Some perturbation results of Kirchhoff type equations via Morse theory

In this paper, we consider the following Kirchhoff type equation: {−(a+b∫Ω|∇u|2dx)Δu=f(x,u)in Ω,u=0on ∂Ω,\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ \textstyle\begin{cases} - (a+b \int _{\varOmega } \vert \nabla u \vert ^{2}\,dx ) \Delta u= f(x,u) &\text{in } \varOmega , \\ u=0 &\text{on } \partial \varOmega , \end{cases} $$\end{document} where a,b>0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$a,b>0$\end{document} are constants and Ω⊂RN\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\varOmega \subset \mathbb{R}^{N}$\end{document} (N=1,2,3\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$N=1,2,3$\end{document}) is a bounded domain with smooth boundary ∂Ω. By applying Morse theory, we obtain some existence and multiplicity results of nontrivial solutions for either a or b being sufficiently small.


Introduction
In this paper, we are concerned with the Kirchhoff equation, which was proposed by Kirchhoff in [13] as a generalization of the well-known d' Alembert wave equation for free vibrations of elastic strings, where ρ is the mass density, ρ 0 is the initial tension, h is the area of the cross section, E is the Young modulus of the material and L is the length of the string. Kirchhoff 's model takes the changes in length of the string produced by transverse vibrations into account. We refer to [13,18] for further references in physics. The stationary analogue of the Kirchhoff equation takes the form where a, b > 0 are constants, and Ω is a bounded domain in R N (N = 1, 2, 3) with smooth boundary ∂Ω. We are interested in the case that f is sub-critical, i.e.
In recent years, many papers study the Kirchhoff type problems by variational methods. When the nonlinearity is 4-superlinear near infinity, the relevant results can be found in [20,24,25,32], and for the case where the nonlinearity is 4-asymptotically linear near infinity, we refer to [8,15,17,23,30,32] for details and further references. For example, if (GC1) holds with μ < λ 1 , then it is shown in [8] that 0 is a local minimizer of I (a,b) . With the condition where δ, C 1 , C 2 are positive constants, it is shown in [25] that the functional I (a,b) has a local linking at zero. Also, using the sequence of eigenvalues constructed in [23], the authors of [24] find nontrivial solutions when the nonlinearity is superlinear near zero but asymptotically 4-linear at infinity by computing the relevant critical groups.
In particular, when the nonlinearity is concave-convex, that is, for λ > 0, 1 < q < 2 < p < 2 * and possibly sign-changing functions g 1 (x), g 2 (x) ∈ C(Ω), by the Nehari manifold and fibering maps, the existence of multiple positive solutions is established in [7]. In [14], a power-type concave-convex nonlinearity with critical exponent is considered, and two positive solutions are found for b being small. For a nonhomogeneous p-Kirchhoff-type equation with nonlinearity as (1.5) in unbounded domains, the existence of multiple solutions for problem (1.1) is studied in [6], by Ekeland's variational principle and the mountain pass theorem. Moreover, without any growth condition on the nonlinear term f at infinity, the paper [28] obtain a sequence of solutions converging to zero for Kirchhoff equation with local sublinear nonlinearities. For more details about the existence of solutions for Kirchhoff-type equation involving concave and convex terms, we refer to [10,16,29] for details and further references. Furthermore, problem (1.1) can be generalized to p-Kirchoff equations and fractional Kirchoff equations. For instance, fractional Kirchoff equations in the case Ω = R have been studied in [22,27], in which the Morse theory were applied to obtain multiple nontrivial solutions. We also refer to [2,3] by Chang for a systematic introduction of Morse theory and various applications to differential equations. Notice that the parameters a > 0 and b > 0 are fixed in all papers cited above. The parameters a and b affect the nature of the equation in the following way. If a > 0, Eq. (1.1) is said to be non-degenerate; and it is called degenerate if a = 0 (see e.g., [9,31]). On the other hand, if b = 0, (1.1) is a usual Laplacian equation. If b > 0, Eq. (1.1) becomes a nonlocal, i.e., Eq. (1.1) is no longer a pointwise equality. This nonlocal nature causes some mathematical difficulties which make the study of such problems particularly interesting. Then it seems rather natural to ask whether it is possible to get some relationships between the two solutions for equations with a = 0 and b = 0, respectively. Motivated by the methods in [5,26], we will give some answer to this question through the estimates of critical groups for critical points of functionals, and we also use Morse theory to obtain the existence of nontrivial solutions of (1.1).
Our results read as follows. For any a > 0, the existence of a nontrivial solution has been proved in [23, Theorem 1.1]. (2) The main novelty of Theorem 1.1 is that no additional assumption on the nonlinearity f near infinity besides (f 0 ) is required. In comparison, the behavior of f near infinity is used in an essential way to get the compactness condition, or derive multiplicity of solutions in the papers we quoted previously. Moreover, let f ∈ C(R) be a function of u, and f (u) = bμu 3 for |u| < 1, and f (u) = |u| p-2 u for |u| > 2, 1 ≤ p < 2 * . Since f may take different forms for 1 < |u| < 2, Theorem 1.1 asserts the existence of nontrivial solution for nonlinearities not only restricted to power-type, which in contrast plays an important role for applying Nehari manifold type arguments. is nontrivial (see [2]). But the approach to proving Theorem 1.1 is no longer able to guarantee the existence of a nontrivial solution of (1.1).
From the arguments in the proof of Theorem 1.1, we can deduce the following results.
Next, we consider nonlinearity f with a perturbation term, where 1 < p < 6 and λ ∈ R. Clearly, if λ = 0, then it returns to the case in Theorem 1.1.
Theorem 1.2 Assume 1 < p < 2 and m ≥ 1 2 (1 -sgn(λ)). If g satisfies (f 0 ) (i.e. replacing f with g in (f 0 )), then there exists ε > 0 such that Eq. (1.1) has at least one nontrivial solution in either of the following cases: Remark 2 In particular, if g(x, u) = bμu 3 , then the existence of nontrivial solution was proved in [7], which provides an example for Theorem 1.2 (i). Moreover, as pointed out in Remark 1 (2), g may take other than power-type forms, thus Theorem 1. This paper is organized as follows. In Sect. 2, we will recall some established results of Morse theory. In Sect. 3, we give the proofs of Theorem 1.1 and Corollary 1.1. The proofs of Theorem 1.2 and Theorem 1.3 are given in Sects. 4 and 5, respectively. In the sequel, the letter C will be used indiscriminately to denote a suitable positive constant whose value may change from line to line.

Preliminaries
In this section, we summarize some well known results that will be used in later sections.
Let I be a C 1 functional defined on a Banach space X, and denote the set of critical points of I by K I . We also assume that I satisfies the Palais-Smale condition. We shall prove the existence of multiple solutions by contradiction, for which the trivial solution will be assumed to be isolated at first. Then to apply the Morse theory, the critical group of isolated critical points needs to be generalized to critical group of compact dynamically isolated critical set using the Gromoll-Meyer pair. Precisely, The following proposition is crucial in applying the perturbation type arguments.
The homotopy invariance of critical group also plays an important role in our proofs. Finally, for any given μ ∈ R, define Φ : If μ / ∈ Σ, then u = 0 is an isolated critical point of Φ. Therefore C * (Φ, 0), the critical groups of Φ at 0, are well-defined; see [3]. The following result was proved by Perera and Zhang in [23].

Proof of Theorem 1.1
We begin with a few lemmas. Proof The proof is partially inspired by [11]. Note that I (0,b) is a C 1 functional and Clearly, by f (x, 0) = 0 we have u = 0 is a critical point of I (0,b) . If the conclusion is not true, then there exists a sequence {u n } ⊂ H 1 0 (Ω)\{0} such that u n → 0 in H 1 0 (Ω) and I (0,b) (u n ) = 0 for any n ∈ N.
Set v n = u n / u n , then v n = 1. Passing to a subsequence we can assume w e a k l yi nH 1 0 (Ω), v n → v, s t r o n g l yi nL 4 then ξ n → μ a.e. in Ω as n → ∞ by (GC3). Moreover, we have this together with (GC3) gives Replace w with v nv in (3.2) and let n → ∞, we get v = 1. Then (3.3) implies that μ is an eigenvalue of (1.3), which is a contradiction since μ ∈ (μ m , μ m+1 )\Σ. The proof is completed.

Lemma 3.2
Assume that (f 0 ) and (GC3) hold, then We argue by contradiction. Assume that there exist sequences {t n } ⊂ [0, 1] and {u n } ⊂ H 1 0 (Ω)\{0} such that u n → 0 in H 1 0 (Ω) as n → ∞ and J t n (u n ) = 0 for any n ∈ N. Set v n = u n / u n , and passing to a subsequence we may assume that t n → t 0 and w e a k l yi nH 1 0 (Ω), v n → v, s t r o n g l yi nL 4 as n → ∞. Similar to (3.2), define ξ n (x) as (3.1), we get Let n → ∞ and recall the condition (GC3), we have Again, replacing w with v nv in (3.4) we get v = 1, which implies that μ is an eigenvalue of (1.3). This contradicts the assumption. The proof is completed.
Combining Proposition 2.4 and Lemma 3.2, we obtain the following. By introducing the quadratic term, the critical group at zero will change.
Lemma 3.4 Assume that f satisfies (f 0 ) and (GC3), then we have where δ is the Kronecker delta.
Proof In fact, by (f 0 ) and (GC3), there exist C > 0 and γ ∈ (2, 2 * ) such that hence, for u > 0 small enough, So 0 is a local minimizer of I (a,b) , and this lemma is true.
Proof of Theorem 1.1 Note that, by (f 0 ), our functionals satisfy the Palais-Smale condition on any closed bounded set. The proof is divided into four steps. (2) Claim: For any ε > 0, setting β = 2ε ρ 2 +2ρ , then Indeed, for any v ∈ H 1 0 (Ω), Therefore then (3.6) holds. Using Proposition 2.2, for ε > 0 small enough, (3.6) implies that (W , W -) is still a Gromoll-Meyer pair for I (a,b) with the critical set Therefore, for 0 < a < β, using (3.5), we have Assume it is not true, then S [a,b] = {0}, which implies that I (a,b) has no critical points in W \W 0 . By the deformation and excision properties of a singular homology (see [2]), we may assume W 0 = W in the above choice of the Gromoll-Meyer pairs for I (a,b) at 0. Therefore, which is a contradiction by (3.7) and (3.8). Therefore, Eq. (1.1) has at least one nontrivial solution. The proof is completed.
Proof of Corollary 1.1 Similar arguments to the proof of Theorem 1.1 yield the existence of the solution, here, we focus on the convergence of the solution.
For the case (i), let u 0 be the only critical point of I (a,0) in the ball B ρ (u 0 ), and u ε n ∈ B ρ (u 0 ) be the critical point of I (a,ε n ) such that ε n → 0 and Then passing to a subsequence we may assume that w e a k l yi nH 1 0 (Ω), u ε n → u * , s t r o n g l yi nL 4 (Ω), Since I (a,ε n ) (u ε n ) -I (a,ε n ) (u * ), u ε nu * = 0, we have, as ε n → 0, which implies that u ε n → u * in H 1 0 (Ω). Let ε n → 0 in (3.9) we get Then u * is a critical point of I (a,0) . But from the isolation of u 0 in B ρ (u 0 ), we must have u * = u 0 . The case (ii) is similar.
Proof We divide it into a few steps.
Step 3 Now we define a mapping T : The proof is completed.
It is worth to point out that both (GC1) and (GC3) satisfy the growth condition in the above lemma. In the proof, the condition 1 < p < 2 plays an essential role. Proof We only need to prove that u = 0 is a local minimizer of I (a,b,λ) in the H 1 0 (Ω) topology.
First we show that u = 0 is a local minimizer of I (a,0,λ) in the C 1 0 (Ω) topology. Indeed, there exist δ > 0 and C > 0 such that Then, for u ∈ C 1 0 (Ω) with |u| ∞ ≤ δ, we have pC . Now, using [1] we know that u = 0 is also a local minimizer of I (a,0,λ) in H 1 0 (Ω) topology. Moreover, since b ≥ 0, we also know that u = 0 is a local minimizer of I (a,b,λ) in H 1 0 (Ω) topology. The proof is completed.
The proof of Theorem 1.2 will be separated into two parts, according to the sign of λ. For any v ∈ H 1 0 (Ω), we have which implies that Then, for any ε > 0, there is β > 0 such that which implies that (W , W -) is still a Gromoll-Meyer pair for I (a,b,λ) with the critical set a,b,λ) , for 0<a, λ < β.
Proof of Theorem 1.