} ‖ u_{λ} (t) ‖$ .

Given a set $N \subset Y$ . In the following we use $S_{\infty} (Φ_{λ}, N)$ to denote the union of all bounded full orbits of $Φ_{λ}$ in N. Let $J \subset R$ be an interval. Set $\begin{matrix} C (J, N) = \overline{⋃_{λ \in J} (S_{\infty} (Φ_{λ}, N) \times {λ})} . \end{matrix}$ If $N = Y$ , we simply write $S_{\infty} (Φ_{λ}, Y) = S_{\infty} (Φ_{λ})$ and $C (J, Y) = C (J)$ .
Theorem 3.1.
Let $μ_{k} \in σ (D)$ . Assume f satisfies the condition (A). Then $(\infty, μ_{k})$ is a bifurcation point of ( 3.1 ). Specifically, there are a sequence $λ_{n}$ with $λ_{n} \to μ_{k}$ ( $n \to \infty$ ) and a bounded full solution $u_{n} = u_{λ_{n}} (t)$ of ( 3.1 ) so that $\begin{matrix} ‖ u_{n} ‖_{\infty} \to \infty, as n \to \infty . \end{matrix}$
Proof.
Let $μ_{k} \in σ (D)$ . Pick two numbers a, b such that $\begin{matrix} μ_{k - 1} < a < μ_{k} < b < μ_{k + 1} . \end{matrix}$ We first consider the following linear equation $\begin{matrix} (3.2) & \frac{d u}{d t} + D u - λ u = 0 . \end{matrix}$ Denote $ϕ_{λ}$ the semiflow in Y generated by equation (3.2). It is trivial to see that if $λ = a, b$ , then the set ${0}$ is an isolated invariant set for $ϕ_{λ}$ . By [31] (see Chap. I, Corollary 11.2) we know that there exist two nonnegative integers p and q with $q - p > 0$ such that $\begin{matrix} (3.3) & h (ϕ_{a}, {0}) = Σ^{p}, h (ϕ_{b}, {0}) = Σ^{q} . \end{matrix}$ (Actually, $q - p$ is the algebraic multiplicity of the eigenvalue $μ_{k}$ of D.)

Next we study the nonlinear equation $\begin{matrix} (3.4) & \frac{d u}{d t} + D u - λ u - ν \tilde{f} (u) = 0, \end{matrix}$ where $ν \in [0, 1]$ is the homotopy parameter. Note that if $ν = 0, 1$ , then the equation (3.4) coincides with (3.2) and (3.1), respectively. Now we claim that for every fixed $ε > 0$ sufficiently small with $\begin{matrix} a < μ_{k} - ε < μ_{k} + ε < b, \end{matrix}$ there is $R_{ε} > 0$ such that if $u_{λ} (t)$ is a bounded full solution of (3.4) for $λ \in [a, μ_{k} - ε] \cup [μ_{k} + ε, b]$ and $ν \in [0, 1]$ , then $\begin{matrix} (3.5) & ‖ u_{λ} ‖_{\infty} < R_{ε} . \end{matrix}$

The techniques involved in the argument are borrowed from [31] (see also the proof of Theorem 3.2 in [37]). We argue by contradiction and suppose the contrary. Then there would exist sequences $λ_{n} \in [a, μ_{k} - ε] \cup [μ_{k} + ε, b]$ and $ν_{n} \in [0, 1]$ such that for each n, the equation $\begin{matrix} (3.6) & \frac{d u}{d t} + D u - λ_{n} u - ν_{n} \tilde{f} (u) = 0, \end{matrix}$ has a bounded full solution $u_{n} (t)$ with $\begin{matrix} ‖ u_{n} ‖_{\infty} \to \infty as n \to \infty . \end{matrix}$ It may be assumed $λ_{n} \to \overline{λ} \in [c, μ - ε] \cup [μ + ε, d]$ .

Set $v_{n} = u_{n} / ‖ u_{n} ‖_{\infty}$ . Then $‖ v_{n} ‖_{\infty} = 1$ for all n. Using some quite standard argument it can be shown that there is a subsequence of ${v_{n}}$ , still denoted by ${v_{n}}$ , such that ${v_{n}}$ converges uniformly on any compact interval of $R$ to a function $v = v (t)$ . Note that $v_{n}$ satisfies $\begin{matrix} (3.7) & \frac{d v_{n}}{d t} + D v_{n} - λ_{n} v_{n} - ν_{n} \tilde{f} (u_{n}) / ‖ u_{n} ‖_{\infty} = 0 . \end{matrix}$ Passing to the limit in (3.7), in view of the boundedness of f one finds that v is a nontrivial bounded full solution of the linear equation (3.2) with $λ = \overline{λ}$ . But this leads to a contradiction, as for $λ = \overline{λ}$ (3.2) has no bounded full solutions other than the trivial one.

Denote by $ϕ_{λ}^{ν}$ the semiflow generated by the equation (3.4) in Y. Thanks to the continuation property of the Conley index, we deduce from (3.3) and (3.5) that $\begin{matrix} (3.8) & \begin{matrix} h (Φ_{λ}, S_{\infty} (Φ_{λ})) & = h (ϕ_{λ}^{1}, S_{\infty} (ϕ_{λ}^{1})) \\ = h (ϕ_{λ}^{0}, S_{\infty} (ϕ_{λ}^{0})) = h (ϕ_{a}, {0}) = Σ^{p} \end{matrix} \end{matrix}$ for $λ \in [a, μ_{k} - ε]$ and $\begin{matrix} (3.9) & \begin{matrix} h (Φ_{λ}, S_{\infty} (Φ_{λ})) & = h (ϕ_{λ}^{1}, S_{\infty} (ϕ_{λ}^{1})) \\ = h (ϕ_{λ}^{0}, S_{\infty} (ϕ_{λ}^{0})) = h (ϕ_{b}, {0}) = Σ^{q} \end{matrix} \end{matrix}$ for $λ \in [μ_{k} + ε, b]$ . Thus by [14, Theorem 3.5], $C ([a, b])$ has an unbounded connected component Υ. Observing that for the $ε > 0$ given in (3.5), we have $\begin{matrix} Υ [λ] \subset B_{Y} (R_{ε}), λ \in [a, μ_{k} - ε] \cup [μ_{k} + ε, b], \end{matrix}$ where $B_{Y} (R_{ε})$ denotes a ball in Y with center 0 and radius $R_{ε}$ . Therefore one can see that the result of the theorem holds true. □

Note that the number $ε > 0$ chosen in the proof of Theorem 3.1 may be arbitrary small. Since the number $a \in (μ_{k - 1}, μ_{k})$ and $b \in (μ_{k}, μ_{k + 1})$ , we conclude from (3.8) and (3.9) that the following basic result holds. Proposition 3.1.
Let $μ_{k} \in σ (D)$ . Assume f satisfies the condition (A). Then $\begin{matrix} (3.10) & h (Φ_{λ}, S_{\infty} (Φ_{λ})) = \{\begin{matrix} Σ^{p}, & λ \in (μ_{k - 1}, μ_{k}); \\ Σ^{q}, & λ \in (μ_{k}, μ_{k + 1}) . \end{matrix} \end{matrix}$

3.2. Basic lemmas

In this subsection we give some detailed estimates on the dynamical behavior of the equation (3.1). To this end we further make the following assumption on f.

(L) The function f satisfies the Landesman-Lazer type condition (1.3).

Let h be a function on Ω. We use $h_{\pm}$ to denote the positive and negative parts of h, respectively. That is, $\begin{matrix} h_{\pm} = max {\pm h (x), 0}, x \in Ω . \end{matrix}$ Clearly, $h = h_{+} - h_{-}$ .

Lemma 3.1 ([14,16]).

Assume the condition (L) holds. Then for each fixed $R > 0$ , $ε > 0$ , there exists $s_{0} > 0$ such that if $s ⩾ s_{0}$ , $\begin{matrix} \int_{Ω} f (x, z + s v) v d x ⩾ \int_{Ω} (\bar{f} v_{+} + \underline{f} v_{-}) d x - ε . \end{matrix}$ for all $z \in {\bar{B}}_{X} (R)$ and $v \in {\bar{B}}_{X} (1)$ , where ${\bar{B}}_{X} (R)$ and ${\bar{B}}_{X} (1)$ denote the closed balls in X centered at 0 with radius R and 1, respectively.

Lemma 3.2.
Let $δ = (μ_{k + 1} - μ_{k}) / 2$ . Assume that $λ ⩽ μ_{k} + δ$ . Then there exists $r_{0} > 0$ (independent of λ) such that if $u = u (t)$ is a solution of ( 3.1 ) for $t ⩾ 0$ , then $\begin{matrix} {‖ u^{+} (t) ‖}^{2} ⩽ {‖ u_{0}^{+} ‖}^{2} e^{- δ t} + r_{0}^{2} (1 - e^{- δ t}), \forall t ⩾ 0, \end{matrix}$ where $u^{+} = P^{+} u$ and $P^{+}$ is the projection from X to $X^{+}$ .
Proof.
Taking the inner product of the system (3.1) with $D u^{+}$ in X gives that $\begin{array}{l} \frac{1}{2} \frac{d}{d t} {‖ u^{+} ‖}^{2} + {| D u^{+} |}^{2} & = λ {‖ u^{+} ‖}^{2} + (D u^{+}, \tilde{f} (u)) \\ ⩽ λ {‖ u^{+} ‖}^{2} + \frac{μ_{k + 1} - μ_{k}}{4 μ_{k + 1}} {| D u^{+} |}^{2} + \frac{μ_{k + 1}}{μ_{k + 1} - μ_{k}} C_{f}^{2}, \end{array}$ where $C_{f} > 0$ depends on the boundedness of f. Thereby $\begin{matrix} (3.11) & \frac{d}{d t} {‖ u^{+} ‖}^{2} + \frac{3 μ_{k + 1} + μ_{k}}{2 μ_{k + 1}} {| D u^{+} |}^{2} ⩽ 2 λ {‖ u^{+} ‖}^{2} + \frac{2 μ_{k + 1}}{μ_{k + 1} - μ_{k}} C_{f}^{2} . \end{matrix}$ Since $| D u^{+} |^{2} ⩾ μ_{k + 1} ‖ u^{+} ‖^{2}$ , we deduce from (3.11) that $\begin{matrix} (3.12) & \frac{d}{d t} {‖ u^{+} ‖}^{2} + (\frac{3 μ_{k + 1} + μ_{k}}{2} - 2 λ) {‖ u^{+} ‖}^{2} ⩽ \frac{2 μ_{k + 1}}{μ_{k + 1} - μ_{k}} C_{f}^{2} . \end{matrix}$ Note that $\begin{matrix} \frac{3 μ_{k + 1} + μ_{k}}{2} - 2 λ ⩾ \frac{3 μ_{k + 1} + μ_{k}}{2} - (μ_{k + 1} + μ_{k}) = δ . \end{matrix}$ Thus $\begin{matrix} (3.13) & \frac{d}{d t} {‖ u^{+} ‖}^{2} + δ {‖ u^{+} ‖}^{2} ⩽ \frac{2 μ_{k + 1}}{μ_{k + 1} - μ_{k}} C_{f}^{2} . \end{matrix}$ Applying the classical Gronwall inequality to (3.13), the results hold true. □

For each $r > 0$ , set $\begin{matrix} B_{r} = {u \in Y : ‖ P^{+} u ‖ ⩽ r} . \end{matrix}$ Then we infer from Lemma 3.2 that $\begin{matrix} (3.14) & S_{\infty} (Φ_{λ}) \subset B_{r_{0}} for all λ ⩽ μ_{k} + δ . \end{matrix}$ Moreover, if $r ⩾ r_{0}$ then $B_{r}$ is positively invariant for the semiflow $Φ_{λ}$ .

Let $Z = Y^{-} \oplus Y^{0}$ and $\begin{matrix} P_{Z} = P^{-} + P^{0} \end{matrix}$ be the projection from Y to Z. For each $0 ⩽ a < b ⩽ \infty$ and each $r > 0$ , denote $\begin{matrix} B_{r} [a, b] = {u \in B_{r} : a ⩽ | P_{Z} u | ⩽ b} . \end{matrix}$ Lemma 3.3.
Assume that δ and $r_{0}$ are the numbers given by Lemma 3.2 . Let $r > r_{0}$ . Then there are $R_{0}, c_{0} > 0$ such that
for every $λ \in [μ_{k}, μ_{k} + δ]$ and every solution $u (t)$ of ( 3.1 ) in $B_{r} [R_{0}, \infty]$ , we have $\begin{matrix} (3.15) & \frac{d}{d t} {| z (t) |}^{2} ⩾ c_{0} | z (t) | \end{matrix}$ where $z (t) = P_{Z} u (t)$ ; and

if $R > R_{0}$ , there exists $0 < θ ⩽ δ$ such that for each $λ \in [μ_{k} - θ, μ_{k})$ and each solution $u (t)$ of ( 3.1 ) in $B_{r} [R_{0}, R]$ , then the differential inequality ( 3.15 ) holds true.

Proof.
Assume $λ \in [μ_{k} - δ, μ_{k} + δ]$ . Let u be a solution of (3.1) in $B_{r}$ . Multiplying the equation (3.1) by $z = P_{Z} u$ and integrating over Ω, it yields $\begin{matrix} (3.16) & \frac{1}{2} \frac{d}{d t} | z |^{2} + ‖ z ‖^{2} = λ | z |^{2} + (\tilde{f} (u), z) . \end{matrix}$ Noticing that $‖ z ‖^{2} ⩽ μ_{k} | z |^{2}$ , we know that $\begin{matrix} (3.17) & \frac{d}{d t} | z |^{2} ⩾ 2 (λ - μ_{k}) | z |^{2} + 2 (\tilde{f} (u), z) . \end{matrix}$ In what follows we estimate the last term in (3.17). Since the norm $‖ \cdot ‖_{L^{1} (Ω)}$ of $L^{1} (Ω)$ is equivalent to that of $L^{2} (Ω)$ in Z, one easily deduces that $\begin{matrix} (3.18) & m : = min {‖ v ‖_{L^{1} (Ω)} : v \in Z, | v | = 1} > 0 . \end{matrix}$ Now we rewrite z as $z = s v$ , where $s = | z |$ . It is easy to see that $v \in \partial B_{X} (1)$ . Take a positive number η with $η ⩽ min {\overline{f}, \underline{f}}$ . Recalling that $‖ u^{+} ‖ ⩽ r$ , one can immediately conclude from Lemma 3.1 that there exists $s_{0} > 0$ (depending upon r, m, η) such that if $s ⩾ s_{0}$ , then $\begin{matrix} (3.19) & (f (x, u^{+} + s v), v) = \int_{Ω} f (x, u^{+} + s v) v d x ⩾ \int_{Ω} (\overline{f} v_{+} + \underline{f} v_{-}) d x - \frac{1}{2} m η . \end{matrix}$ Hence $\begin{matrix} (3.20) & (\tilde{f} (u), z) = s (f (x, u^{+} + s v), v) ⩾ s (\int_{Ω} (\overline{f} v_{+} + \underline{f} v_{-}) d x - \frac{1}{2} m η), \end{matrix}$ provided $s ⩾ s_{0}$ . By (3.18), we see that $\begin{array}{l} \int_{Ω} (\overline{f} v_{+} + \underline{f} v_{-}) d x - \frac{1}{2} m η ⩾ η \int_{Ω} | v | d x - \frac{1}{2} m η ⩾ \frac{1}{2} m η . \end{array}$ Then it follows from (3.20) that if $s ⩾ s_{0}$ , $\begin{matrix} (3.21) & (\tilde{f} (u), z) ⩾ \frac{1}{2} m η s = \frac{1}{2} m η | z | . \end{matrix}$ Combining (3.17) and (3.21), it yields that $\begin{matrix} (3.22) & \frac{d}{d t} {| z (t) |}^{2} ⩾ 2 (λ - μ_{k}) {| z (t) |}^{2} + m η | z (t) | \end{matrix}$ as long as $| z (t) | ⩾ s_{0}$ .

Now choose $R_{0} = s_{0}$ and $c_{0} = m η / 2$ . Then if $λ \in [μ_{k}, μ_{k} + δ]$ , one can see from (3.22) that $\begin{matrix} \frac{d}{d t} {| z (t) |}^{2} ⩾ m η | z (t) | > c_{0} | z (t) | \end{matrix}$ at every point t where $| z (t) | ⩾ R_{0}$ , which completes the proof of assertion (1).

Let $λ < μ_{k}$ . Assume $R > R_{0}$ . Pick a positive number θ such that $\begin{matrix} θ R^{2} ⩽ m η s_{0} / 4 . \end{matrix}$ Thus if $λ \in [μ_{k} - θ, μ_{k})$ and $u (t)$ is a solution of (3.1) in $B_{r} [R_{0}, R]$ , we conclude from (3.22) that $\begin{array}{l} \frac{d}{d t} {| z (t) |}^{2} ⩾ & - 2 | λ - μ_{k} | R^{2} + m η | z (t) | \\ (3.23) & ⩾ & (c_{0} | z (t) | - 2 θ R^{2}) + c_{0} | z (t) | ⩾ c_{0} | z (t) | . \end{array}$ Therefore assertion (2) holds true. □

3.3. Dynamic bifurcation from infinity

Now we will study the dynamic bifurcation from infinity of the equation (3.1). Let $Φ_{λ}$ be the semiflow associated with (3.1).

Proposition 3.2.
Let the assumptions (A) and (L) hold. Then $S_{\infty} (Φ_{λ})$ is uniformly bounded in Y with respect to $λ \in [μ_{k}, μ_{k} + δ]$ , and $\begin{matrix} (3.24) & h (Φ_{μ_{k}}, S_{\infty} (Φ_{μ_{k}})) = h (Φ_{λ}, S_{\infty} (Φ_{λ})) = Σ^{q} . \end{matrix}$
Proof.
Let $r > r_{0}$ , where $r_{0}$ is the number given by Lemma 3.2. Then we infer from Lemma 3.3 (1) that there are positive numbers $c_{0}$ and $R_{0}$ such that (3.15) holds true. Hence $\begin{matrix} (3.25) & S_{\infty} (Φ_{λ}) \subset B_{r} [0, R_{0}], λ \in [μ_{k}, μ_{k} + δ] . \end{matrix}$ By the continuation of Conley index and Proposition 3.1, one can conclude that $\begin{matrix} h (Φ_{μ_{k}}, S_{\infty} (Φ_{μ_{k}})) = h (Φ_{λ}, S_{\infty} (Φ_{λ})) = Σ^{q} . \end{matrix}$ □

Next we present a detailed description on the dynamic bifurcation of the equation (3.1) near the eigenvalue $μ_{k}$ . Theorem 3.2.
Let the assumptions (A) and (L) hold. Then for each $λ \in R$ , $S_{λ} : = S_{\infty} (Φ_{λ})$ is nonempty. Moreover, there is $ε > 0$ so that the following assertions hold true.
If $λ \in I_{-} : = [μ_{k} - ε, μ_{k})$ , then $S_{λ}$ has a Morse decomposition $M = {S_{λ}^{\infty}, S_{λ}^{1}}$ . Moreover, there exists one connecting trajectory γ between $S_{λ}^{\infty}$ and $S_{λ}^{1}$ .

The Morse set $S_{λ}^{\infty}$ satisfies $\begin{matrix} (3.26) & lim_{λ \to μ_{k}^{-}} min_{u \in S_{λ}^{\infty}} ‖ u ‖ = \infty, \end{matrix}$ while $S_{λ}^{1}$ remains bounded uniformly with respect to $λ \in I_{-}$ .

Proof.
Firstly, by virtue of Propositions 3.1 and 3.2, we find that the Conley index of $S_{λ}$ is nontrivial, from which it can be seen that $S_{λ} \neq \emptyset$ for $λ \in R$ .

(i) Secondly, we show the validity of assertion (1). Let $r_{0}$ and δ be the numbers given in Lemma 3.2. Then by (3.14), we see that $\begin{matrix} S_{λ} \subset B_{r_{0}}, λ ⩽ μ_{k} + δ . \end{matrix}$ Assume $R_{0}$ and $c_{0}$ are the numbers given in Lemma 3.3. Fix a number $r > r_{0}$ . Since $S_{μ_{k}}$ is isolated, one can pick an isolating neighborhood $N_{1}$ of $S_{μ_{k}}$ such that $\begin{matrix} (3.27) & B_{r} [0, R_{0}] \subset N_{1} . \end{matrix}$ Then there exists $ε > 0$ such that $N_{1}$ is also an isolating neighborhood of $Φ_{λ}$ for $λ \in I : = [μ_{k} - ε, μ_{k} + ε]$ . Denote $S_{λ}^{1} = S_{\infty} (Φ_{λ}, N_{1})$ . We have $\begin{matrix} (3.28) & h (Φ_{λ}, S_{λ}^{1}) \equiv const., λ \in I . \end{matrix}$ Thanks to Proposition 3.2 we conclude that $\begin{matrix} (3.29) & h (Φ_{λ}, S_{λ}^{1}) = h (Φ_{μ_{k}}, S_{μ_{k}}^{1}) = h (Φ_{μ_{k}}, S_{μ_{k}}) = Σ^{q}, λ \in I . \end{matrix}$ Note that $\begin{matrix} (3.30) & S_{λ}^{1} \subset N_{1} \cap B_{r} : = {\tilde{N}}_{1}, λ \in I . \end{matrix}$ Take a number $R_{1} > 2 R_{0}$ sufficiently large so that $\begin{matrix} (3.31) & {\tilde{N}}_{1} \subset B_{r} [0, R_{1} / 2], λ \in I . \end{matrix}$ Then by Lemma 3.3 (2), there exists $θ_{1} > 0$ such that for each $λ \in [μ_{k} - θ_{1}, μ_{k})$ , if $u (t)$ is a solution of (3.1) in $B_{r} [R_{0}, R_{1}]$ , then we have $\begin{matrix} (3.32) & \frac{d}{d t} {| z (t) |}^{2} ⩾ c_{0} | z (t) |, where z (t) = P_{Z} u (t) . \end{matrix}$ It can be assumed that $ε ⩽ θ_{1}$ . Then one can immediately conclude from (3.32) that if $λ \in I_{-} = [μ_{k} - ε, μ_{k})$ and $u (t)$ is a bounded full solution of (3.1) in $B_{r}$ with $u (t_{0}) \in B_{r} [R_{0}, R_{1}]$ for some $t_{0} \in R$ , then there exists $T > 0$ so that $\begin{matrix} (3.33) & u (t) \in B_{r} [0, R_{0}] (for t < - T) and u (t) \in B_{r} [R_{1}, \infty] (for t > T) . \end{matrix}$ Hence one can deduce from (3.30), (3.31) and (3.33) that $\begin{matrix} (3.34) & S_{λ}^{1} \subset B_{r} [0, R_{0}], λ \in I_{-} . \end{matrix}$ This along with (3.27) implies that $S_{λ}^{1}$ is the maximal compact invariant set of $Φ_{λ}$ in $B_{r} [0, R_{0}]$ .

Set $\begin{matrix} (3.35) & S_{λ}^{\infty} = S_{\infty} (Φ_{λ}, B_{r} [R_{1}, \infty]), λ \in I_{-} . \end{matrix}$ By the choice of r and (3.32), one can easily see that $U : = B_{r} [R_{1}, \infty]$ is an isolating neighborhood of $S_{λ}^{\infty}$ . Assume $λ \in I_{-} : = [μ_{k} - ε, μ_{k})$ . We claim that $\begin{matrix} M = {S_{λ}^{\infty}, S_{λ}^{1}} \end{matrix}$ forms a Morse decomposition of $S_{λ}$ , provided $S_{λ}^{\infty}$ is nonempty. In order to show our claim, we first prove that $S_{λ}^{\infty} \neq \emptyset$ for $λ \in I_{-}$ . Indeed, if $S_{λ_{1}}^{\infty} = \emptyset$ for some $λ_{1} \in I_{-}$ , then one can easily deduce from (3.33) that $S_{λ_{1}} = S_{λ_{1}}^{1}$ . Therefore by (3.10) we have $\begin{matrix} h (Φ_{λ_{1}}, S_{λ_{1}}^{1}) = h (Φ_{λ_{1}}, S_{λ_{1}}) = Σ^{p}, \end{matrix}$ which contradicts (3.29).

Now we verify our claim. Assume that $u (t)$ is a full solution in $S_{λ} ∖ (S_{λ}^{1} \cup S_{λ}^{\infty})$ for $λ \in I_{-}$ . Note that $S_{λ} \subset B_{r} [0, \infty]$ . Since $S_{λ}^{1}$ and $S_{λ}^{\infty}$ are the maximal compact invariant sets of $Φ_{λ}$ in $B_{r} [0, R_{0}]$ and $B_{r} [R_{1}, \infty]$ , respectively, we necessarily have $u (t_{0}) \in B_{r} [R_{0}, R_{1}]$ for some $t_{0} \in R$ . Thus (3.33) implies that $\begin{matrix} (3.36) & ω^{*} (u) \subset S_{λ}^{1}, ω (u) \subset S_{λ}^{\infty}, \end{matrix}$ which proves our claim.

In the following we verify the existence of the connecting trajectory between $S_{λ}^{1}$ and $S_{λ}^{\infty}$ for $λ \in I_{-}$ . We consider the Morse equation of M: $\begin{matrix} (3.37) & p (t, S_{λ}^{1}) + p (t, S_{λ}^{\infty}) = p (t, S_{λ}) + (1 + t) Q (t), \end{matrix}$ where $p (t, M_{λ}^{1})$ and $p (t, M_{λ}^{\infty})$ are Poincaré polynomials of $S_{λ}^{1}$ and $S_{λ}^{\infty}$ , respectively, and $Q (t)$ is a formal polynomial. Because $\begin{matrix} h (Φ_{λ}, S_{λ}^{1}) = Σ^{q}, h (Φ_{λ}, S_{λ}) = Σ^{p}, λ \in I_{-}, \end{matrix}$ we see that $\begin{matrix} p (t, S_{λ}^{1}) = t^{q}, p (t, S_{λ}) = t^{p} . \end{matrix}$ Thus the above equation (3.37) reads $\begin{matrix} (3.38) & t^{q} + p (t, S_{λ}^{\infty}) = t^{p} + (1 + t) Q (t), \end{matrix}$ from which it can be easily seen that $Q (t) \neq 0$ . Thanks to the basic knowledge in the Morse theory of invariant sets (see [31, Chap. III, Theorem 3.5]), we conclude that there exists at least one connecting trajectory between $S_{λ}^{\infty}$ and $S_{λ}^{1}$ . This completes the proof of assertion (1).

(ii) Finally, we prove that the assertion (2) holds true. It is trivial to see that $S_{λ}^{1}$ remains uniformly bounded with respect to $λ \in I$ .

Let $R_{1}$ be chosen as in (3.31). We infer from Lemma 3.3 (2) that for each $R > R_{1}$ , there exists $0 < θ < θ_{1}$ such that if $λ \in [μ_{k} - θ, μ_{k})$ and $u (t)$ is a solution of (3.1) in $B_{r} [R_{0}, R]$ , then the differential inequality (3.32) holds true. Thus we conclude that if $u (t)$ is a bounded full solution outside $B_{r} [0, R_{0}]$ , then it is necessarily contained in $B_{r} [R, \infty]$ . Therefore $S_{λ}^{\infty} \subset B_{r} [R, \infty]$ . This asserts that (3.26) holds true, which completes the proof of the theorem. □

4. Bifurcation from infinity and multiplicity of solutions

In this section we study the bifurcation from infinity and multiplicity of solutions of the elliptic system (1.1) by considering dynamic bifurcations from infinity and the multiplicity of stationary solutions of the dynamic equation (3.1). Let $Φ_{λ}$ be the semiflow generated by (3.1).

Theorem 4.1.

Let the hypotheses (A) and (L) hold. Assume that $ε > 0$ is the number given in Theorem 3.2 . Then we have

for each $λ \in R$ , ( 1.1 ) has at least one pair of solution $(u_{λ}, v_{λ})$ ;

for every $λ \in I_{-} = [μ_{k} - ε, μ_{k})$ , ( 1.1 ) has at least two distinct pairs of solutions $(u_{λ}^{1}, v_{λ}^{1})$ and $(u_{λ}^{\infty}, v_{λ}^{\infty})$ satisfying $\begin{matrix} (4.1) & lim_{λ \to μ_{k}^{-}} {({‖ u_{λ}^{\infty} ‖}^{2} + {‖ v_{λ}^{\infty} ‖}^{2})}^{\frac{1}{2}} = \infty, \end{matrix}$ whereas $(u_{λ}^{1}, v_{λ}^{1})$ remains bounded with respect to $λ \in I_{-}$ ;

there exists an open dense subset J of $R$ such that if $λ \in I_{-} \cap J$ , then ( 1.1 ) has at least three distinct pairs of solutions.

Proof.

(1) Theorem 3.2 asserts that $Φ_{λ}$ has a nonempty compact invariant set $S_{λ}$ for $λ \in R$ . Thus for each $λ \in R$ , there is at least one stationary solution of $Φ_{λ}$ in $S_{λ}$ . This implies that (1.1) has at least one pair of solution $(u_{λ}, v_{λ})$ .

(2) By Theorem 3.2 (2), we know that $S_{λ}$ has a Morse decomposition $M = {S_{λ}^{\infty}, S_{λ}^{1}}$ for $λ \in I_{-}$ . We observe from (3.34) and (3.35) that $\begin{matrix} S_{λ}^{\infty} \cap S_{λ}^{1} = \emptyset, λ \in I_{-} . \end{matrix}$ Pick two stationary solutions $e_{λ}^{\infty} \in S_{λ}^{\infty}$ and $e_{λ}^{1} \in S_{λ}^{1}$ for $λ \in I_{-}$ . Then Theorem 3.2 (2) shows that $e_{λ}^{\infty}$ and $e_{λ}^{1}$ satisfy the requirements in (2). Thus asserts (2) follows from the fact that the stationary solutions $e_{λ}^{\infty}$ and $e_{λ}^{1}$ are actually the solutions of the equivalent system (1.6).

(3) Using the same argument as in the proof of [32, Theorem 2.1], one can easily verify that there exists an open dense subset J of $R$ such that if $λ \in J$ , then all the equilibria of the semiflow $Φ_{λ}$ are hyperbolic. Hence to verify the validity of assertion (3), it suffices to prove that for each $λ \in I_{-} \cap J$ , there exists another equilibrium $e_{λ}^{2 \infty}$ in $S_{λ}^{\infty}$ with $e_{λ}^{2 \infty} \neq e_{λ}^{\infty}$ .

We argue by contradiction and suppose that $Φ_{λ}$ has only one hyperbolic stationary solution $e_{λ}^{\infty}$ in $S_{λ}^{\infty}$ . Then one has $p (t, S_{λ}^{\infty}) = t^{l}$ for some $l ⩾ 0$ . Thus the corresponding Morse equation (3.37) reads $\begin{matrix} t^{q} + t^{l} = t^{p} + (1 + t) Q (t) . \end{matrix}$ Observing that for every formal polynomial $Q (t)$ , the sum of the coefficients of the left-hand side does not equal that of the right-hand side. This leads to a contradiction, and completes the proof of the theorem. □

Finally, we consider a particular but important case where

(F) $f (x, s) = o (| s |)$ as $| s | \to 0$ uniformly with respect to $x \in \overline{Ω}$ .

Hence 0 is an equilibrium of $Φ_{λ}$ generated by the equation (3.1). Next we first show some bifurcation and multiplicity results on stationary solutions of (3.1) near each eigenvalue $μ_{k}$ .

Let $W_{loc}^{c} (0)$ and $W_{loc}^{c u} (0)$ denote the local center manifold and local center-unstable manifold of $Φ_{μ_{k}}$ at 0, and denote by φ and ϕ the restriction of $Φ_{μ_{k}}$ on $W_{loc}^{c} (0)$ and $W_{loc}^{c u} (0)$ , respectively. Let J be the open dense subset of $R$ given by Theorem 4.1 (3). Then we first have the following main results for (3.1).

Theorem 4.2.

Let f satisfy the conditions (A), (L) and (F). Assume that 0 is an isolated equilibrium of $Φ_{μ_{k}}$ . Then there is $ε > 0$ such that one of the following assertions holds true.

If 0 is an attractor of φ, then for every $λ \in I_{-} = [μ_{k} - ε, μ_{k})$ (resp., $I_{+} = (μ_{k}, μ_{k} + ε]$ ), $Φ_{λ}$ has at least two (resp., three) distinct nontrivial equilibria; and for every $λ \in (I_{-} \cup I_{+}) \cap J$ , it has at least four distinct nontrivial equilibria.

If 0 is a repeller of φ, then for every $λ \in I_{-}$ (resp., $I_{-} \cap J$ ), $Φ_{λ}$ has at least three (resp., four) distinct nontrivial equilibria.

If 0 is neither an attractor nor a repeller of φ, then for every $λ \in I_{-}$ (resp., $I_{+}$ ), $Φ_{λ}$ has at least three (resp., two distinct) nontrivial equilibria; and for every $λ \in I_{-} \cap J$ , it has at least four distinct nontrivial equilibria.

Proof.

Let $ε > 0$ be the number sufficiently small such that Theorems 3.2 and 4.1 remain valid.

(1) Let 0 be an attractor of φ. Assume that $N_{1}$ is the isolating neighborhood of $S_{μ_{k}}$ chosen in the proof of Theorem 3.2 (see (3.27)). Note that $S_{λ}^{1} = S_{\infty} (Φ_{λ}, N_{1})$ for $λ \in I = [μ_{k} - ε, μ_{k} + ε]$ . By (3.30) and (3.31), we deduce that $\begin{matrix} S_{λ}^{1} \subset B_{r} [0, R_{1} / 2], λ \in I_{-} . \end{matrix}$ Because $S_{λ}^{\infty}$ is the maximal compact invariant set of $Φ_{λ}$ in $B_{r} [R_{1}, \infty]$ for $λ \in I_{-}$ (see (3.35)), one can conclude from the proof of Theorem 4.1 that for every $λ \in I_{-}$ (resp., $λ \in I_{-} \cap J$ ) the system $Φ_{λ}$ has at least one equilibrium (resp. two distinct equilibria) outside $N_{1}$ .

Since 0 is isolated, one can pick an isolating neighborhood $N_{0}$ of 0 such that $\begin{matrix} N_{0} \subset B_{Y} (β) \subset N_{1} \end{matrix}$ for some $β > 0$ . It can be assumed that the ε is sufficiently small such that both $N_{0}$ and $N_{1}$ are isolating neighborhoods of $Φ_{λ}$ for $λ \in I = [μ_{k} - ε, μ_{k} + ε]$ . Write $\begin{matrix} K_{λ}^{i} = S_{\infty} (Φ_{λ}, N_{i}), i = 0, 1 . \end{matrix}$ Then for $i = 0, 1$ , one has $\begin{matrix} (4.2) & h (Φ_{λ}, K_{λ}^{i}) \equiv const., λ \in I . \end{matrix}$ As $N_{0}$ is an isolating neighborhood of 0, one can easily verify that $\begin{matrix} (4.3) & d_{H} (K_{λ}^{0}, {0}) \to 0 as λ \to μ_{k} . \end{matrix}$ We may assume that the isolating neighborhood $N_{0}$ is chosen sufficiently small such that the product formula of Conley index in [31, Chap. II, Theorem 3.1] holds. Thus $\begin{matrix} (4.4) & h (Φ_{μ_{k}}, {0}) = Σ^{p} \land h (φ, {0}) . \end{matrix}$ Choose an isolating block B of 0 in $W_{loc}^{c} (0)$ such that B is contained in the attraction region of 0 (with respect to φ). Then we necessarily have $B^{-} = \emptyset$ . We prove that B is contractible. Indeed, $B^{-} = \emptyset$ implies that B is positively invariant. Define $\begin{matrix} H (s, x) = \{\begin{matrix} φ (s / (1 - s)) x, & x \in B, s \in [0, 1); \\ x, & x \in B, s = 1 . \end{matrix} \end{matrix}$ One trivially verifies that H is a strong deformation retraction. Hence B is contractible. By the definition of Conley index, we find that $\begin{matrix} h (φ, {0}) = [(B / B^{-}, [B^{-}])] = Σ^{0} . \end{matrix}$ Thus we deduce from (4.4) that $\begin{matrix} (4.5) & h (Φ_{μ_{k}}, {0}) = Σ^{p} \land Σ^{0} = Σ^{p} . \end{matrix}$ Combining (4.5) with (4.2), it yields $\begin{matrix} (4.6) & h (Φ_{λ}, K_{λ}^{0}) = h (Φ_{μ_{k}}, K_{μ_{k}}^{0}) = h (Φ_{μ_{k}}, {0}) = Σ^{p}, λ \in I . \end{matrix}$ On the other hand, by (4.2) and Proposition 3.2 we obtain $\begin{matrix} (4.7) & h (Φ_{λ}, K_{λ}^{1}) = h (Φ_{μ_{k}}, K_{μ_{k}}^{1}) = h (Φ_{μ_{k}}, S_{μ_{k}}) = Σ^{q}, λ \in I . \end{matrix}$ Therefore (4.6) and (4.7) show that $K_{λ}^{1} \neq K_{λ}^{0}$ , from which it can be seen that $\begin{matrix} (4.8) & K_{λ}^{1} ∖ N_{0} \neq \emptyset, λ \in I . \end{matrix}$ Now we pick a $z_{λ} \in K_{λ}^{1} ∖ N_{0}$ for every $λ \in I$ . Then there exists a bounded full trajectory $u_{λ} (t)$ of $Φ_{λ}$ in $K_{λ}^{1}$ so that $u_{λ} (0) = z_{λ}$ . In what follows we prove that if the ε is chosen sufficiently small, then $\begin{matrix} (4.9) & either ω (u_{λ}) ∖ N_{0} \neq \emptyset, or ω^{*} (u_{λ}) ∖ N_{0} \neq \emptyset . \end{matrix}$

We argue by contradiction and suppose that there would exist a sequence $λ_{n} \to μ_{k}$ (as $n \to \infty$ ) so that both $ω (u_{n})$ and $ω^{*} (u_{n})$ are contained in $N_{0}$ , then $\begin{matrix} (4.10) & ω (u_{n}) \cup ω^{*} (u_{n}) \subset K_{λ_{n}}^{0}, \end{matrix}$ where $u_{n} = u_{λ_{n}}$ . Denote $L$ the Lyapunov function of (3.1). Hence we deduce from (4.3) and (4.10) that $\begin{matrix} (4.11) & lim_{n \to \infty} max_{z \in ω (u_{n})} | L (z) | = 0 = lim_{n \to \infty} max_{z \in ω^{*} (u_{n})} | L (z) | . \end{matrix}$ Let $\begin{matrix} Γ_{n} = \overline{orb (u_{n})} = orb (u_{n}) \cup ω (u_{n}) \cup ω^{*} (u_{n}) . \end{matrix}$ It is easy to see that $\begin{matrix} min_{z \in Γ_{n}} L (z) = min_{z \in ω (u_{n})} L (z), max_{z \in Γ_{n}} L (z) = max_{z \in ω^{*} (u_{n})} L (z) . \end{matrix}$ Then one can see from (4.11) that $\begin{matrix} (4.12) & max_{z \in Γ_{n}} | L (z) | \to 0 as n \to \infty . \end{matrix}$ Note that $Γ_{n} \subset K_{λ_{n}}^{1} \subset N_{1}$ and $N_{1}$ is bounded. Thus we conclude from the asymptotic compactness of the family ${Φ_{λ}}$ that $⋃_{λ_{n} \in I} Γ_{n}$ is precompact. Thanks to [17, Lemma 2.2] or [1], there exists a nonempty compact invariant set $Γ_{0}$ of $Φ_{μ_{k}}$ such that $Γ_{n} \to Γ_{0}$ as $n \to \infty$ . Since each $Γ_{n}$ is connected, one easily sees that $Γ_{0}$ is connected as well. By (4.10), we know that $Γ_{n} \cap K_{λ_{n}}^{0} \neq \emptyset$ . Hence $0 \in Γ_{0}$ . We infer from (4.12) that $L (z) \equiv 0$ on $Γ_{0}$ . Thereby $Γ_{0}$ consists of equilibrium points of $Φ_{μ_{k}}$ . On the other hand, noticing that $\begin{matrix} (4.13) & u_{n} (0) \in Γ_{n} ∖ N_{0} \end{matrix}$ for all n, we conclude that $Γ_{0} ∖ int N_{0} \neq \emptyset$ . As $N_{0}$ is an isolating neighborhood of 0, we deduce that $Γ_{0}$ is not connected. This contradicts the fact that $Γ_{0}$ is connected, and completes the proof of (4.9).

Now, by (4.9), for every $λ \in I$ , we pick an equilibrium $\begin{matrix} e_{λ}^{c} \in (ω (v_{λ}) \cup ω^{*} (v_{λ})) ∖ N_{0} \end{matrix}$ of $Φ_{λ}$ . Noticing that $e_{λ}^{c} \in K_{λ}^{1} \subset N_{1}$ , we know that $Φ_{λ}$ has at least two nontrivial equilibria $e_{λ}^{\infty}$ and $e_{λ}^{c}$ for every $λ \in I_{-}$ ,

Moreover, by the attractor bifurcation theory (see e.g., Ma and Wang [19, Theorem 6.1] or Li and Wang [17, Theorem 4.2]), we deduce that if ε is sufficiently small, then for every $λ \in I_{+}$ , $K_{λ}^{0}$ contains at least two distinct equilibria $e_{λ}^{1}$ and $e_{λ}^{2}$ . Consequently, $Φ_{λ}$ has at least three distinct nontrivial equilibria $e_{λ}^{c}$ , $e_{λ}^{1}$ , $e_{λ}^{2}$ for $λ \in I_{+}$ .

In the following we assume $λ \in I \cap J$ , and verify that there are at least two distinct nontrivial equilibria of $Φ_{λ}$ in $N_{1} ∖ N_{0}$ , provided $ε > 0$ is sufficiently small.

We argue by contradiction and suppose the contrary. Then there would be a sequence $λ_{n} \to μ_{k}$ so that $Φ_{λ}$ has only one equilibrium $e_{n} = e_{λ_{n}}^{c}$ in $N_{1} ∖ N_{0}$ for every n. Hence two possibilities may occur.

Case (i) ${lim}_{n \to \infty} | L (e_{n}) | = 0$ . In this case, let us we first prove that if n is sufficiently large, then there are no connecting orbits between $e_{n}$ and $K_{n}^{0} = K_{λ_{n}}^{0}$ . Then using this result we further show that $\begin{matrix} (4.14) & K_{n}^{1} : = K_{λ_{n}}^{1} = {e_{n}} \cup K_{n}^{0} . \end{matrix}$ Therefore, $K_{n}^{1}$ has a Morse decomposition $M = {{e_{n}}, K_{n}^{0}}$ .

Suppose the contrary. Then there exists a subsequence of ${n}_{n = 1}^{\infty}$ , still denoted by ${n}_{n = 1}^{\infty}$ , such that for every n, there exists a connecting orbit $γ_{n}$ between $e_{n}$ and $K_{n}^{0}$ . Write $Γ_{n} = {\overline{γ}}_{n}$ . Then one can easily see that $Γ_{n}$ is a connected compact invariant set satisfying $\begin{matrix} e_{n} \in Γ_{n} and Γ_{n} \cap K_{n}^{0} \neq \emptyset . \end{matrix}$ Let ${\tilde{K}}_{n} = {e_{n}} \cup K_{n}^{0}$ . Then we find that $\begin{matrix} (4.15) & min_{z \in {\tilde{K}}_{n}} L (z) ⩽ min_{z \in Γ_{n}} L (z) ⩽ max_{z \in Γ_{n}} L (z) ⩽ max_{z \in {\tilde{K}}_{n}} L (z) . \end{matrix}$ In view of (4.3), by virtue of the assumption that ${lim}_{n \to \infty} | L (e_{n}) | = 0$ , we have $\begin{matrix} max_{z \in {\tilde{K}}_{n}} | L (z) | \to 0 as n \to \infty . \end{matrix}$ Hence it follows from (4.15) that $\begin{matrix} (4.16) & lim_{n \to \infty} max_{z \in Γ_{n}} | L (z) | = 0 . \end{matrix}$ Using the argument below (4.12) and replacing $u_{n} (0)$ in (4.13) by $e_{n}$ , we can similarly obtain a contradiction.

Now we further verify that if n is sufficiently large, then (4.14) holds. Suppose that there exists a subsequence of ${n}_{n = 1}^{\infty}$ , still denoted by ${n}_{n = 1}^{\infty}$ , so that for every n, $K_{n}^{1} \neq {e_{n}} \cup K_{n}^{0}$ . Thus we deduce that $\begin{matrix} K_{n}^{1} ∖ (N_{0} \cup {e_{n}}) \neq \emptyset, n ⩾ 1 . \end{matrix}$ Pick a sequence $z_{n} \in K_{n}^{1} ∖ (N_{0} \cup {e_{n}})$ for $n ⩾ 1$ . Then there exists a bounded full trajectory $u_{n} (t)$ of $Φ_{λ_{n}}$ in $K_{n}^{1}$ such that $u_{n} (0) = z_{n}$ . Observing that there are no connecting orbits between $e_{n}$ and $K_{n}^{0}$ , one can easily find that $\begin{matrix} either ω (u_{n}) = ω^{*} (u_{n}) = {e_{n}}, or ω (u_{n}) \cup ω^{*} (u_{n}) \subset K_{n}^{0} . \end{matrix}$ It is easy to see that the first case cannot occur. Therefore $\begin{matrix} ω (u_{n}) \cup ω^{*} (u_{n}) \subset K_{n}^{0} . \end{matrix}$ Set $Γ_{n} = \overline{orb} (u_{n})$ . Then using the same argument as obtaining (4.16), one can easily verify that ${lim}_{n \to \infty} {max}_{z \in Γ_{n}} | L (z) | = 0$ . Moreover, as the argument below (4.12), we can similarly obtain a contradiction.

For every fixed n, by the proof of Theorem 4.1 (3), we see that $e_{n}$ is hyperbolic. Then there is $m ⩾ 0$ such that $p (t, e_{n}) = t^{m}$ . We infer from (4.6) and (4.7) that $\begin{matrix} p (t, K_{n}^{0}) = t^{p}, p (t, K_{n}^{1}) = t^{q} . \end{matrix}$ Hence the corresponding Morse equation of $M$ reads $\begin{matrix} t^{m} + t^{p} = t^{q} + (1 + t) Q (t) \end{matrix}$ for some formal polynomial $Q (t)$ . This is impossible and completes the proof of this case.

Case (ii) Either ${lim sup}_{n \to \infty} L (e_{n}) > 0$ , or ${lim inf}_{n \to \infty} L (e_{n}) < 0$ .

Without loss of generality it can be assumed that $\begin{matrix} \underset{n \to \infty}{lim sup} L (e_{n}) : = 2 L_{+} > 0 . \end{matrix}$ Thus one can pick a subsequence of $e_{n}$ , still denoted by $e_{n}$ , so that $\begin{matrix} L (e_{n}) > L_{+}, n ⩾ 1 . \end{matrix}$ In what follows we prove that if $γ (t)$ is a full trajectory of $Φ_{λ}$ in $K_{n}^{1} ∖ ({e_{n}} \cup K_{n}^{0})$ for n sufficiently large, then $\begin{matrix} ω^{*} (γ) = {e_{n}}, ω (γ) \subset K_{n}^{0}, \end{matrix}$ from which it can be deduced that $K_{n}^{1}$ has a Morse decomposition $M = {K_{n}^{0}, {e_{n}}}$ . We also argue by contradiction and suppose that this was false. Then there would be a subsequence of ${n}_{n = 1}^{\infty}$ , still denoted by ${n}_{n = 1}^{\infty}$ , so that there exists a full trajectory $γ_{n} (t)$ in $K_{n}^{1} ∖ ({e_{n}} \cup K_{n}^{0})$ for every n, satisfying $\begin{matrix} either ω (γ_{n}) = ω^{*} (γ_{n}) = {e_{n}}, or ω (γ_{n}) \cup ω^{*} (γ_{n}) \subset K_{n}^{0} . \end{matrix}$ Clearly, the first case cannot occur. Thus $\begin{matrix} ω (γ_{n}) \cup ω^{*} (γ_{n}) \subset K_{n}^{0} . \end{matrix}$ Recalling that $γ_{n}$ is not contained in $K_{n}^{0} = K_{\infty} (Φ_{λ_{n}}, N_{0})$ , one can easily find that there is some $t_{0} \in R$ , such that $\begin{matrix} γ_{n} (t_{0}) \notin N_{0} . \end{matrix}$ Write $Γ_{n} = \overline{orb} (γ_{n})$ . Then by the same argument as that in Case (i), it can be shown that ${max}_{z \in Γ_{n}} | L (z) | = 0$ as $n \to \infty$ . Furthermore, using the same argument following (4.12), we can obtain a contradiction.

Similar to Case (i), we can also check the corresponding Morse equation of the Morse decomposition $M$ of $K_{n}^{1}$ , from which one can get a contradiction. Thus for every fixed $λ \in I \cap J$ , there are at least two distinct nontrivial equilibria of $Φ_{λ}$ in $N_{1} ∖ N_{0}$ . Because there are at least two distinct nontrivial equilibria of $Φ_{λ}$ outside $N_{1}$ if $λ \in I_{-} \cap J$ , and in $N_{0}$ if $λ \in I_{+} \cap J$ , respectively, one can immediately conclude that if $λ \in (I_{-} \cup I_{+}) \cap J$ , the system $Φ_{λ}$ has at least four distinct nontrivial equilibria.

(2) Let 0 be a repeller of φ. Similar to Case (i), by the attractor bifurcation theory (see Ma and Wang [19, Theorem 6.1] or Li and Wang [17, Theorem 4.2]), we deduce that if $λ \in I_{-}$ , then $K_{λ}^{0}$ contains at least two distinct equilibria $e_{λ}^{1}$ and $e_{λ}^{2}$ . Because for $λ \in I_{-}$ (resp., $λ \in I_{-} \cap J$ ), the system $Φ_{λ}$ has at least one (resp., two distinct) nontrivial equilibria outside $N_{1}$ , we know that if $λ \in I_{-}$ (resp., $λ \in I_{-} \cap J$ ), then $Φ_{λ}$ has at least three (resp., four) distinct nontrivial equilibria.

(3) Finally, we discuss the case where 0 is neither an attractor nor a repeller of φ. When this case occurs, we first infer from Li and Wang [17, Theorem 4.4] that there exists $ε > 0$ such that for each $λ \in I = [μ_{k} - ε, μ_{k} + ε]$ , the system $Φ_{λ}$ has a nonempty compact invariant set $K_{λ}$ in $N_{0}$ with $0 \notin K_{λ}$ and $\begin{matrix} d_{H} (K_{λ}, {0}) \to 0 as λ \to μ_{k} . \end{matrix}$ Clearly, one can see that there is at least one nontrivial equilibrium $e_{λ}$ in $K_{λ}$ .

Now we claim that $\begin{matrix} (4.17) & h (Φ_{μ_{k}}, {0}) \neq Σ^{q}, \end{matrix}$ from which one can conclude that there exists a two-sided neighborhood I of $μ_{k}$ such that $Φ_{λ}$ has another equilibrium $e_{λ}^{c} \in N_{1} ∖ N_{0}$ .

Indeed, it is clear to see that $\begin{matrix} (4.18) & h (Φ_{μ_{k}}, {0}) = h (ϕ, {0}), \end{matrix}$ where ϕ denotes the restriction of $Φ_{μ_{k}}$ on $W_{loc}^{c u} (0)$ . Denote by $H_{*} (\cdot)$ and $H^{*} (\cdot)$ the singular homology and cohomology functors, respectively. Then in order to verify (4.17), it suffices to show that $\begin{matrix} (4.19) & H_{*} (h (ϕ, {0})) \neq H_{*} (Σ^{q}) . \end{matrix}$ Suppose the contrary. Then one has $\begin{matrix} H_{q} (h (ϕ, {0})) = H_{q} (Σ^{q}) = Z . \end{matrix}$ Thus we deduce from the Poincaré-Lefschetz duality theory of the Conley index (see e.g., McCord [22, Theorem 2.1] and Mrozek and Srzednicki [25, pp. 164]) that $\begin{matrix} (4.20) & H^{0} (h (ϕ^{- 1}, {0})) = H_{q} (h (ϕ, {0})) = Z, \end{matrix}$ where $ϕ^{- 1}$ denotes the inverse flow of ϕ.

On the other hand, by [6, Theorem 1.5], one can pick a pass-connected isolating block $B \subset W_{loc}^{c u} (0)$ of $S_{0} = {0}$ for the inverse flow $ϕ^{- 1}$ . Due to the fact that $S_{0}$ is not an attractor of $ϕ^{- 1}$ on the center-unstable manifold $W_{loc}^{c u} (0)$ , we know that $B^{-} \neq \emptyset$ . Therefore one can conclude from the basic knowledge in the theory of algebraic topology that $H^{0} (B, B^{-}) = 0$ , from which it can be seen that $\begin{matrix} H^{0} (h (ϕ^{- 1}, {0})) = H^{0} (B, B^{-}) = 0 . \end{matrix}$ This contradicts (4.20) and completes the proof of our claim.

By (4.7), we have $\begin{matrix} h (Φ_{λ}, K_{λ}^{1}) = Σ^{q}, λ \in I = [μ_{k} - ε, μ_{k} + ε] . \end{matrix}$ Because $\begin{matrix} h (Φ_{λ}, K_{λ}^{0}) = h (Φ_{μ_{k}}, K_{μ_{k}}^{0}) = h (Φ_{μ_{k}}, {0}) \neq Σ^{q}, \forall λ \in I, \end{matrix}$ one can deduce that $K_{λ}^{1} \neq K_{λ}^{0}$ , which implies $\begin{matrix} K_{λ}^{1} ∖ N_{0} \neq \emptyset, λ \in I . \end{matrix}$ Hence by using the same argument following (4.8), one can easily prove that there exists another equilibrium $e_{λ}^{c}$ of $Φ_{λ}$ in $N_{1} ∖ N_{0}$ .

Therefore, for each $λ \in I ∖ {μ_{k}}$ , the system $Φ_{λ}$ has at least two distinct nontrivial equilibria in $N_{1}$ . Consequently, by the proof of the Case (i) at the beginning, one can easily see that the assertion (3) holds true. □

Because the stationary solution of $Φ_{λ}$ generated by (3.1) is actually the solution of the equivalent system (1.6), by Theorem 4.2, we immediately obtain the following main results for the elliptic system (1.1).

Theorem 4.3.

Let f satisfy the conditions (A), (L) and (F). Assume that 0 is an isolated equilibrium of $Φ_{μ_{k}}$ . Then there is $ε > 0$ such that one of the following assertions holds true.

If 0 is an attractor of φ, then for every $λ \in I_{-} = [μ_{k} - ε, μ_{k})$ (resp., $I_{+} = (μ_{k}, μ_{k} + ε]$ ), the elliptic equation ( 1.1 ) has at least two (resp., three) distinct pairs of solutions; and for every $λ \in (I_{-} \cup I_{+}) \cap J$ , it has at least four distinct pairs of solutions.

If 0 is a repeller of φ, then for every $λ \in I_{-}$ (resp., $I_{-} \cap J$ ), ( 1.1 ) has at least three (resp., four) distinct pairs of solutions.

If 0 is neither an attractor nor a repeller of φ, then for every $λ \in I_{-}$ (resp., $I_{+}$ ), ( 1.1 ) has at least three (resp., two distinct) nontrivial pairs of solutions; and for every $λ \in I_{-} \cap J$ , it has at least four distinct nontrivial pairs of solutions.

Remark 4.1.

If we assume that $\begin{matrix} (4.21) & \underset{s \to + \infty}{lim sup} f (x, s) ⩽ - \overline{f} < 0, \underset{s \to - \infty}{lim inf} f (x, s) ⩾ \underline{f} > 0 \end{matrix}$ uniformly with respect to $x \in \overline{Ω}$ , then dual versions of all the results in this section hold true.

Footnotes

Acknowledgements

This work was supported by Zhejiang Provincial Natural Science Foundation of China under Grant No. LQ22A010002.

References

Castaing and

Valadier, Convex Analysis and Measurable Multifunctions, Springer-Verlag, Berlin, 1977.

Chang and

Li, Existence and multiplicity of nontrivial solutions for semilinear elliptic Dirichlet problems across resonance, Topol. Methods Nonlinear Anal. 36 (2010), 285–310.

Chiappinelli and

D.G.

de Figueiredo, Bifurcation from infinity and multiple solutions for an elliptic system, Differential Integral Equations 6(4) (1993), 757–771. doi:10.57262/die/1370032232.

Chiappinelli,

Mawhin and

Nugari, Bifurcation from infinity and multiple solutions for some Dirichlet problems with unbounded nonlinearities, Nonlinear Anal. TMA 18 (1992), 1099–1112. doi:10.1016/0362-546X(92)90155-8.

Conley, Isolated Invariant Sets and the Morse Index, Regional Conference Series in Mathematics, Vol. 38, Amer. Math. Soc., Providence RI, 1978.

Conley and

Easton, Isolated invariant sets and isolating blocks, Trans. Amer. Math. Soc. 158 (1971), 35–61. doi:10.1090/S0002-9947-1971-0279830-1.

D.G.

de Figueiredo and

Mitidieri, A maximum principle for an elliptic system and applications to semilinear problems, SIAM J. Math. Anal. 17 (1986), 836–849. doi:10.1137/0517060.

F.O.

de Paiva and

Massa, Semilinear elliptic problems near resonance with a nonprincipal eigenvalue, J. Math. Anal. Appl. 342 (2008), 638–650. doi:10.1016/j.jmaa.2007.12.053.

Du and

Ouyang, Bifurcation from infinity induced by a degeneracy in semilinear equations, Advanced Nonlinear Studies 2(2) (2002), 117–132. doi:10.1515/ans-2002-0202.

10.

Henry, Geometric Theory of Semilinear Parabolic Equations, Lecture Notes in Math., Vol. 840, Springer-Verlag, Berlin, 1981.

11.

Koga and

Kuramoto, Localized patterns in reaction-diffusion systems, Progress of Theoretical Physics 63(1) (1980), 106–121. doi:10.1143/PTP.63.106.

12.

A.C.

Lazer and

P.J.

McKenna, On state solutions of a system of reaction-diffusion equations from biology, Nonlinear Anal., TMA 6 (1982), 523–530. doi:10.1016/0362-546X(82)90045-1.

13.

C.Q.

Li,

D.S.

Li and

J.T.

Wang, A remark on attractor bifurcation, Dynamics of PDE 18(2) (2021), 157–172.

14.

C.Q.

Li,

D.S.

Li and

Z.J.

Zhang, Dynamic bifurcation from infinity of nonlinear evolution equations, SIAM J. Appl. Dyn. Syst. 16 (2017), 1831–1868. doi:10.1137/16M1107358.

15.

C.Q.

Li and

J.T.

Wang, Bifurcation from infinity of the Schrödinger equation via invariant manifolds, Nonlinear Anal. 213 (2021), 112490. doi:10.1016/j.na.2021.112490.

16.

D.S.

Li,

G.L.

Shi and

X.F.

Song, Linking theorems of local semiflows on complete metric spaces, 2015, arXiv:1312.1868.

17.

D.S.

Li and

Z.Q.

Wang, Local and global dynamic bifurcations of nonlinear evolution equations, Indiana Univ. Math. J. 67(2) (2018), 583–621. doi:10.1512/iumj.2018.67.7292.

18.

Liang and

Su, Multiple solutions for semilinear elliptic boundary value problems with double resonance, J. Math. Anal. Appl. 354 (2009), 147–158. doi:10.1016/j.jmaa.2008.12.053.

19.

Ma and

S.H.

Wang, Bifurcation Theory and Applications, World Sci. Ser. Nonlinear Sci. Ser. A, Vol. 53, World Scientific, Hackensack, NJ, 2005.

20.

Mawhin and

Schmitt, Landesman-Lazer type problems at an eigenvalue of odd multiplicity, Results Math. 14 (1988), 138–146. doi:10.1007/BF03323221.

21.

Mawhin and

Schmitt, Nonlinear eigenvalue problems with the parameter near resonance, Ann. Polon. Math. 51 (1990), 241–248. doi:10.4064/ap-51-1-241-248.

22.

McCord, Poincaré-Lefschetz duality for the homolopy Conley index, Trans. Amer. Math. Soc. 329 (1992), 233–252.

23.

Mischaikow and

Mrozek, Conley index, in: Handb. Dynam. Syst. 2, Elsevier, New York, 2002, pp. 393–460.

24.

Mottoni and

Rothe, A singular perturbation analysis for a reaction-diffusion system describing pattern formation, Stud. Appl. Math. 63(3) (1980), 227–247. doi:10.1002/sapm1980633227.

25.

Mrozek and

Srzednicki, On time-duality of the Conley index, Results Math. 24 (1993), 161–167. doi:10.1007/BF03322325.

26.

N.S.

Papageorgiou and

Papalini, Multiple solutions for nearly resonant nonlinear Dirichlet problems, Potential Anal. 37 (2012), 247–279. doi:10.1007/s11118-011-9255-8.

27.

P.H.

Rabinowitz, On bifurcation from infinity, J. Differential Equations 14 (1973), 462–475. doi:10.1016/0022-0396(73)90061-2.

28.

Rothe, Some analytical results about a simple reaction-diffusion system for morphogenesis, J. Math. Biol. 7(4) (1979), 375–384. doi:10.1007/BF00275155.

29.

Rothe, Global existence of branches of stationary solutions for a system of reaction-diffusion equations from biology, Nonlinear Anal., TMA 5 (1981), 487–798. doi:10.1016/0362-546X(81)90097-3.

30.

Rothe and

de Morttoni, A simple system of reaction-diffusion equations describing morphogenesis: Asymptotic behavior, Annali di Matematica Pura ed Applicata 122 (1979), 141–157. doi:10.1007/BF02411692.

31.

K.P.

Rybakowski, The Homotopy Index and Partial Differential Equations, Springer-Verlag, Berlin, 1987.

32.

J.C.

Saut and

Temam, Generic properties of nonlinear boundary value problems, Comm. Partial Differential Equations 4 (1979), 293–319. doi:10.1080/03605307908820096.

33.

Schmitt and

Z.Q.

Wang, On bifurcation from infinity for potential operators, Differential Integral Equations 4 (1991), 933–943.

34.

Su, Existence and multiplicity results for classes of elliptic resonant problems, J. Math. Anal. Appl. 273 (2002), 565–579. doi:10.1016/S0022-247X(02)00274-3.

35.

Teng and

Wu, Multiplicity results for semilinear resonant elliptic problems with discontinuous nonlinearities, Nonlinear Anal. TMA 68(6) (2008), 1652–1667. doi:10.1016/j.na.2006.12.050.

36.

J.T.

Wang,

D.S.

Li and

J.Q.

Duan, On the shape Conley index theory of semiflows on complete metric spaces, Discrete Contin. Dyn. Syst. – A 36(3) (2016), 1629–1647. doi:10.3934/dcds.2016.36.1629.

37.

J.R.

WardJr., A global continuation theorem and bifurcation from infinity for infinite-dimensional dynamical systems, Proc. Roy. Soc. Edinburgh Sect. A 126 (1996), 725–738. doi:10.1017/S0308210500023039.

38.

Zhou and

D.S.

Li, Global dynamic bifurcation of local semiflows and nonlinear evolution equations, J. Differential Equations 300 (2021), 625–659. doi:10.1016/j.jde.2021.07.035.