Nontrivial solutions for Kirchhoff type equations via Morse theory

In this paper, the existence of nontrivial solutions is obtained for 
a class of Kirchhoff type problems with Dirichlet boundary 
conditions by computing the critical groups and Morse theory.

The condition (f * ) implies that finding weak solutions of (1) in X := H 1 0 (Ω) is equivalent to finding the critical points of the C 1 functional given by where F (x, t) = t 0 f (x, s)ds and X is the usual Hilbert space endowed with the norm u = ( Ω |∇u| 2 dx) 1/2 .
Assume that f (x, 0) ≡ 0, then u ≡ 0 is a trivial solution of problem (1). We are interested in finding nontrivial solutions for problem (1). For this purpose, we should consider the behaviors of the nonlinearity f (x, t) or its primitive F (x, t) near zero and infinity. To state our results, we recall some results on the eigenvalue problems: −∆u = λu in Ω, u = 0 on ∂Ω, 484 JIJIANG SUN AND SHIWANG MA and − u 2 ∆u = µu 3 in Ω, u = 0 on ∂Ω.
Recently, the solvability of the Kirchhoff type problem (1) has been paid much attention by various authors. Alves et al. [1], Ma and Rivera [16] and Chen et al. [5] for example, obtained positive solutions via variational methods. Zhang and Perera [28] and Mao and Zhang [17] using variational methods and invariant sets of descent flow, obtained sign changing solutions of such problem. Sun and Tang [22] obtained the existence and multiplicity of solutions for problem (1) by minimax theorems. To the best of our knowledge, the first result on resonance problems for Kirchhoff type equations is due to Sun and Tang [23]; they considered the existence of weak solutions for Kirchhoff type equations with Dirichlet boundary conditions which are resonant at an arbitrary eigenvalue under a Landesman-Lazer type condition by the minimax methods. He and Zou [7] studied the existence, multiplicity and concentration behavior of positive solutions for a singular perturbation problem of (1) in R 3 by using the variational methods. To our konwledge, few papers in the literature studied the existence of nontrivial solutions for problem (1) via Morse theory (see [20,27]). In [20] Perera and Zhang obtained nontrivial solutions of problem (1) for the case that f (x, t) satisfies the following assumption: where λ ∈ (λ l , λ l+1 ), µ ∈ (µ m , µ m+1 ) and l = m. Motivated by [20], in this paper, we will consider the cases that λ may be equal to some λ l or µ equal to some µ m in (4), and obtain the existence of nontrivial solutions of the problem (1) by using Morse theory. So our results are new. Firstly, we consider the case that the nonlinearity f (x, t) is asymptotically 2-linear near zero and 4-superlinear at infinity. More precisely, we have the following result: Theorem 1.1. Suppose f (x, t) satisfies (f * ) and the following conditions: (f 1 0 ) There exist δ > 0 and λ l <λ < λ l+1 with l ∈ N such that = +∞ uniformly for x ∈ Ω. Then problem (1) has at least one nontrivial weak solution in X. Remark 1. Clearly, the condition (f 1 0 ) contains the situation lim t→0 2F (x, t)/at 2 = λ ∈ (λ l , λ l+1 ) ang λ → λ l+1 from the left. But here we do not need to assume that the limit exists. The same conclusion holds if we replace the condition (f 1 0 ) by the following stronger condition: (f 1 0 ) There exists δ > 0, and λ l <λ < λ l+1 with l ∈ N such that aλt 2 ≤ f (x, t)t ≤ aλ l+1 t 2 , for |t| ≤ δ, x ∈ Ω.
There is an analogous result in our previous work [22], where the problem (1) was studied via local linking theorem under the following stronger condition for the case p = 4: In the case p = 2, the condition (f 1 ) was first introduced by Jeanjean in [8]. Later, it was used by Liu and Li [15] to deal with the superlinear p-Laplacian equations for the general case p > 1. When θ = 1, we can easily prove that the condition ( is nondecreasing in t ≥ 0, and nonincreasing in t ≤ 0 for x ∈ Ω, (see Proposition 2.3 in [15]). To our knowledge, (f 1 ) is widely used by many authors (see for instance [6,11,24]) to ensure the Euler-Lagrange functional satisfies the Cerami condition (see Section 2 for details). Both (f 1 ) and (f 2 ) are global condition on f (x, t). In [14], Liu showed that the functional satisfies the Cerami condition under the local condition near infinity: is nondecreasing in t ≥ R, and nonincreasing in t ≤ −R. And the author also showed (f 2 ) implies (see Lemma 2.3 in [14] also Lemma 2.4 in It is easy to see that when θ = 1, the condition (f 3 ) implies (f 1 ) when p = 4. Hence, our condition (f 1 ) is more general and concludes the global condition and also the local condition near infinity. In Lemma 2.1 we will show that under the condition (f 1 ), the functional I satisfies the Cerami condition.
Under our assumption (f 1 0 ), the functional I has a local linking at 0 (see Lemma 2.3), then 0 is not a local minimizer of I. We may pay attention to finding nontrivial solutions for the case that 0 is a local minimizer of I. We have the following result.
) and the following conditions: Then problem (1) has at least one nontrivial weak solution in X.
We now consider the dual case that the nonlinearity f (x, t) is asymptotically 4-linear at infinity and sublinear at zero. We have the following theorem.
and the following conditions: Then problem (1) has at least one nontrivial solution in X.
Remark 3. (i) It is easy to check that (f 3 0 ) implies that lim t→0 F (x, t)/t 2 = +∞ which means that the nonlinearity f (x, t) is sublinear near zero. (ii) The condition (f 2 ∞ ) contains the situation lim |t|→∞ 4F (x, t)/bt 4 = µ ∈ (µ m , µ m+1 ) and µ → µ m+1 from the left. But here we do not need to assume that the limit exists. The same conclusion holds if we replace the condition (f 2 ∞ ) by the following stronger condition: Finally, we consider the case that λ may be equal to some λ l and µ equal to some µ m in (4) simultaneously. We have the following theorem.
Then problem (1) has at least one nontrivial solution in X. , we can obtain two nontrivial solutions by using the theorem 2.1 in [13], since the functional I is coercive in X. But here we consider the case that m ≥ 1.
This paper is divided into three sections. In Section 2, we first recall the definition and some basic properties of the critical groups, and then discuss the compactness issue and compute the relevant critical groups of the functional I. In Section 3, we give the proof of our main results.

Preliminaries.
We denote by · L r the usual L r -norm. The Lebesgue measure of Ω is denoted by |Ω|. c i , will denote a positive constant unless specified, Since Ω is a bounded domain, X → L r (Ω) continuously for r ∈ [1, 2 * ], compactly for r ∈ [1, 2 * ), and there exists γ r > 0 such that Let I be a C 1 functional defined on a Hilbert space X, then the k-th critical group of I at an isolated critical point u with I(u) = c is defined by where I c = {u ∈ X|I(u) ≤ c}, U is a neighborhood of u, containing the unique critical point and H * is the singular relative homology with coefficients in an Abelian group G. According to the excision property of the singular homology theory, the critical groups do not depend on a special choice of the neighborhood U .
We denote a subsequence of a sequence {u n } as {u n } to simplify the notation unless specified. Now we recall the definition of some compactness conditions. We say that I satisfies the Cerami condition at the level c ∈ R ((Ce) c for short), if any sequence {u n } ⊂ X with I(u n ) → c and (1 + u n ) I (u n ) → 0 in X * as n → ∞ possesses a convergent subsequence in X; such a sequence is then called a Cerami sequence; I satisfies the (Ce) condition if I satisfies the condition (Ce) c for all c ∈ R. The Cerami condition introduced by Cerami [3] is a weak version of the (P S) condition: any sequence {u n } in X such that {I(u n )} is bounded and I (u n ) → 0 in X * as n → ∞ has a convergent subsequence in X.
If I satisfies (Ce) condition or (P S) condition and the critical values of I are bounded from below by some α > −∞, then the critical groups of I at infinity introduced by Bartsch and Li [2] as Note that by the second deformation lemma and the homotopy invariance of homology groups, H k (X, I α ) dose not depend on the choice of α.
We refer the readers to [4,18,19] for more details on Morse theory. In the proof of our theorems we shall use the following results. Proposition 1. Suppose that I ∈ C 1 (X, R) satisfies (Ce) condition or (P S) condition and I has only finitely many critical points, then: (i) If for some k ≥ 0 we have C k (I, ∞) = 0, then I has a critical point u with C k (I, u) = 0; (ii) Let 0 be an isolated critical point of I. If for some k ≥ 0 we have C k (I, ∞) = C k (I, 0), then I has a nontrivial critical point.
Proposition 2 (see [12]). Suppose the I ∈ C 1 (X, R) has a critical point u = 0 with I(0) = 0. If I has a local linking at 0 with respect to i.e., there exists ρ > 0 small such that Then C d (I, 0) = 0; that is, 0 is a homological nontrivial critical point of I.
Proposition 3 (see Proposition 3.8 in [2]). Suppose X splits as X = V ⊕ W such that I is bounded from below on W and I(u) → −∞ for u ∈ V as u → ∞. Then In order to apply the Morse theory to I, we must show that I satisfies the compactness condition. We have the following lemmas. Proof. We suppose that {u n } is the (Ce) sequence, that is for c ∈ R and (1 + u n )I (u n ) → 0 (n → ∞).
From (6) and (7), for n large enough, we have (8) If {u n } is unbounded, there exists a subsequence of {u n } satisfying u n → ∞ as n → ∞. Set w n = u n / u n , then w n = 1. Going if necessary to a subsequence, also denoted by {w n }, there is w ∈ X such that w n w weakly in X, If w = 0. From (7), we obtain since f (x, u)u ≥ 0. For x ∈ Θ := {x ∈ Ω : w(x) = 0}, we have |u n (x)| → +∞ as n → ∞.
This contradicts with (9). If w = 0, set a sequence {t n } of real numbers such that I(t n u n ) = max t∈[0,1] I(tu n ). For any integer m > 0, set w m n = (8m/b) 1/4 w n . By (f * ), one has |F (x, t)| ≤ C(|t| q + 1). Since w m n → 0 in L q (Ω), we see that F (·, w m n ) → 0 in L 1 (Ω) (see Proposition B.1 in [21]). Thus So for n large enough, one has 0 ≤ (8m/b) 1/4 / u n ≤ 1 and we obtain which implies that I(t n u n ) → +∞, as n → ∞. Noting that I(0) = 0 and I(u n ) → c, so 0 < t n < 1 when n is large enough. It follows that a + b Ω |∇t n u n | 2 dx Ω |∇t n u n | 2 dx − Ω f (x, t n u n )t n u n dx = I (t n u n ), t n u n = t n dI(tu n ) dt | t=tn = 0.
Therefore, by (f 1 ), we have as n → ∞, which contradicts (8). Hence {u n } is bounded, that is, there exists a positive constant M such that u n ≤ M, for all n ∈ N.
Now, we prove that {u n } has a convergent subsequence in X. Indeed, since {u n } is bounded in X, by the reflexivity of X we can assume that there exists u ∈ X such that u n u weakly in X, (11) u n → u strongly in L r (Ω) (1 ≤ r < 2 * ).
Proof. We suppose that {u n } is the (Ce) sequence, that is for c ∈ R and (1 + u n )I (u n ) → 0 (n → ∞). (14) From (13) and (14), for n large enough, we have (15) Assume by contradiction that {u n } is unbounded. There exists a subsequence of {u n } satisfying u n → ∞ as n → ∞. Set w n = u n / u n , then w n = 1. Going if necessary to a subsequence, also denoted by {w n }, there is w ∈ X such that w n w weakly in X, w n → w strongly in L r (Ω) (1 ≤ r < 2 * ), By (f 2 ∞ ), there exists R > 0, such that bμt 4 ≤ 4F (x, t) ≤ bµ m+1 t 4 , for |t| ≥ R and x ∈ Ω. Then there exists M R > 0, such that bμt 4 −M R ≤ 4F (x, t) ≤ bµ m+1 t 4 +M R , for t ∈ R and x ∈ Ω. Hence from (13), we have for n large enough, Dividing the above inequality by u n 4 , and taking n → ∞, we conclude that 1 ≤ µ m+1 w 4 L 4 . Therefore the set Θ := {x ∈ Ω : w(x) = 0} has positive Lebesgue measure. For x ∈ Θ, we have |u n (x)| → +∞. Hence by (f 2 ∞ ), we deduce f (x, u n (x))u n (x) − 4F (x, u n (x)) → +∞ as n → ∞. Therefore via Fatou lemma we have as n → ∞. This contradicts with (15). Hence {u n } is bounded in X. The proof that {u n } has a convergent subsequence in X is similar to lemma 2.1, we omit it here. The proof is completed. Now, we compute the critical groups C k (I, 0) and C k (I, ∞).

2.2.
Critical groups at zero. Assume that the problem (1) has finitely many solutions. Then 0 is an isolated critical point of I and the critical group of I at zero is defined.
Using the conditions (f * ) and (f 1 0 ), we show that the functional I has a local linking at 0 with respect to X = V ⊕ W .
Thus the mapping T is well-defined and by the implicit function theorem we see that the mapping T is continuous in u. Similarly to the proof of the Proposition 2.1 in [9], we can construct a strong deformation retract η : [0, 1] × B ρ (0) → B ρ (0) by η(s, u) = (1 − s)u + sT (u)u, for s ∈ [0, 1] and u ∈ B ρ (0) which satisfies η(0, u) = u and η(1, u) ∈ I 0 for any u ∈ B ρ (0). By the homotopy invariance of homology group, The proof is completed.
2.3. Critical groups at infinity. Assume that the problem (1) has finitely many solutions. Since I satisfies the (Ce) condition, the critical group of I at infinity make sense.
Proof. Let ϕ i be the normalized eigenfunction corresponding to the eigenvalue µ i . Define V = span{ϕ 1 , ϕ 2 , · · · , ϕ m } and W = V ⊥ , then we can split the Hilbert X with X = V ⊕ W . And we have u 4 ≤ µ m u 4 L 4 for u ∈ V ; u 4 ≥ µ m+1 u 4 L 4 for u ∈ W .
For u ∈ V with u ≥ M := R/c 11 , we have |u| ≥ c 11 u ≥ R since V is finite dimensional. Thus by (f 2 ∞ ) and (5), for u ∈ V , we have Since µ m <μ, one has I(u) → −∞ for u ∈ V as u → ∞. By (f 2 ∞ ), there exists M R > 0 such that F (x, t) ≤ b 4 µ m+1 t 4 + M R , for (x, t) ∈ Ω × R. Then for u ∈ W , by (5), one has which implies I is bounded from below on W . Now the desired result follows from Proposition 3. The proof is completed.
3. Proof of the theorems. In this section we prove our main results.