EXISTENCE OF SOLUTIONS FOR KIRCHHOFF TYPE PROBLEMS WITH RESONANCE AT HIGHER EIGENVALUES

. We study the following Kirchhoﬀ type problem: Note that F ( x,t ) = (cid:82) 10 f ( x,s ) ds is the primitive function of f . In the ﬁrst result, we prove the existence of solutions by applying the G − Linking Theorem when the quotient 4 F ( x,t ) bt 4 stays between µ k and µ k +1 allowing for resonance with µ k +1 at inﬁnity. In the second result, for the case that the quotient 4 F ( x,t ) bt 4 stays between µ 1 and µ (cid:48) 2 allowing for resonance with µ (cid:48) 2 at inﬁnity, we ﬁnd a nontrivial solution by using the classical Linking Theorem and argument of the characterization of µ (cid:48) 2 . Meanwhile, similar results are obtained for degenerate problem.

1. Introduction and main results. We consider the following nonlocal Kirchhoff type problem with Dirichlet boundary condition − a + b Ω |∇u| 2 dx ∆u = f (x, u), in Ω, where Ω is a bounded domain in R N (N = 1, 2, 3) with a smooth boundary ∂Ω, a ≥ 0, b > 0 are real constants and f :Ω × R → R is a Caratheodory function that satisfies the subcritical growth condition: where C is a positive constant, 2 * = 6, N = 3, +∞, N = 1, 2. Let F (x, t) = t 0 f (x, s)ds be the primitive function of f . The Euler functional corresponding to problem (1) is In particular, if a = 0, then Since f satisfies the subcritical growth condition (f 0 ), we know at once that I a ∈ C 1 (H, R) and finding weak solutions of problem (1) is equivalent to finding critical points of functional I a in H. Note that Ω is a bounded domain in R N which means the embedding H → L s (Ω) is continuous for s ∈ [1, 2 * ], compact for s ∈ [1, 2 * ). Hence, for s ∈ [1, 2 * ], there exists τ s > 0 such that where u L s denotes the norm of L s (Ω). Problem (1) is related to the stationary analogue of the equation which was first proposed by Kirchhoff [9] in 1883 to describe the transversal oscillations of a stretched string, where u denotes the displacement, f is the external force, b represents the initial tension, and a is related to the intrinsic properties of the string. Some early classical studies of Kirchhoff equations were given by Bernstein [2] and Pohožaev [16]. However, (1) received much attention only after Lions [12] proposed an abstract framework to the problem. It is pointed out in Chipot-Lovat [5] that (1) models several physical and biological systems where u describes a process which depends on the average of itself (for example, the population density).
To state the assumptions, we now recall the definition and some basic properties of eigenvalues about the following two problems: and − u 2 u = µu 3 , in Ω, u = 0, on ∂Ω.
Denote by 0 < λ 1 < λ 2 ≤ · · · ≤ λ k · · · the eigenvalues of (3) and by ϕ 1 , ϕ 2 , ϕ 3 · · · the normalized eigenfunctions. It is well known that the sequence {ϕ k } k∈N is an orthonormal basis of L 2 (Ω) and { ϕ k √ λ k } k∈N is an orthonormal basis of H. Moreover, λ 1 is the smallest, called the principal eigenvalue. It is simple, isolated and it is the only eigenvalue with an eigenfunction of constant sign ϕ 1 > 0. So, in terms of the Lagrange multiplier rule, we know that λ 1 can be characterized as For u ∈ H, one has Nevertheless, H can be split with and As shown in [10], µ 1 > 0 is the principal eigenvalue of (4) and ψ 1 > 0 is an eigenfunction corresponding to µ 1 with ψ 1 L 4 = 1.
Meanwhile, as shown in [17], problem (4) has a sequence of eigenvalues with the variational characterization where Σ k = {Λ ⊂ Σ : there exists a continuous, odd and surjective h : S k−1 → Λ} and S k−1 denotes the unit sphere in R k . Besides µ 1 = µ 1 , it is not known whether the sequence {µ k } k∈N contains all eigenvalues of problem (4). Let We may call f is "3-sublinear " at infinity if µ = 0, f is "3-superlinear " at infinity if µ = ∞ and f is"asymptotically 3-linear" at infinity if 0 < µ < +∞. Especially, we say problem (1) is resonant near infinity at µ k if µ = µ k in (7). We mention the works for the case of "3-superlinear" at infinity in [4,13,18,19,21,24]; for the case of "3-sublinear" at infinity in [24]. The case of "asymptotically 3-linear" at infinity can be found in [4,3,8,10,11,14,17,18,22,23,24]. In these papers, we may see, the Euler functional I a associated with problem (1) is coercive when µ < µ 1 , see [22,23,24]. It is still true that the functional I a is coercive if (7) is satisfied with µ = µ 1 and the following condition: [22,23]). In such case, problem (1) is resonant near infinity at µ 1 from the left side. So, if µ ≤ µ 1 , the existence of a global minimum is obtained because I a is coercive and weakly lower semicontinuous on H. Furthermore, if f satisfies some conditions near zero, multiplicity of solutions for problem (1) was obtained by using Local Linking Theorem or invariant sets of descent flow method, see [22,23,24]. It is interesting to inspect what happens if we allow µ > µ 1 in (7). See, for example, in [3], the author handled the case that 2F (x, t) at 2 < λ 1 uniformly in x ∈ Ω and obtained the existence of at least one nontrivial solution by the Mountain Pass Theorem. We also refer readers to [8,11] for similar results. If µ ∈ (µ k , µ k+1 ) in (7) (nonresonance at infinity), nontrivial solutions of problem (1) were obtained by using the Yang index and critical groups in [10,18]. We refer to [17] for the case of resonance at an arbitrary eigenvalue under a Landesman-Lazer type condition.
The main goal of this paper is to establish the existence of weak solutions for problem (1) when the quotient 4F (x,t) bt 4 exhibits between µ k and µ k+1 at infinity. We introduce the main results as follows: Assume that a > 0 and µ k < µ k+1 are two consecutive eigenvalues of problem (4). If the nonlinearity f satisfies (f 0 ) and the following conditions: then problem (1) has at least a weak solution in H.

Remark 1.
We do not assume in (f 2 ) that the limit as |t| → ∞ of the quotient necessarily exists and moreover, we assume that lim inf t→∞ is bigger that µ k , allowing for resonance with respect to µ k+1 and nonresonance with respect to µ k which implies the Euler functional I a is not coercive at infinity. The possibility of resonance with respect to µ k+1 necessitates an additional asymptotic condition, and this is (f 3 ). We point out the characterization of variational eigenvalues µ k defined by (6) plays an essential role in the geometry of the Euler functional associated to problem (1) under (f 2 ). But it becomes invalid in such case that the quotient 4F (x,t) bt 4 stays between µ k and µ k+1 , allowing for resonance with respect to µ k and nonresonance with respect to µ k+1 , that is , The next result is related to a second eigenvalue of problem (4). For our purpose, the characterization of µ 2 defined by (6) is not convenient. Instead, we will use an alternative one defined as follows which will be proved in the next section. Let We are now ready to state our results.

Remark 2.
If the left inequality of (f 4 ) is strict, we see, (f 5 ) is redundant. Moreover, from (f 4 ), the following two conditions are permitted at the same time which means problem (1) may be doubly resonant at infinity.

Remark 3.
There are functions f (x, t) satisfying the assumptions in Theorem 1.2.
For example, let In the following, we consider problem (1) with a = 0, that is, Problem (8) is called degenerate elliptic Kirchhoff equation. Such degenerate partial differential equations arise naturally because nonlinear elliptic equations often may have vanishing coefficients or small parameters in higher order partial derivatives. We state the results as follows: Theorem 1.3. Assume µ k < µ k+1 are two consecutive eigenvalues of problem (4). Problem (8) has at least a weak solution in H if the nonlinearity f satisfies (f 0 ), (f 1 ) and (f 2 ).
Besides, assume that f satisfies

problem (8) has at least a weak solution in H.
Furthermore, if f satisfies f (x, 0) = 0 and the following conditions: then problem (8) has at least a nontrivial solution in H.

Remark 4.
It is clear that the assumption (f 4 ) implies problem (8) may be doubly resonant at infinity. If the left inequality of (f 4 ) is strict, then (f 8 ) is redundant.
2. Construction of a second eigenvalue. This section is devoted to the construction of a second eigenvalue µ 2 which has been introduced ahead.
then {u n } possesses a convergent subsequence. Note that the norm of the derivative at u ∈ Σ of Φ| Σ is defined as where . H * denotes the norm on the dual space H * . Let {u n } ⊂ Σ and {t n } ⊂ R be sequences such that, for some constant c, and for all v ∈ H and ε n → 0. From (11), we see that {u n } is bounded in H. Consequently, for a subsequence, u n u weakly in H and u n → u strongly in L 4 (Ω).
The relation in (12) (with v = u n ) implies that {t n } remains bounded. Thus, from (2) and letting v = u n − u in (12), one has So, we deduce from (14) that In addition, from (13), we know Then, we have (15) and (16). Next, we turn to the study of the geometry of Φ| Σ . It is clear that Γ 0 defined by (10) is nonempty (take, e.g., ψ ∈ H with ψ ∈ span{ψ 1 }, consider the path For this purpose, we introduce the continuous map ϑ : H → R defined by which is a closed linear subspace of H. We see We define the following quantity We claim µ V > µ 1 . and It follows from (19) that the sequence {u n } is bounded and so we may assume u n u weakly in H, u n → u strongly in L 4 (Ω).
3. Proof of the Theorems. We introduce the symmetric cone [20]) Let Q be a submanifold of a Banach space X with relative boundary ∂Q, S be a closed subset of a Banach space Y and G be a subset of C 0 (∂Q, Y \ S). We say S and ∂Q are G−linking if for any map h ∈ C 0 (Q, Y ) such that h | ∂Q ∈ G there holds h(Q) ∩ S = ∅. (c) I satisfies the Cerami condition which is stated in [1]. Then, the number defines a critical values c ≥ β of I.
Remark 5. Similar conclusions can be found in [20] if we replace the Cerami condition by the Palais-Smale condition. The conclusion in [20] is based on a linking structure and on a deformation lemma which can be ensured by the Palais-Smale condition. It was shown in [1] that the Cerami condition suffices to get the deformation lemma. Thus, we deduce immediately that Theorem 3.2 is still correct. With the same reasoning, the classical Linking Theorem (see [15] Theorem 8.4) holds true under the Cerami condition.
Our approach is variational based on the critical point theory for C 1 −functionals. Specifically, we shall look for some weak solutions by applying the classical Linking Theorem and Theorem 3.2 (G−Linking theorem). Furthermore, in order to show the weak solutions are nonzero, we shall make more accurate estimate on the energy functional in the proofs of Theorem 1.2 and Theorem 1.4. We divide the proofs into two cases.
Notation: Throughout the paper, we denote by c and c i various positive constants which may vary from place to place. Case I: a > 0.
In order to prove our theorems, we first verify that I a satisfies the compactness condition. Proof. Let {u n } ⊂ H be a bounded Cerami sequence of I a . By the reflexivity of H, we can assume that there exists a function u ∈ H such that u n u weakly in H, u n → u strongly in L p (Ω), u n (x) → u(x) a.e. x ∈ Ω. (23) By (f 0 ) and (2), we see Since {u n } is a bounded Cerami sequence of I a , we obtain | I a (u n ), u − u n | ≤ I a (u n ) H * u − u n ≤ I a (u n ) H * u + I a (u n ) H * u n → 0 as n → ∞.
Then, we have (24) and (25). Hence, u n → u in H due to the uniform convexity of H.

Remark 6.
It is easy to see Lemma 3.3 is still correct for a = 0.
Due to Lemma 1, it suffices to show that {u n } is bounded in H. For this we suppose by contradiction that, passing to subsequence, u n → ∞ as n → ∞.
(27) On the other hand, set u n = φ n + w n where φ n ∈ E 1 and w n ∈ E ⊥ 1 . From (f 0 ) and (f 3 ), there exists a constant c 1 > 0 such that for all t ∈ R and x ∈ Ω. By the definition of E 1 and E ⊥ 1 , one has u n 2 = φ n 2 + w n 2 , u n 2 L 2 = φ n 2 L 2 + w n 2 L 2 . Then, combining (26) and (28), we see where |Ω| denotes the Lebesgue measure of Ω. The above inequality shows w n ≤ c.
Proof of Theorem 1.1. By (f 0 ) and the left inequality of (f 2 ), there exist an ε 1 > 0 with µ k + ε 1 < µ k+1 and a positive constant c = c(ε 1 ) > 0 such that It follows from the definition of µ k in (6) that there exists some Λ 1 ∈ Σ k such that From (29) and (30), for any u ∈ Λ 1 and t > 0, we have On the other hand, defining for all x ∈ Ω and |s| > C M . Hence, for s > C M , from the above inequality, we have d ds Integrating the above expression over the interval [t, T ] ⊂ (C M , +∞), one has for t ≤ −C M and x ∈ Ω. Hence, we have by the arbitrariness of M , that Combining (f 0 ) with the above equality, we find a positive constant c such that Then, for any u ∈ C k+1 , one has Notice that (31) implies I a (tu) → −∞ as t → +∞ uniformly for u ∈ Λ 1 , which, combining (32), shows immediately that there exists a constant ρ 1 > 0 large enough such that where S k−1 is the boundary of the closed unit ball B k in R k , that is S k−1 = ∂B k .
We claim that C k+1 and S k−1 are G−Linking. In fact, for any h ∈ G, by (33), we get h(S k−1 ) ∩ C k+1 = ∅, which shows that G is a subset of C(S k−1 , H \ C k+1 ). On the other hand, letting we have the following conclusion: Γ 1 is nonempty and h(B k ) ∩ C k+1 = ∅ for all h ∈ Γ 1 . The proof may be found in [7]. We include it for completeness.
Moreover, for any h ∈ Γ 1 , if 0 ∈ h(B k ), one has h(B k ) ∩ C k+1 = ∅. Otherwise, let π : H\{0} → Σ be defined by (the radial retraction of H\{0} onto Σ). Consider the maph : Furthermore, (b) of the G−Linking Theorem is satisfied from (33), and (c) holds by Lemma 3.4. As to (a) of the G−Linking Theorem, it is easy to verify from the compactness of B k . Therefore, Theorem 1.1 holds with the critical value Proof of Theorem 1.2. From Lemma 3.4, we need only to study the "geometry" of I a . Define one has lim |t|→∞ G 2 (x, t) = −∞ uniformly for x ∈ Ω and then G 2 (x, t) < c for some c > 0. For any u ∈ D, we deduce easily that On the other hand, we deduce from (f 5 ) that I a (tψ 1 ) → −∞ as t → +∞, ∀u ∈ span{ψ 1 }.
Case II: a=0. Proof. Note that Let u n ⊂ H be a Cerami sequence of I 0 . It needs only to show that u n is bounded in H due to Remark 6. We may assume by contraction that I 0 (u n ) → c, (1 + u n )I 0 (u n ) → 0 and u n → ∞ as n → ∞.
Thus, one has for n large enough, From the right inequality of (f 2 ), fixed an ε 2 > 0, there exists a C ε2 > 0 such that Hence, by (f 0 ) and F (x, t) Thus, for n large enough, we have where c 2 = 4(c + 1) + 4c 1 |Ω|. Let v n = un un , then there is v ∈ H such that v n v weakly in H and v n → v strongly in L 4 (Ω). Dividing (44) by u n 4 and letting n → ∞, we have 1 ≤ (µ k+1 + ε 2 ) v 4 L 4 , which shows that {x ∈ Ω : v(x) = 0} has a positive measure. Now, by (f 0 ) and (f 1 ), there exists a constant c > 0 such that Hence, An application of Fatou's lemma yields Ω [f (x, u n )u n − 4F (x, u n )]dx → +∞ as n → ∞, Combining (f 0 ) with the above equality, we find a positive constant c such that G 1 (x, t) < c ∀(x, t) ∈ Ω × R.
For any u ∈ C k+1 , from the above inequality, one has From the left inequality of (f 0 ) and (f 2 ), one has for some c 1 > 0 and ε 3 ∈ (0, µ k+1 −µ k 2 ). Nevertheless, from the definition of the variational eigenvalue µ k , it follows that there exists some Λ 2 ∈ Σ k such that sup u∈Λ2 u 4 ≤ µ k + ε 3 .
For any h ∈ G , by (50), we get h(S k−1 ) ∩ C k+1 = ∅, which shows that G is a subset of C(S k−1 , H \ C k+1 ). Let Then, proceeding exactly as in the previous Theorem 1.1, we may verify h(B k ) ∩ C k+1 = ∅ for all h ∈ Γ 1 . Thus, we have shown that C k+1 and S k−1 are G −Linking.
Since the conditions of G−Linking theorem are satisfied by the compactness of B k , (50) and Lemma 3.5, the conclusion in Theorem 1.3 is correct.