A note on multiplicity of solutions near resonance of semilinear elliptic equations

In this paper we are concerned with the multiplicity of solutions near resonance for the following nonlinear equation: \begin{document}$ -\Delta u = \lambda u+f(x,u) $\end{document} associated with the Dirichlet boundary condition, where \begin{document}$ f $\end{document} satisfies some appropriate conditions. We will treat this problem in the framework of dynamical systems. It will be shown that there exist a one-sided neighborhood \begin{document}$ \Lambda_- $\end{document} of the eigenvalue \begin{document}$ \mu_k $\end{document} of the Laplacian operator and a dense subset \begin{document}$ {\mathcal D} $\end{document} of \begin{document}$ \mathbb{R} $\end{document} such that the equation has at least four distinct nontrivial solutions generically for \begin{document}$ \lambda\in\Lambda_- \cap {\mathcal D} $\end{document} .


1.
Introduction. This note is concerned with the following equation: x ∈ Ω, where Ω ⊂ R n is a bounded domain with smooth boundary ∂Ω, f is a bounded function, and λ ∈ R is a parameter. We are mainly interested in the bifurcation from infinity and multiplicity of solutions near resonance of (1). The bifurcation and multiplicity of semilinear elliptic equations near resonance is an interesting topic and has attracted much attention in the past decades, see [2,7,19,24,15,14,6,22], etc. This problem (1) can be traced back to the earlier work [15] by Mawhin and Schmitt, where the authors considered the case when λ crosses an eigenvalue of odd multiplicity. Later Schmitt and Wang [22] developed a theory on bifurcation from infinity for potential operators, through which they extended the results in [15] to the case when λ crosses any eigenvalue µ k of the Laplacian. More specifically, under an abstract Landesman-Laser type condition on the Nemitski operatorf : H 1 0 (Ω) → L 2 (Ω) corresponding to the function f (x, s), the authors proved the following result and its "dual" version: there exists δ > 0 such that for each λ ∈ [µ k − δ, µ k ) =: Λ − the equation (1) has at least two solutions, and for each λ ∈ [µ k , µ k + δ] at least one.
Note that a solution of (1) can be regarded as a stationary solution of the following evolution equation: x ∈ Ω, u(x) = 0, x ∈ ∂Ω.
Moreover, there exists a Lyapunov function for system (2). Then the problem (1) can be studied via the method of dynamical systems. Very recently, the equation (2) was studied in [9], where the authors established a general theorem on dynamic bifurcation from infinity in the framework of local semiflows on a Banach space. As an application of their theoretical results, they considered the equation (2), and gave some static bifurcation and multiplicity results of (2) under the following Landesman-Lazer type conditions on f : uniformly for x ∈ Ω. Specifically, they proved that for each eigenvalue µ k of the Laplacian (associated with the Dirichlet boundary condition), there exists δ = δ k > 0 such that for each λ ∈ Λ − = [µ k − δ, µ k ), the equation has at least two distinct stationary solutions e 1 λ , e ∞ λ with whereas e 1 λ remains bounded on Λ − . Furthermore, they also paid some attention to the case where uniformly for x ∈ Ω. They showed that there is a one-sided neighborhood Λ 1 of µ k such that (2) has at least three distinct equilibria. For such a case on f , Chiappinelli, Mawhin and Nugari [3] studied the multiplicity of solutions of the problem near the first eigenvalue µ 1 . (Note that the nonlinearity in [3] was allowed to be unbounded.) Motivated by the above works, we further discuss the bifurcation and multiplicity of the stationary solutions of (2). Roughly speaking, for any eigenvalue µ k , we will prove that there exist a dense subset D of R and a one-sided neighborhood Λ 1 of µ k such that the equation (1) has at least four distinct solutions for λ ∈ D ∩ Λ 1 under the Landesman-Lazer type conditions and (4) on f . It is worth mentioning that dual versions of our results mentioned above hold true if, instead of (3), we assume that uniformly for x ∈ Ω. This work is organized as follows. In Section 2 we make some preliminaries. Section 3 is devoted to our main results.

2.
Preliminaries. In this section we introduce some basic concepts and results. The interested reader is referred to [9,11,20] for details.
2.1. Basic topological notions and results. Let X be a complete metric space with metric d(·, ·).
Let A and B be two subsets of X. The Hausdorff semidistance and Hausdorff distance between A and B are defined by Lemma 2.1. (see [1]). Let X be a compact metric space. Denote K (X) the family of compact subsets of X which is equipped with the Hausdorff metric δ H (·, ·). Then K (X) is a compact metric space.

2.2.
Wedge/smash product of pointed spaces. Let (X, x 0 ) and (Y, y 0 ) be two pointed spaces. The smash product (X, We denote by [(X, x 0 )] the homotopy type of a pointed space (X, x 0 ). Since the operation "∧ " preserves homotopy equivalence relations, it can be naturally extended to the homotopy types of pointed spaces. Let 0 and Σ 0 be the homotopy types of the pointed spaces ({p}, p) and ({p, q}, q), respectively, where p and q are two distinct points. By Σ m we denote the homotopy type of pointed m-dimensional sphere. One easily shows that Σ m ∧ Σ n = Σ m+n , ∀ m, n ≥ 0.

2.3.
Local semiflows on metric spaces. For the reader's convenience, in this subsection we collect some fundamental notions and facts concerning local semiflows on metric spaces.
2.3.1. Local semiflows. Let X be a complete metric space. A local semiflow Φ on X is a continuous mapping from an open set D(Φ) ⊂ R + × X to X and satisfies the following properties: 1. for each x ∈ X, there exists 0 < T x ≤ ∞, called the escape time of Φ(t, x), such that ) for all x ∈ X and t, s ∈ R + with t + s ≤ T x . Let Φ be a given local semiflow on X. For notational simplicity, we usually rewrite Φ(t, x) as Φ(t)x.
Let I ⊂ R be an interval. A trajectory of Φ on I is a continuous mapping γ : I → X that satisfies ∀t, s ∈ I, t ≥ s.
A trajectory γ on R is called a full trajectory. Let γ be a full trajectory. The ω-limit set ω(γ) and ω * -limit set of γ are defined, respectively, by ω(γ) = {y ∈ X : there exists t n → ∞ such that γ(t n ) → y}, ω * (γ) = {y ∈ X : there exists t n → −∞ such that γ(t n ) → y}. Let γ be a trajectory on I ⊂ R. Set We call orb(γ) the orbit of γ on I. The orbit of a full trajectory is simply called a full orbit.
For the sake of convenience, given U ⊂ X, denote K ∞ (Φ, U ) the union of all bounded full orbits in U . In the case U = X, we will simply rewrite [20]). N ⊂ X is said to be admissible, if for any sequences x n ∈ N and t n → ∞ with Φ([0, t n ])x n ⊂ N for all n, the sequence Φ(t n )x n has a convergent subsequence.
N is said to be strongly admissible, if it is admissible and moreover, Φ does not explode in N .
be a family of semiflows on X, where Λ is a metric space. We say that Φ λ depends on λ continuously, if for any λ ∈ Λ, x ∈ X, t ∈ R + and any sequence (t n , x n , λ n ), whenever (t n , x n , λ n ) → (t, x, λ) as n → ∞ and Φ λ (t)x is defined, then Φ λn (t n )x n is also defined for all n sufficiently large, and furthermore, Suppose the family Φ λ (λ ∈ Λ) depends on λ continuously. Set Then Π is a local semiflow on the product space X × Λ. For convenience, we call Π the skew-product flow of the family Φ λ (λ ∈ Λ). The family Φ λ (λ ∈ Λ) is said to be λ-locally uniformly asymptotically compact (λ-l.u.a.c. in short), if the skew-product flow Π is asymptotically compact.

Invariant sets, attractors, and Morse decompositions. Let Φ be a local semi-
Remark 1. A full trajectory satisfying (6) will be referred to as a connecting trajectory between M i and M j .

Conley index.
In this subsection we briefly recall the definition of Conley index. The interested reader is referred to [4,17] and [20], etc. for details. Let Φ be a local semiflow on X. Since X may be an infinite dimensional space, we will always assume Φ is asymptotically compact.
Let N, E be two closed subsets of X. E is called an exit set of N , if the two properties hold: An important example for isolating neighborhoods is the so called isolating block, which plays a crucial role in the Conley index theory.
where τ ≥ 0, s > 0, the following properties hold: 1. there exists 0 < ε < s such that 2. if τ > 0, then there exists 0 < δ < τ such that Denote by B e (resp. B i , B b ) the set of all strict egress (resp. strict ingress, bounceoff) points of the closed set B, and set  hoping that this will not cause any confusion.
Let S be a compact isolated invariant set. Denote H * and H * the singular homology and cohomology theories with coefficient group Z, respectively. Applying H * and H * to h(Φ, S) one immediately obtains the homology and cohomology Conley indices of S.
The Poincaré polynomial of S, denoted by p(t, S), is the formal polynomial Suppose that S has a Morse decomposition M = {M 1 , · · · , M l }. Then the following Morse equation 3. Multiplicity of stationary solutions of evolution equations. In this section we consider the following boundary value problem: where Ω ⊂ R n is a bounded domain with smooth boundary, λ ∈ R, and f ∈ 3.1. Mathematical setting. Let H = L 2 (Ω) and V = H 1 0 (Ω). By (·, ·) and | · | we denote the usual inner product and norm on H, respectively. The norm · on V is defined as Denote A the operator −∆ associated with the homogenous Dirichlet boundary condition. A is a sectorial operator and has a compact resolvent. Denote 0 < µ 1 < µ 2 < · · · < µ k < · · · the eigenvalues of A.
The equation (7) can be reformulated in an abstract form on V : wheref (u) is the Nemitski operator from V to H defined bỹ By the basic theory on evolution equations, see [8,23], the equation (8) is well-posed on V . Specifically, for any u 0 ∈ V , there exists a unique solution u(t) of (8) on the interval [0, T ) for some T > 0.
Denote Φ λ the semiflow generated by (8). Then it follows from (A1) that Φ λ is a global semiflow. Namely, for each u 0 , is the solution of the equation on R + with initial value u(0) = u 0 . It is trivial to see that Φ λ depends on λ continuously. Moreover, using a standard argument (see [20, Chapter I, Theorem 4.4]), one can easily verify that the family Φ λ is λ-l.u.a.c.
Let L = A−µ k , where µ k is an eigenvalue of A. The space H can be decomposed into the orthogonal direct sum of its subspaces H − , H 0 and H + corresponding to the negative, zero and positive eigenvalues of L, respectively. It is trivial to see that both H − and H 0 are finite-dimensional. Denote P σ (σ ∈ {0, ±}) the projection from H to H σ . Set Then one can see that V − and V 0 coincide with H − and H 0 , respectively, as H − and H 0 are both finite-dimensional. We also have Set W := V − ⊕ V 0 , and let P W = P − + P 0 be the projection from V to W . Lemma 3.1. (see [9]). Assume λ ≤ µ k + η, where η = (µ k+1 − µ k )/2. Then there exists ρ 0 > 0 (independent of λ) such that for any solution u = u(t) of (8) on R + , Here u + = P + u.
Lemma 3.3. (see [9]). Let the condition (A1) hold. Then K ∞ (Φ λ ) is uniformly bounded in V for λ ∈ [µ k , µ k + η], and where p is the sum of the multiplicities of the eigenvalues µ i (0 ≤ i ≤ k − 1) of A, and r the multiplicity of µ k .
Furthermore, there is at least one connecting trajectory γ be- 3. Each of the sets C 1 and C ∞ has a connected component Γ with Γ[λ] = ∅ for all λ ∈ Λ − , where

3.2.
Static bifurcation and multiplicity of stationary solutions. Now we pay our attention to the static bifurcation and multiplicity of stationary solutions of (8). Note that the equation (7) has a natural Lyapunov function J(u) defined by Then this problem can be treated via the method of dynamical systems. Proof. In fact, the above result is just a consequence of conclusion 3 in [9, Theorem 5.8]. Here we give a sketch of the proof for the reader's convenience.
where R 0 is the number given in Lemma 3.2. We may restrict δ such that N 1 is also an isolating neighborhood of Φ λ for all λ ∈ Λ : Thus by Lemma 3.3 we deduce that By slightly modifying the proof of [21, Theorem 2.1], it can be shown that there is an open dense subset D of R such that all the equilibria of Φ λ are hyperbolic if λ ∈ D. Now assume λ ∈ Λ − ∩ D. Let M = {M ∞ λ , M 1 λ } be the Morse decomposition of S λ as in Lemma 3.4. Hence, there is at least one equilibrium e ∞ λ ∈ M ∞ λ . We will show that there is another equilibrium z ∞ λ ∈ M ∞ λ with z ∞ λ = e ∞ λ . We argue by contradiction and suppose M ∞ λ consists of exactly one hyperbolic stationary solution e ∞ λ . Then p(t, M ∞ λ ) = t m for some m ≥ 0. Now we consider the Morse equation of M: Since h(Φ λ , M 1 λ ) = Σ p+r and h(Φ λ , S λ ) = Σ p (see (65) in [9]), one has t p+r + t m = t p + (1 + t)Q(t).
But this is impossible for any formal polynomial Q(t) with coefficients in Z + , as the sum of the coefficients of the left-hand side does not equal that of the right-hand side.
Theorem 3.7. Assume f satisfies the hypotheses (A1)-(A2). Denote W c loc (0) the local center manifold of Φ µ k at the equilibrium point 0, and let φ be the restriction of Φ µ k on W c loc (0). Suppose 0 is an isolated equilibrium of Φ µ k (i.e., an isolated stationary solution of (8) at λ = µ k ). Then there exists δ > 0 such that one of the following assertions holds: 1. 0 is an attractor of φ. In this case, for each λ ∈ (Λ − ∪ Λ + ) ∩ D, it has at least four distinct nontrivial equilibria; 2. 0 is a repeller of φ (i.e., an attractor of the inverse flow φ −1 ). When this occurs, for each Λ − ∩ D, Φ λ has at least four distinct nontrivial equilibria; 3. 0 is neither an attractor nor a repeller of φ. In this case, for each λ ∈ Λ − ∩ D, it has at least four distinct nontrivial equilibria.
Proof. In the following argument, we always assume that δ > 0 is sufficiently small so that Lemma 3.4 and 3.5 remain valid.
1. Let N 1 be the isolating neighborhood of S µ k given in the proof of Lemma 3.6. Then by Remark 5, for each λ ∈ Λ − ∩ D, the system Φ λ always has at least two distinct equilibria outside N 1 .
Pick an isolating neighborhood N 0 of 0 with for some β > 0, where B V (β) denotes the ball in V centered at 0 with radius β.
We may restrict δ so that both N 0 and N 1 are isolating neighborhoods of Φ λ for all As N 0 is an isolating neighborhood of 0, one easily verifies that We may assume N 0 is chosen sufficiently small so that the product formula of Conley index given in [20, Chap. II, Theorem 3.1] holds true. Therefore, we have where p is given in Lemma 3.3. By Example 2.10 in [9], one sees that By (15) and Lemma 3.3 we also have Therefore K 1 λ = K 0 λ , and hence For each λ ∈ Λ, pick a v λ ∈ K 1 λ \ N 0 . Let u λ (t) be a bounded full trajectory of Φ λ in K 1 λ with u λ (0) = v λ . We claim that if δ is chosen small enough then either ω(u λ ) Indeed, if not, there would exist a sequence λ n → µ k (as n → ∞) such that both ω(u n ) and ω * (u n ) are contained in N 0 and hence in K 0 λn , where u n = u λn . Thus by (16) we deduce that Then min v∈Γn J(v) = min v∈ω(un) It follows by (22) that max v∈Γn |J(v)| → 0 as n → ∞.
We infer from the asymptotic compactness of the skew-product flow of the family {Φ λ } and Γ n ⊂ K 1 λn ⊂ N 1 that λn∈Λ Γ n is precompact. Hence by Lemma 2.1 it can be assumed that Γ n converges to a nonempty compact invariant set K of Φ µ k (in the sense of Hausdorff distance). Clearly 0 ∈ K. Since each Γ n is connected, K is connected as well. (23) implies that J(v) ≡ 0 on K. Thereby K consists of equilibrium points of Φ µ k . On the other hand, because u n (0) ∈ Γ n \ N 0 (24) for all n, we deduce that K \ int N 0 = ∅. Further by the connectedness of K one concludes that K ∩ ∂V = ∅ for any small neighborhood V of 0, which contradicts the hypothesis that 0 is an isolated equilibrium of Φ µ k , which completes the proof of our claim. By virtue of (21), for each λ ∈ Λ we can pick an equilibrium e c λ ∈ (ω(u λ ) ∪ ω * (u λ )) \ N 0 of Φ λ . Note that e c λ ∈ K 1 λ ⊂ N 1 \ N 0 . Hence for each λ ∈ Λ − , we conclude that Φ λ has at least two nontrivial equilibria e c λ and e ∞ λ . We also infer from the attractor bifurcation theory (see e.g. Ma and Wang [13,Theorem 4.3], [12, Theorem 6.1] or Li and Wang [11,Theorem 4.2]) that K 0 λ contains at least two distinct equilibrium points e 1 λ and e 2 λ for λ ∈ Λ + , provided δ is sufficiently small. In conclusion, Φ λ has at least three distinct nontrivial equilibria for λ ∈ Λ + .
We now assume λ ∈ Λ ∩ D and prove that if δ > 0 is small enough, then Φ λ has at least two distinct nontrivial equilibrium points in N 1 \ N 0 .
Suppose the contrary. Then there would exist a sequence λ n → µ k such that for each n, Φ λ has exactly one equilibrium e n = e c λn in N 1 \ N 0 . There are two possibilities.
(i) lim n→∞ |J(e n )| = 0. When this occurs we first show that there are no connecting orbits between e n and K 0 n = K 0 λn , provided n is sufficiently large. Based on this fact we further verify that and hence M = {{e n }, K 0 n } forms a Morse decomposition of K 1 n . Suppose that there is a subsequence of {n} ∞ n=1 , still denoted by {n} ∞ n=1 , such that for each n, there is a connecting orbit γ n between e n and K 0 n . Let Γ n = γ n . Then Γ n is a connected compact invariant set with e n ∈ Γ n , Γ n ∩ K 0 n = ∅.