Multiple Sign-Changing Solutions for Kirchhoff-Type Equations

We study the following Kirchhoff-type equations −(a+b ∫ Ω |∇u| 2 dx)Δu+V(x)u = f(x, u), inΩ, u = 0, in ∂Ω, whereΩ is a bounded smooth domain of R (N = 1, 2, 3), a > 0, b ≥ 0, f ∈ C(Ω × R,R), and V ∈ C(Ω,R). Under some suitable conditions, we prove that the equation has three solutions of mountain pass type: one positive, one negative, and sign-changing. Furthermore, iff is odd with respect to its second variable, this problem has infinitely many sign-changing solutions.

When () = 0, problem (1) is related to the stationary analogue of the Kirchhoff equation proposed by Kirchhoff in [1] as an existence 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.Some interesting studies by variational methods can be found in [2][3][4][5][6][7][8][9][10][11].In these papers, many achievements had been obtained on the existence of ground states, infinitely many radial solutions, soliton solutions, and high energy solution for (1) by using the Fountain Theorem, the mountain pass theorem, using the variational methods and the local minimum methods, and the invariant sets of descent flow.Particularly, in [12], the authors consider the following problem: where Ω is a bounded smooth domain of R  ( = 1, 2 or 3),  > 0,  ≥ 0, and  : Ω × R → R is a continuous function which is 3-superlinear.The unbounded sequence of signchanging solutions of ( 3) is obtained by using some variants of the mountain pass theorem.In [13], authors considered the following -Laplacian equation coupled with the Dirichlet boundary condition: where  > , the parameter  > 0,  ∈  1 (Ω) is a nonzero potential, and  ∈ ([0, +∞), R) with (0) = 0.By using variational method, they proved that for every  > 1 problem (4) has at least two nonzero, nonnegative weak solutions, while there exists λ > 1 such that problem (4) has at least three nonzero, nonnegative weak solutions.In [14], Ricceri proved that there were at least three distinct weak solutions in  1 0 (Ω) for the following equation: by using and improving the three critical points' theorem, where ,  ∈ (R, R); let   be an open interval with   ⊂ [0, +∞),  ∈   .
In this paper, we study the sign-changing solutions of problem (1).We need the following assumptions: We need the following several notations.Let  fl  1 0 (Ω) with the inner produce and norm Recall that a function  ∈  is called a weak solution of problem (1) if Seeking a weak solution of problem ( 1) is equivalent to finding a critical point of the  1 -functional Since Ω is a bounded domain, it is well known that the embedding  →   (Ω) is continuous for all  ∈ [1, 2 * ] and the embedding  →   (Ω) is compact for all  ∈ [1, 2 * ).Furthermore, there is another norm and we know that ‖ ⋅ ‖ and ‖ ⋅ ‖ 0 are equivalent on ; that is, there exist constants  > 0,  > 0 such that By Lemma 1 in [9], we know that, under the conditions (), ( 1 ), and ( 2 ),  ∈  1 (, R) and for each  ∈ , for all  ∈ .
Our main result of this paper is the following.
Theorem 1. Suppose that () and ( 1 )-( 3 ) are satisfied.Then (1) has three solutions of mountain pass type: one positive, one negative, and one sign-changing.If moreover  is odd with respect to its second variable (i.e., ( 4 ) holds), then problem (1) has infinitely many sign-changing solutions.
Throughout the paper, → and ⇀ denote the strong and weak convergence, respectively., ,   , and   express distinct constants.For 1 ≤  < ∞, the usual Lebesgue space is endowed with the norm The paper is organized as follows.In Section 2, we introduce some notions and results of some critical theorem.In Section 3, we complete the proof of Theorem 1.

Some Critical Point Theorems
Let us begin by recalling some notions and results of some critical point theorems (see [15]).
In the following,  will denote a Hilbert space endowed with the norm ‖ ⋅ ‖  ,  ⊂ , which is a closed convex cone.
For  > 0, we denote by   () the -neighborhood of  ⊂ ; that is, Let  ∈  1 (, R).We denote by  the set of critical points of  and  =  \ .
For  0 > 0, we consider the following situation.
Theorem 4 (see [15]).Let  ∈  1 (, R).Assume there exists  0 > 0 such that (  0 ) is satisfied.Assume also that there exists a continuous map  0 : Δ →  such that, for any  ∈ (0,  0 ), the following conditions are satisfied: Then there exists a sequence where If in addition  satisfies the ()  condition for any  > 0, then  has a sign-changing critical point.
In the following, we assume that  is of the form and that there is another norm ‖ ⋅ ‖ * on  such that (, ‖ ⋅ ‖  ) embeds continuously into (, ‖ ⋅ ‖ * ).
Then, for  large enough there exists a sequence where If in addition  satisfies the ()  condition for any  > 0, then it possesses a sequence {  } of sign-changing critical points such that (  ) → ∞, as  → ∞.

Proof of Theorem 1
We divide the proof of Theorem 1 into the following lemmas.
For  ∈  fixed, we consider the functional It is easy to prove that Ĩ is of class  1 , coercive, bounded below, weakly lower semicontinuous, and strictly convex in .Therefore, by Theorem 1.1 in [16], Ĩ admits a unique global minimizer in  which is the unique solution to the problem Here we introduce an auxiliary operator , which will be used to construct the descending flow for the functional (⋅).We define an operator  :  → : for  ∈ ,  ∈  is the unique solution of (31).Then the set of fixed points of  coincide with the set  of critical point of .
Furthermore, the operator  has the following important properties.

Lemma 6.
(1)  is continuous and maps bounded sets to bounded sets.
On the other hand, for any  ∈ , taking  =  ∈  in (34), we obtain Using the Hölder inequality, the Sobolev embedding theorem, ( 1 ), and the fact  ≥ 0, we obtain where  > 0 is constant.This shows that  is bounded in  whenever  is bounded in .
Notice that the vector field  is not locally Lipschitz.However, it can be used as [18] to construct a locally Lipschitz vector field which will satisfy condition (  0 ).More precisely, we have the following result.