DOI 10.1515/anona-2014-0038 Ë Adv. Nonlinear Anal. 2015; 4 (1):13–23

Research Article

Kanishka Perera, Marco Squassina and Yang Yang A note on the Dancer–Fučík spectra of the fractional p-Laplacian and Laplacian operators

Abstract: We study the Dancer–Fučík spectrum of the fractional -Laplacian operator. We construct an unbounded sequence of decreasing curves in the spectrum using a suitable minimax scheme. For , we present a very accurate local analysis. We construct the minimalp and maximal curves of the spectrum locally near the points where it intersects the main diagonal of the plane. We give a sucient conditionp = 2 for the region between them to be nonempty and show that it is free of the spectrum in the case of a simple eigenvalue. Finally, we compute the critical groups in various regions separated by these curves. We compute them precisely in certain regions and prove a shifting theorem that gives a nite-dimensional reduction in certain other regions. This allows us to obtain nontrivial solutions of perturbed problems with nonlinearities crossing a curve of the spectrum via a comparison of the critical groups at zero and innity.

Keywords: Fractional -Laplacian, Dancer–Fučík spectrum, critical groups

MSC 2010: 35R11, 35P30, 35A15 p

ËË Kanishka Perera: Department of Mathematical Sciences, Florida Institute of Technology, Melbourne, FL 32901, USA, e-mail: [email protected] Marco Squassina: Dipartimento di Informatica, Università degli Studi di Verona, 37134 Verona, Italy, e-mail: [email protected] Yang Yang: School of Science, Jiangnan University, Wuxi, 214122, China, e-mail: [email protected]

1 Introduction

s For , and , the fractional -Laplacian p is the nonlinear nonlocal operator dened on smooth functions by p ∈ (1, ∞) s ∈ (0, 1) N > sp p p−2 (−Δ) s N p ù↘0 N+sp ℝ \B (x) |u(x) − u(y)| (u(x) − u(y)) (−Δ) u(x) = 2 lim N X dy, x ∈ ℝ . This denition is consistent, up to a normalizationù constant|x − y| depending on and , with the usual denition of the linear fractional Laplacian operator s when . There is currently a rapidly growing literature on problems involving these nonlocal operators. In particular, fractional N-eigenvalues problems have been studied in Lindgren and Lindqvist [29], Iannizzotto(−Δ) andp = Squassina2 [25] and Franzina and Palatucci [20], regularity of fractional -minimizers in Di Castro, Kuusi and Palatucci [15]p and existence via Morse theory in Iannizzotto, Liu, Perera and Squassina [24]. We refer to Caarelli [6] for the motivations that have lead to their study. p Let be a bounded domain in N with Lipschitz boundary . The Dancer–Fučík spectrum of the s s 2 operator p in is the set p of all points such that the problem Ω ℝ àΩ s + p−1 − p−1 in (−Δ) Ω Σ (Ω) p (a, b) ∈ ℝ (1.1) in N (−Δ) u = b(u ) − a(u ) Ω, where ± are the positive® and negative parts of , respectively, has a nontrivial weak solution. u = 0 ℝ \ Ω, Let us recall the weak formulation of (1.1). Let

u = max{±u, 0} pu 1/p

s,p N+sp ℝ |u(x) − u(y)| [u] = ¤ X2N dx dy¥ |x − y| 14 Ë K. Perera, M. Squassina and Y. Yang, A note on the Dancer–Fučík spectra be the Gagliardo seminorm of the measurable function N and let

s,p N p N u : ℝ →s,pℝ be the fractional Sobolev space endowedW (ℝ with) = the{u ∈ normL (ℝ ) : [u] < ∞}

p p 1/p s,p p s,p

p N where p is the norm in . We work‖u‖ in the= closed(|u| + linear[u] ) subspace,

s s,p N N | ⋅ | L (ℝ ) p a.e. in equivalently renormed by settingX (Ω) = {u ∈s,pW (see(ℝ Di) : Nezza, u = 0 Palatucciℝ \ andΩ} Valdinoci [16, Theorem 7.1]). s A function p is a weak solution of problem (1.1) if ‖ ⋅ ‖ = [ ⋅ ] p−2 u ∈ X (Ω) + p−1 − p−1 s N+sp for all p (1.2) ℝ |u(x) − u(y)| (u(x) − u(y))(v(x) − v(y)) Ω X2N dx dy = X(b(u ) − a(u ) )v dx v ∈ X (Ω). This notion of spectrum|x − y| for linear local elliptic partial dierential operators has been introduced by Dancer [10, 11] and Fučík [21], who recognized its signicance for the solvability of related semilinear boundary value problems. In particular, the Dancer–Fučík spectrum of the Laplacian in with the Dirichlet boundary condition is the set of all points 2 such that the problem Ω + − in Σ(Ω) (a, b) ∈ ℝ (1.3) on −Δu = bu − au Ω, ® has a nontrivial solution. Denoting by k u =the0 Dirichlet eigenvaluesàΩ, of in , the spectrum clearly contains the sequence of points k k . For , where is an interval, Fučík [21] showed that with the periodic boundary condition consistsë ↗ of + a∞ sequence of hyperbolic-like curves−Δ Ω passing throughΣ(Ω) the points

k k , with one or two curves(ë going, ë through) N = each1 point.Ω For , the spectrum consistsΣ(Ω) locally of curves emanating from the points k k (see Gallouët and Kavian [22], Ruf [42], Lazer and McKenna [27], Lazer(ë , ë [26],) Các [5], Magalhães [32], Cuesta and Gossez [9], de FigueiredoN ≥ 2 and GossezΣ(Ω) [14] and Margulies and

Margulies [33]). Schechter [43] showed(ë , ë that) in the square k−1 k+1 k−1 k+1 , the spectrum con- tains two strictly decreasing curves, which may coincide, such that the points in the square that are either below the lower curve or above the upper curve are not in (ë , ë while) × the(ë points, ë between) them mayΣ(Ω) or may not belong to when they do not coincide. p−2 2 The Dancer–Fučík spectrum of the -Laplacian p Σ(Ω) is the set p of all such that the problemΣ(Ω) + p−1 − p−1 p p Δ u = div(|∇u|in ∇u) Σ (Ω) (a, b) ∈ ℝ on −Δ u = b(u ) − a(u ) Ω, has a nontrivial solution. For ® , the Dirichlet spectrum of in consists of a sequence of u = 0 p àΩ, p simple eigenvalues k and p has the same general shape as (see Drábek [17]). For , the rst eigenvalue 1 of p Nis positive,= 1 simple and has anò( associated−Δ ) − eigenfunctionΔ Ω that is positive in (see Anane [2] and Lindqvistë ↗ +∞ [30, 31]),Σ (Ω) so p contains the two lines Σ(Ω)1 and 1. Moreover,N ≥1 2is isolated in the spectrum,ë so−Δ the second eigenvalue 2 p 1 is well-dened (see Anane andΩ Tsouli [3]), and a rst nontrivial curve in pΣ (Ω)passing through 2 2 andë × asymptoticℝ ℝ × toë 1 and ë 1 at innity was constructed using the mountain passë = theoreminf ò(−Δ by) ∩ Cuesta,(ë , ∞) de Figueiredo and Gossez [8].

Although a complete description of Σp (Ω)is not yet available,(ë an, increasingë ) and unboundedë × ℝ sequenceℝ × ë of eigenvalues can be constructed via a standard minimax scheme based on the Krasnosel’ski˘ı genus, or via nonstandard schemes based on the cogenusò(−Δ ) as in Drábek and Robinson [18] and the cohomological index as in Perera [35]. Unbounded sequences of decreasing curves in p , analogous to the lower and upper curves of Schechter [43] in the semilinear case, have been constructed using various minimax schemes by Cuesta [7], Micheletti and Pistoia [34], and Perera [36]. Σ (Ω) K. Perera, M. Squassina and Y. Yang, A note on the Dancer–Fučík spectra Ë 15

Goyal and Sreenadh [23] recently studied the Dancer–Fučík spectrum for a class of linear nonlocal elliptic operators that includes the fractional Laplacian s. As in Cuesta, de Figueiredo and Gossez [8], they con- structed a rst nontrivial curve in the Dancer–Fučík spectrum that passes through 2 2 and is asymptotic to 1 and 1 at innity. Very recently, in [4],(−Δ) the authors proved, among other things, that the second variational eigenvalue 2 is larger than 1 and 1 2 does not contain any other eigenvalues.(ë , ë ) ë The× ℝ purposeℝ × ofë this note is to point out that the general theories developed in Perera, Agarwal and O’Regan [37] and Pereraë and Schechterë [41] apply(ë , ë to) the fractional -Laplacian and Laplacian operators, respectively, and draw some conclusions about their Dancer–Fučík spectra. We construct an unbounded s sequence of decreasing curves in p using a suitable minimax scheme.p For , we present a very accu- rate local analysis. We construct the minimal and maximal curves of the spectrum locally near the points where it intersects the main diagonalΣ (Ω) of the plane. We give a sucient conditionp for= 2 the region between them to be nonempty and show that it is free of the spectrum in the case of a simple eigenvalue. Finally, we compute the critical groups in various regions separated by these curves. We compute them precisely in certain regions and prove a shifting theorem that gives a nite-dimensional reduction in certain other regions. This allows us to obtain nontrivial solutions of perturbed problems with nonlinearities crossing a curve of the spectrum via a comparison of the critical groups at zero and innity.

2 The Dancer–Fučík spectrum of the fractional p-Laplacian

The general theory developed in Perera, Agarwal and O’Regan [37] applies to problem (1.1). Indeed, the odd s s s ∗ s ∗ s -homogeneous operator p p p , where p is the dual of p , dened by

p−2 (p − 1) s A ∈ C(X (Ω), X (Ω) ) X (Ω) X (Ω)s p N+sp p ℝ |u(x) − u(y)| (u(x) − u(y))(v(x) − v(y)) A (u) v = X2N dx dy, u, v ∈ X (Ω), that is associated with the left-hand side of equation|x − y| (1.2) satises

s p s p−1 s p p for all p (2.1)

1 and is the Fréchet derivativeA (u) of u the= ‖u‖-functional, |A (u) v| ≤ ‖u‖ ‖v‖ u, v ∈ X (Ω) p s s p C N+sp p 1 ℝ |u(x) − u(y)| I (u) = X2N dx dy, u ∈ X (Ω). s p |x − y| s Moreover, since p is uniformly convex, it follows from (2.1) that p is of type (S), i.e., every sequence s j p such that s X (Ω) j p j j A (u ) ⊂ X (Ω) has a subsequence that converges strongly to (see [37, Proposition 1.3]). Hence, the operator s satises the u ⇀ u, A (u )(u − u) → 0 p structural assumptions of [37, Chapter 1]. When , problem (1.1) reduces tou the nonlinear eigenvalue problem A

s p−2 in a = b = ë p (2.2) in N (−Δ) u = ë|u| u Ω, ® Eigenvalues of this problem coincide with criticalu = 0 values of theℝ functional\ Ω. −1 p

Ω Ψ(u) = ¤X |u| dx¥ on the manifold s M p

= {u ∈ X (Ω) : ‖u‖ = 1}. 16 Ë K. Perera, M. Squassina and Y. Yang, A note on the Dancer–Fučík spectra

The rst eigenvalue

1 u∈M is positive, simple, isolated and has an associated eigenfunction that is positive in (see Lindgren and ë = infs Ψ(u) Lindqvist [29] and Franzina and Palatucci [20]), so p contains the two lines 1 and 1. Let F denote the class of symmetric subsets of M, let denote the 2-cohomological indexΩ of F (see Fadell and Rabinowitz [19]) and set Σ (Ω) ë × ℝ ℝ × ë

k i(M) ℤ M ∈ M∈F u∈M i(M)≥k s Then, k is a sequence of eigenvaluesë := ofinf problemsup Ψ(u), (2.2) (see k ≥ 2. [37, Theorem 4.6]), so p contains the sequence of points k k . s Followingë ↗ +∞ [37, Chapter 8], we now construct an unbounded sequence of decreasingΣ curves(Ω) in p . For , let (ë , ë ) −1 Σ (Ω) + p − p M t > 0 t Ω s Then, the point p if andΨ (u) only= ¤ ifX((uis a) critical+ t(u value) ) dx of¥ ,t (see u ∈ [37,. Lemma 8.3]). For each such that k k−1, let ë ë ë (c, ct) ∈ Σ (Ω) t c t t Ψ k ≥ 2 ë > ë ë k−1 k−1 k−1 ë be the cone on the sublevel set t M t k−1 , let k denote the class of maps t M k−1 CΨ = (Ψ × [0, 1])/(Ψ × {1}) k−1 such that Ψt is the identity and set

ëk−1 Ψ = {u ∈s :Ψ (u) ≤ ë } Γ ã ∈ C(CΨ , ) t ã| k ã∈Γ u∈ã(CΨ ) k ëk−1 We have the following theorem as a specialc (t) case= inf of [37,sup Theoremt Ψ (u). 8.8].

Theorem 2.1. Let s s s k−1 k Ck k k k k−1 Then, C is a decreasing continuous curve in s and ës . ë k = –(c (t),p c (t)t) : k < tk< —. ë ë Σ (Ω) c (1) ≥ ë 3 The Dancer–Fučík spectrum of the fractional Laplacian

The Dancer–Fučík spectrum of the operator s in is the set s of all points 2 such that the problem s (−Δ)+ Ω− in Σ (Ω) (a, b) ∈ ℝ (3.1) in N (−Δ) u = bu − au Ω, has a nontrivial weak solution. The® general theory developed in Perera and Schechter [41] applies to prob- s s u =s 0 ℝ \ Ω, lem (3.1). Indeed, set 2 and let be the inverse of the solution operator 2 2 s X (Ω) = X (Ω) A of the problem S:L (Ω) →s S(L (Ω)) ⊂inX (Ω), f Ü→ u, in N (−Δ) u = f(x) Ω, Then, s is a self-adjoint operator on ®2 and we have u = 0 ℝ \ Ω. s/2 s/2 for all s A 2 L (Ω) N+2s ℝ (u(x) − u(y))(v(x) − v(y)) and (u, v) = (A u, A v) = X2N dx dy u, v ∈ X (Ω) |x − y|2 1/2 s/2 s 2 N+2s for all ℝ |u(x) − u(y)| ‖u‖ = ‖A u‖ = ¤ X2N dx dy¥ u ∈ X (Ω), |x − y| K. Perera, M. Squassina and Y. Yang, A note on the Dancer–Fučík spectra Ë 17

s 2 where and 2 are the inner products in and , respectively. Moreover, its spectrum s and s −1 2 2 is a compact operator since the embedding s 2 is com- s pact. Thus,( ⋅ , ⋅ ) consists( ⋅ , ⋅ ) of isolated eigenvalues k,X (Ω), of niteL (Ω) multiplicities satisfying 1 2 . ⊥ s Theò(A rst) ⊂ (0, eigenvalue ∞) (A1)is simple:L (Ω) and→ L has(Ω) an associated eigenfunction 1 and if X (Ω)1 í→ L (Ω) , then ò(A ) ë k ≥ 1 0 < ë < ë < ⋅ ⋅ ⋅ s s ë 1 1 2 1 2 ÿ1 > 0 1 2 w ∈ ((ℝÿ ) ∩ X (Ω)) \ {0} so ± . Hence, the operator s satises all the assumptions of [41, Chapter 4]. 0 = (w, ÿ ) = (A w, ÿ ) = (w, A ÿ ) = ë (w, ÿ ) , Now, we describe the minimal and maximal curves of s in the square w ̸= 0 A 2 k k−1 k+1 Σ (Ω)

1 constructed in [41]. Weak solutions of problemQ = (ë (3.1), coincide ë ) , with k ≥ 2, critical points of the -functional 2 + 2 − 2 Cs N+2s 1 ℝ |u(x) − u(y)| 1 Ω I(u, a, b) = X2N dx dy − X(b(u ) + a(u ) )dx, u ∈ X (Ω). Denote by k the eigenspace2 of k and|x − sety| 2

k E ë ⊥ s k j k k j=1

s N = L E ,M = N ∩ X (Ω). Then, k k is an orthogonal decomposition with respect to both and 2. When k, k−1 k k is strictly concave in and strictly convex in , i.e., if 1 2 k−1, Xk−1(Ω), then= N ⊕M ( ⋅ , ⋅ ) ( ⋅ , ⋅ ) (a, b) ∈ Q I(v + y + w), v + y + w ∈ N ⊕ E ⊕ M v w v ̸= v ∈ N 1 2 1 2 for all w ∈ M and if , , then k 1 I((12 − kt)v + tv + w) > (1 − t)I(v + w) + tI(v + w) t ∈ (0, 1)

v ∈ N w ̸= w ∈ M 1 2 1 2 for all

(see [41, PropositionI(v 4.6.1]).+ (1 − t)w + tw ) < (1 − t)I(v + w ) + tI(v + w ) t ∈ (0, 1)

Proposition 3.1 ([41, Proposition 4.7.1, Corollary 4.7.3, Proposition 4.7.4]). Let k. (i) There is a positive homogeneous map k−1 k−1 such that is the unique solution of (a, b) ∈ Q ὔ è( ⋅ , a, b) ∈ C(M ,N ) k−1 v = è(w) v ∈N

ὔ ὔ I(v + w) = supk−1 I(v + w), w ∈ M . Moreover, k−1 if and only if . Furthermore, the map is continuous on k−1 k and satises k k for all k−1. (ii) There is a positiveI (v + w) homogeneous⊥ N map v = è(w) k k such that è is the uniqueM solution× Q of è(w, ë , ë ) = 0 w ∈ M ὔ ó( ⋅ , a, b) ∈ C(N ,M ) kw = ó(v) w ∈M

ὔ ὔ I(v + w) = inf k I(v + w ), v ∈ N . Moreover, k if and only if . Furthermore, the map is continuous on k k and satises k k for all k. I (v + w) ⊥ M w = ó(v) ó N × Q For , let ó(v,k ë , ë ) = 0 v ∈ N k−1 k k−1 (a, b) ∈ Q k k k ò(w, a, b) = è(w, a, b) + w, w ∈ M ,S (a, b) = ò(M , a, b), k Then, k and are topological manifolds modeled on k−1 and k, respectively. Thus, k is innite-dimen- k æ(v, a, b) = v + ó(v, a, b), v ∈ N ,S s(a, b) = æ(N , a, b). sional, while is k-dimensional, where k k. For , set . We say that is a radialS setS if . Since and areM positive homogeneous,N so are Sand and hence k and k are radialS manifolds.d d = dim N B ⊂ X (Ω) B̃ = {u ∈ B : ‖u‖ = 1} B B = {tu : u ∈ B,̃ t ≥ 0} è ó ò æ S S 18 Ë K. Perera, M. Squassina and Y. Yang, A note on the Dancer–Fučík spectra

Let s ὔ be the set of critical points of . Since ὔ is positive homogeneous, it follows that is a radial set. K(a, b) = {u ∈ X (Ω) : I (u, a, b) = 0} As ὔ , we have I( ⋅ , a, b) I for all K (3.2) I(u) = (I (u), u)/2 Since s , Proposition 3.1 implies k−1 k k I(u) = 0 u ∈ K. k ὔ (3.3) X (Ω) = N ⊕ E ⊕ M k k Together with (3.2), it also implies K = {u ∈ S ∩ S k:I (u) ⊥ E }. k (3.4) Set K ⊂ {u ∈ S ∩ S : I(u) = 0}. k−1 k ̃ w∈M w∈M v∈N v∈Ñ

k−1 k−1 k Since is nonincreasingn (a, b) = inf in supfor xedI(v + w,and a, b),and m (a, in b)for= sup xedk inf andI(v + w,, it a, follows b). that k−1 and k are nonincreasing in for xed and in for xed . By Proposition 3.1, I(u, a, b) a u b b u a n (a, b) k−1 k m (a, b) a ̃ b b a w∈M v∈Ñ

n (a, b) = infk−1 I(ò(w, a, b), a, b), m (a, b) = supk I(æ(v, a, b), a, b).ὔ ὔ Proposition 3.2 ([41, Proposition 4.7.5, Lemma 4.7.6, Proposition 4.7.7]). Let k. (i) Assume that k−1 . Then, (a, b), (a , b ) ∈ Q for all n (a, b) = 0 k k and s . ὔ ὔ I(u, a, b) ≥ὔ 0 ὔ u ∈ S (a, b),ὔ K(a,ὔ b) = {u ∈ S (a, b) : I(u, a, b) = 0} (a) If , and , then k−1 , (a, b) ∈ Σ (Ω) ὔ ὔ for all ὔ ὔ a ≤ a b ≤ b (a , b ) ̸= (a, b) n (a , b ) > 0 k and ὔ ὔ s . ὔ ὔ ὔ ὔ I(u, a , b ) > 0 ὔ ὔ u ∈ S (a , b ) \ {0} ὔ ὔ (b) If , and , then k−1 and there is some k such that (a , b ) ∉ Σ (Ω) ὔ ὔ a ≥ a b ≥ b (a , b ) ̸= (a, b) n (a , b ) < 0 u ∈ S (a , b ) \ {0} Furthermore, is continuous on and . k−1 k k−1 I(u,k ak , b ) < 0. (ii) Assume that k . Then, n Q n (ë , ë ) = 0 for all k k m (a, b) = 0 and s . ὔ ὔ I(u, a, b) ≤ὔ 0 ὔ u ∈ S (a, b),ὔ K(a,ὔ b) = {u ∈ S (a, b) : I(u, a, b) = 0} (a) If , and , then k , (a, b) ∈ Σ (Ω) ὔ ὔ for all k ὔ ὔ a ≥ a b ≥ b (a , b ) ̸= (a, b) m (a , b ) < 0 and ὔ ὔ s . ὔ ὔ ὔ ὔ I(u, a , b ) < 0 ὔ ὔ u ∈ S (a , b ) \ {0} k ὔ ὔ (b) If , and , then k and there is some such that (a , b ) ∉ Σ (Ω) ὔ ὔ a ≤ a b ≤ b (a , b ) ̸= (a, b) m (a , b ) > 0 u ∈ S (a , b ) \ {0} Furthermore, is continuous on and . k k k k I(u,k a , b ) > 0. For , set k−1 k+1m Q m (ë , ë ) = 0

a ∈ (ë k−,1 ë ) k−1 k+1 k−1 k k−1 k+1 k Then, í (a) = sup{b ∈ (ë , ë ) : n (a, b) ≥ 0}, ì (a) = inf{b ∈ (ë , ë ) : m (a, b) ≤ 0}. k−1 k−1 k k (see [41, Lemma 4.7.8]). b = í (a) ⇐⇒ n (a, b) = 0, b = ì (a) ⇐⇒ m (a, b) = 0 K. Perera, M. Squassina and Y. Yang, A note on the Dancer–Fučík spectra Ë 19

Theorem 3.3 ([41, Theorem 4.7.9]). Let k. (i) The function k−1 is continuous, strictly decreasing and satises (a) k−1 k k, (a, b) ∈ Q s (b) k−1 í , s (c) í (ëk−1) = ë . (ii) The functionb = í (a)k isâ⇒ continuous,(a, b) ∈ Σ strictly(Ω) decreasing and satises (a) bk< ík (a)k,â⇒ (a, b) ∉ Σ (Ω) s (b) k ì , s (c) ì (ë k) = ë . (iii) k−1 b = ì k(a) .â⇒ (a, b) ∈ Σ (Ω) b > ì (a) â⇒ (a, b) ∉ Σ (Ω) Thus, í (a) ≤ ì (a) k k k−1 k are strictly decreasing curves in that belong to s . They both pass through the point and may k C : b = í (a), C : b = ì (a) k k coincide. The region

Q k Σ (Ω)k k−1 (ë , ë ) below the lower curve and the region k I = {(a, b) ∈ Q : b < í (a)} k C k k

k s s above the upper curve are free of I .= They{(a, b) are∈ theQ : minimal b > ì (a)} and maximal curves of in k in this sense. Points in the region

C kΣ (Ω) k k−1 k Σ (Ω) Q between and k, when it is nonempty, may or may not belong to s . k II = {(a, b) ∈ Q : í (a) < b < ì (a)} For k, let k C C Nk k Σ (Ω) (a, b) ∈ Q Since and k are radial sets, so is N . The next two propositions show that N is a topological manifold k k (a, b) = S (a, b) ∩ S (a, b). k modeled on k and hence S S Nk k k−1 E We will call it the null manifold of . dim = d − d . Proposition 3.4 . Let . ([41, Proposition 4.8.1,I Lemma 4.8.3, Proposition 4.8.4]) k (i) There is a positive homogeneous map k k−1 such that is the unique solution of (a, b) ∈ Q ὔ ç( ⋅ , a, b) ∈ C(E ,N ) vk= ç(y) v ∈N

ὔ ὔ I(æ(v + y)) = supk−1 I(æ(v + y)), y ∈ E . Moreover, k−1 if and only if . Furthermore, the map is continuous on k k and satises k k for all k. (ii) There is a positiveI (æ(v + homogeneousy)) ⊥ N map v = ç(y) k k such that ç is the unique solutionE × Q of ç(y, ë , ë ) = 0 y ∈ E ὔ î( ⋅ , a, b) ∈ C(E ,M ) w =k î(y) w ∈M

ὔ ὔ I(ò(y + w)) = inf k I(ò(y + w )), y ∈ E . Moreover, k if and only if . Furthermore, the map is continuous on k k and satises k k for all k. (iii) For all I (ò(yk, + w)) ⊥ M w = î(y) î E × Q î(y, ë , ë ) = 0 y ∈ E y ∈ E i.e., and . æ(ç(y) + y) = ò(y + î(y)), Let ç(y) = è(y + î(y)) î(y) = ó(ç(y) + y) k

ÿ(y) = æ(ç(y) + y) = ò(y + î(y)), y ∈ E . 20 Ë K. Perera, M. Squassina and Y. Yang, A note on the Dancer–Fučík spectra

Proposition 3.5 ([41, Proposition 4.8.5]). Let k. s (i) k is a positive homogeneous map such that (a, b) ∈ Q ÿ( ⋅ , a, b) ∈ C(E ,X (Ω)) k w∈M v∈N v∈N w∈M

k k ὔ I(ÿ(y)) = inf supk−1 I(v + y + w) = supk−1 inf I(v + y + w), y ∈ E , and k for all k. ὔ ὔ ὔ ὔ (ii) If k with and , then I (ÿ(y)) ∈ E y ∈ E ὔ ὔ ὔ ὔ (a , b ) ∈ Q a ≥ a b ≥ b for all k

(iii) is continuous on k kI(ÿ(y,. a , b ), a , b ) ≤ I(ÿ(y, a, b), a, b) y ∈ E . (iv) k k for all k. (v) ÿNk E × Q k . (vi) ÿ(y,Nk ëk , ëk ) = yk. y ∈ E (a, b) = {ÿ(y, a, b) : y ∈ E } By (3.3) and (3.4), (ë , ë ) = E ὔ Nk k Nk (3.5) The following theorem shows that the curves and k are closely related to . K = {u ∈ :I (u)k ⊥ E } ⊂ {u ∈ : I(u) = 0}. N Theorem 3.6 . Let . k ([41, Theorem 4.8.6]) C k C Ĩ = I| (i) If k−1 , then (a, b) ∈ Q for all Nk b < í (a) (ii) If , then k−1 I(u,̃ a, b) > 0 u ∈ (a, b) \ {0}.

b = í (a) for all Nk Nk

(iii) If k−1 I(u,k̃ a,, then b) ≥ there0 are ui ∈ Nk(a, b), K(a,, b) = ,{u such∈ that(a, b) : I(u,̃ a, b) = 0}.

í (a) < b < ì (a) u ∈ 1(a, b) \ {0} i = 1,2 2

(iv) If k , then I(ũ , a, b) < 0 < I(ũ , a, b).

b = ì (a) for all Nk Nk

(v) If k , thenI(u,̃ a, b) ≤ 0 u ∈ (a, b), K(a, b) = {u ∈ (a, b) : I(u,̃ a, b) = 0}. for all Nk b > ì (a) By (3.5), solutions of (3.1) are in Nk. Thẽ set of solutions is all of Nk exactly when k is on k I(u, a, b) < 0 u ∈ (a, b) \ {0}. both k and (see [41, Theorem 4.8.7]). When k is a simple eigenvalue, Nk is -dimensional and hence this implies that is on exactly one of thoseK(a, curves b) if and only if (a, b) (a, b) ∈ Q C C ë 1 (a, b) 0 for some 0 k (see [41, CorollaryK(a, 4.8.8]). b) = {tÿ(y , a, b) : t ≥ 0} The following theorem gives a sucient condition for the region k to be nonempty. y ∈ E \ {0} Theorem 3.7 . If there is a function such that + − , then there is a neighbor- ([41, Theorem 4.9.1]) k II 2 2 hood k of k k such that every point k k with k is in k. y ∈ E |y | ̸= |y | For the local problem (1.3), this result is due to Li, Li and Liu [28]. When is a simple eigenvalue, the region N ⊂ Q (ë , ë ) (a, b) ∈ N \ {(ë , ë )} ak + b = 2ë II k is free of s (see [41, Theorem 4.10.1]). For problem (1.3), this is due to Gallouët and Kavian [22]. When s , is the only critical point of and its criticalë groups are given by II Σ (Ω) 0 0 (a, b) ∉ Σ (Ω) 0 q q I 0 s where and C (I,denotes 0) = H singular(I ,I \homology. {0}), q ≥ 0, We take the coecient group to be the eld 2. The following theorem gives our main results on the critical groups. I = {u ∈ X (Ω) : I(u) ≤ 0} H ℤ K. Perera, M. Squassina and Y. Yang, A note on the Dancer–Fučík spectra Ë 21

s Theorem 3.8 ([41, Theorem 4.11.2]). Let k . (i) If k, then (a, b) ∈ Qq \Σ (Ω)qd 2 (a, b) ∈ I (ii) If k, then k−1 C (I, 0) ≈ ä ℤ . q qd 2 (a, b) ∈ I (iii) If , then k k C (I, 0) ≈ ä ℤ . q k−1 or k (a, b) ∈ II and C (I, 0) = 0, q ≤ d q ≥ d q q−d −1 Nk k−1 k

where denotes the reduced homologyk groups.−1 In particular, for all when is simple. C (I, 0) ≈ H̃ ({u ∈ : I(u) < 0}),q d < q < d , k For the local problem (1.3), this result is due to Dancer [12, 13] and Perera and Schechter [38–40]. It can be H̃ C (I, 0) = 0 q ë used, for example, to obtain nontrivial solutions of perturbed problems with nonlinearities that cross a curve of the Dancer–Fučík spectrum, via a comparison of the critical groups at zero and innity. Consider the problem s in (3.6) in N (−Δ) u = f(x, u) Ω, where is a Carathéodory function on® . u = 0 ℝ \ Ω, Theorem 3.9 . If f ([41, Theorem 5.6.1]) Ω × ℝ

+ − 0 0 as + − as .b t − a t + o(t) t → 0, f(x, t) = > uniformly a.e. in for some and bt −inat + o(t)s that|t| are→ on∞, opposite sides of or k, then 0 0 F k k problem (3.6) has a nontrivial weak solution. Ω (a , b ) (a, b) Q \Σ (Ω) C C For problem (1.3), this was proved in Perera and Schechter [39]. It generalizes a well-known result of Amann and Zehnder [1] on the existence of nontrivial solutions for problems crossing an eigenvalue.

Acknowledgement: This work was completed while the third-named author was visiting the Department of Mathematical Sciences at the Florida Institute of Technology and she is grateful for the kind hospitality of the department.

Funding: The second-named author was supported by the 2009 MIUR project “Variational and Topological Methods in the Study of Nonlinear Phenomena”. The third-named author was supported by NSFC-Tian Yuan Special Foundation (No. 11226116), the Natural Science Foundation of the Jiangsu Province of China for Young Scholars (No. BK2012109) and the China Scholarship Council (No. 201208320435).

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Received October 1, 2014; accepted October 6, 2014.