A multiplicity result for asymptotically linear Kirchhoff equations

In this paper, we study the following Kirchhoff type equation: −(1 + b ∫ RN |∇u|2 dx)∆u + u = a(x)f(u) inRN , u ∈ H1(RN), where N ≥ 3, b > 0 and f(s) is asymptotically linear at infinity, that is, f(s) ∼ O(s) as s → +∞. By using variational methods, we obtain the existence of a mountain pass type solution and a ground state solution under appropriate assumptions on a(x).


Introduction and main result
In this paper, we study the following Kirchhoff type equations in ℝ N (N ≥ 3): : |x| ≥ R 0 }.
Throughout this paper, we denote by H := H 1 (ℝ N ) the usual Sobolev space equipped with the following inner product and norm: Define the energy functional I b : H → ℝ by where F(s) = ∫ s 0 f(t) dt. The functional I b is well defined for each u ∈ H and belongs to C 1 (H, ℝ). Moreover, for any u, φ ∈ H, we have Clearly, the critical points of I b are the weak solutions for problem (1.1).
In recent years, much attention has been paid to Kirchhoff type equations. Let Ω ⊂ ℝ N be a bounded domain. The following Kirchhoff problem with zero boundary data: (1.2) which is related to the stationary analogue of the equation was proposed by Kirchhoff [10] as an extension of the classical D'Alembert's wave equation for free vibrations of elastic strings. Kirchhoff's model takes into account the changes in length of the string produced by transverse vibrations. In (1.2), u denotes the displacement, f(x, u) the external force and b the initial tension, while a is related to the intrinsic properties of the string, such as Young's modulus. We would like to point out that such nonlocal problems also appear in other fields such as biological systems, where u describes a process which depends on the average of itself, for example, population density. It is worth mentioning that Fiscella and Valdinoci [5] proposed a stationary fractional Kirchhoff model, in bounded regular domains of ℝ N , which takes into account the nonlocal aspect of the tension arising from nonlocal measurements of the fractional length of the string. For some recent results about stationary Kirchhoff problems involving the fractional Laplacian, we refer to [18-20, 22, 23, 25, 30, 31, 34] and the references therein. For more mathematical and physical background for problem (1.2), we refer the readers to [1,2,5,24] and the references therein.
Recently, problems like type (1.2) in bounded domains have been investigated by many authors, see, for instance, [6,7,16,17,21,26,32,33]. In [16], Ma and Muñoz Rivera obtained positive solutions via variational methods. In [21], Perera and Zhang obtained a nontrivial solution via the Yang index and the critical group. Zhang and Perera [33], and Mao and Zhang [17] obtained multiple and sign-changing solutions via the invariant sets of descent flow. Shuai [26] obtained one least energy sign-changing solution via a constraint variational method and the quantitative deformation lemma. He and Zou [6,7] obtained infinitely many solutions via the local minimum method and the fountain theorems.
Equations of type (1.1) in the whole space ℝ N (N ≥ 3), but with the nonlinear term a(x)f(u) being replaced by a more general nonlinear term f(x, u), have also been studied extensively, see, for example, [8,9,12,13,15,28,35] and the references therein. More recently, the researchers paid their attention on asymptotically linear Kirchhoff equations. In [11], for a special type of a Kirchhoff equation with asymptotically linear term, in which the nonlocal term is like 1 + b ∫ ℝ N (|∇u| 2 + V(x)u 2 ) dx (this makes the functional contain a term like 2 ), Li and Sun only needed to verify the (PS) condition in order to apply the mountain pass theorem to obtain the existence and multiplicity of solutions. Moreover, we noticed that the potential V(x) in [11] was assumed to be radially symmetric, in order to consider the problem in a radial function Sobolev space in which the compact Sobolev embedding holds, and thus being easy to verify the (PS) condition. In some cases, this symmetric condition may be replaced by other compact conditions which make the compact embedding holds; here we just quote [3]. In [29], Wu and Liu studied the existence and multiplicity of nontrivial solutions for a Kirchhoff equation in ℝ 3 with asymptotically linear term via Morse theory and local linking. To overcome the loss of compactness, the usual strategy is to restrict the functional to a subspace of H 1 (ℝ 3 ), which embeds compactly into L 2 (ℝ 3 ) with certain assumptions on radially symmetric functions. In this paper, we will study directly problem (1.1) in H, not in any subspaces.
To overcome the loss of compactness, motivated by [14] in which Liu, Wang and Zhou studied an asymptotically linear Schrödinger equation, we will make a careful prior estimate for the Cerami sequence (defined later). Unlike the problem in [11], we not only show the convergence of the Cerami sequence, but also show that ∫ ℝ N |∇u n | 2 dx → ∫ ℝ N |∇u| 2 dx as n → ∞, where {u n } is a Cerami sequence. This makes the study of our problem more difficult. Through a careful observation, we show that this limit holds only by assuming that (A1), (A2) and (A3) hold, without any more assumptions. Now let us state the main result of this paper.
then there existsb > 0 such that for any b ∈ (0,b ), problem (1.1) admits at least two nontrivial nonnegative solutions, in which one is a mountain pass type solution and the other is a ground state solution.
We use the following notation: • C, C 1 , C 2 , etc. will denote positive constants whose exact values are not essential.
denotes the open ball centered at x having radius R.

Preliminary lemmas
To prove Theorem 1.1, we use a variant version of the mountain pass theorem, which allows us to find a so-called Cerami type (PS) sequence. The properties of this kind of (PS) sequence are very helpful in showing the boundedness of the sequence in the asymptotically linear case.

Theorem 2.1 ([4])
. Let E be a real Banach space with its dual space E * , and suppose that I ∈ C 1 (E, R) satisfies for some μ < η, ρ > 0 and e ∈ E with ‖e‖ > ρ. Let c ≥ η be characterized by This kind of sequence is usually called a Cerami sequence.

Proof of Theorem 1.1
Now we are in a position to give the proof of Theorem 1.1.
Proof of Theorem 1.1. By Lemma 2.3, the sequence {u n } defined in (2.4) is bounded in H. Since H is a reflexive space, going if necessary to a subsequence, u n ⇀ u in H for some u ∈ H. In order to prove the theorem, we need to show that ∫ ℝ N |∇u n | 2 dx → ∫ ℝ N |∇u| 2 dx and that the sequence {u n } has a strong convergence subsequence in H, that is, ‖u n ‖ → ‖u‖ as n → ∞. Note that, by (2.4), and Since u n ⇀ u in H, we have Since the embedding H → D 1,2 (ℝ N ) is continuous, u n ⇀ u in D 1,2 (ℝ N ), and thus For any ϵ > 0, by Lemma 2.4, (A3) and Hölder's inequality, for n large enough, we have This and the compactness of the embedding If λ = ∫ ℝ N |∇u| 2 dx, from (3.1)-(3.5), it is also easy to see that ‖u n ‖ → ‖u‖ in H as n → ∞, hence the proof is completed.
If λ > ∫ ℝ N |∇u| 2 dx, from (3.1)-(3.5), we obtain By Fatou's Lemma, it is easy to see from our assumption that Moreover, we get ‖u n ‖ → ‖u‖ as n → ∞. Now we show that the solution u is nonnegative. Multiplying equation (1.1) by u − and integrating over ℝ N , where u − = min{u(x), 0}, we find Hence, u − = 0 and u is a nonnegative solution of problem (1.1).
To get a ground state solution, we denote by K the nontrivial critical set of I b . Set It is easy to see that K is nonempty. For any u ∈ K, we have Now we choose ϵ ∈ (0, C −1 1 ) as in the proof of Lemma 2.1 and use (2.1), (2.3) and the Sobolev embedding theorem to get Therefore, for any u ∈ K, we have 0 ≥ ‖u‖ 2 − ϵC 1 ‖u‖ 2 − C 2 C ϵ ‖u‖ 2 * . (3.8) We recall that u ̸ = 0 whenever u ∈ K, and (3.8) implies > 0 for all u ∈ K. (3.9) Hence, any limit point of a sequence in K is different from zero. We claim that I b is bounded from below on K, i.e., there exists M > 0 such that I b (u) ≥ −M for all u ∈ K. Otherwise, there exists {u n } ⊂ K such that I b (u n ) < −n for all n ∈ ℕ. (3.10) It follows from (2.3) that This and (3.10) imply that ‖u n ‖ → +∞ as n → ∞. Let ω n = u n ‖u n ‖ . There exists ω ∈ H such that (2.5) holds. Note that I b (u n ) = 0 for u n ∈ K. As in the proof of Lemma 2.3, we obtain that ‖u n ‖ → +∞ is impossible. Then I b is bounded from below on K. So m ≥ −M. Let {ū n } ⊂ K be such that I b (ū n ) → m as n → ∞. Then (2.4) holds for the sequence {ū n } and m. Following almost the same procedures as in the proofs of Lemmas 2.3 and 2.4, and using the above arguments, we can show that {ū n } is bounded in H and, going if necessary to a subsequence, ‖ū n ‖ → ‖ū ‖, whereū ∈ H \ {0} and ∫ ℝ N |∇ū n | 2 dx → ∫ ℝ N |∇ū | 2 dx as n → ∞. There is only one difference in showing that (3.6) does not hold. Based on the aforementioned discussions, we know that the possible critical value c > 0. Here we do not know if m > 0, but (3.9) holds, and so λ > 0. Moreover, I b (ū ) = m and I b (ū ) = 0. Therefore,ū ∈ H \ {0} is a ground state solution of problem (1.1). Finally, we can also show that the ground state solutionū is nonnegative.