Nodal Solution for a Kirchhoff-Type Problem in N 

In this paper, we study the existence of nodal solutions of the following general Schödinger-Kirchhoff type problem: where , 0 a b > , : g + →   is an even differential function and for all 0 s ≥ , : →   is an odd differential function. These equations are related to the generalized quasilinear Schödinger equations: Because the general Schödinger-Kirchhoff type problem contains the nonlocal term, it implies that the equation (KP1) is no longer a pointwise identity and is very different from classical elliptic equations. By introducing a variable re-placement, we first prove that (KP1) is equivalent to the following problem:

Because the general Schödinger-Kirchhoff type problem contains the nonlocal term, it implies that the equation (KP1) is no longer a pointwise identity and is very different from classical elliptic equations. By introducing a variable replacement, we first prove that (KP1) is equivalent to the following problem: G − is the inverse of G. Next, we prove that (KP2) is equivalent to the following system with respect to ( ) For every integer 0 k > , radial solutions of (KP1) with exactly k nodes are obtained by dealing with the system (S) under some appropriate assumptions. Moreover, this paper established the nonexistence results if 4 N ≥ and b is sufficiently large.

Introduction
In this paper, we consider the existence of nodal solutions for the following generalized quasilinear Schödinger-Kirchhoff type problem: : g + →   is an even differential function and h →   is an odd differential function, the potential It is necessary pointing out that we only con- These equations are related to the quasilinear Schödinger (QS) equations and u is a real function, (QS) can be reduce to the corresponding equation of elliptic type (see [1]): The form of the above equation has derived as models of several physical phenomena corresponding to various types of ( ) l s . For instance, the case ( ) l s s = models the time evolution of the condensate wave function in super-fluid film and are called the superfluid film equation in the fluid mechanics by Kurihara [2]. In the case ( ) ( ) which is related to the model of the self-channeling of high-power ultrashort laser in matter.
. We observe that the natural variational functional Then we get Under suitable assumptions on g and h, we conclude that J is well defined in If u is a nontrivial solution of (1), then it should satisfy for all  . Therefore, in order to find nodal solutions of (1), it suffices to study the following equation Now, we consider the existence of nodal solutions of (3). Nonlocal problems like (3) have drawn a great deal of attention in recent years (see [11] [12] [13] [14] [15]). To begin with, Equation (3) can be derived as a nonlocal model for proposed by Kirchhoff in [16] 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. Here, ρ is the mass density, 0 P h is the initial tension, E is related to the intrinsic properties of the string, such as the Young's modulus of material and L is the length of the string. In [17], it was pointed out that such problems as (3) may be applied to describe the growth and movement of a particular species.
Different from the above mentioned literatures, for example [12] [13] [15], we provide a new viewpoint motivated by [18] for solving the generalized quasilinear Schödinger-Kirchhoff type problem. In this paper, some suitable algebraic techniques are used to find solutions. Precisely, we derive nodal solutions of (1) by transforming it into (3) and establish Equation (2) is equivalent to a system.
Then, we occur the existence of nodal solutions of this system. We not only prove multiplicity result for 3 N = , but also give information about the case that 4 N ≥ . We achieve our purpose by solving the following system respect to Recall that a node of a radial solution of (1) is a radius 0 ρ > such that The main purpose in this paper is to prove the equivalent of (3) and (4) if and only if problem (3) has at least one solution such that v is radial. Moreover, v and u have the same number of nodes.
In order to state our main result, we need the following hypotheses: (h 3 ) There exists 0 δ > such that for any 0 t > , there holds  is an even positive function and Under the assumptions (h 1 )-(h 2 ), g and h possess many important properties.
Readers can find them in [19].
Applying Proposition 1 and 2, we can prove the following theorems. Moreover, each solution k v obtained in i)-iii) has exactly k nodes Moreover, each solution k u obtained in i)-iii) has exactly k nodes

Existence and Nonexistence of Solutions of (4)
In this section, we devote to solve system (4). Firstly, we will show an essential result which will be used to conclude Theorem 3.

Moreover, the first component k v of every solution obtained in i)-iii) is ra-
dially symmetric and has exactly k nodes Clearly, ( ) 0, a is a trivial solution of (4). Here a solution In order to obtain conclusion of Proposition 5, we first establish the result about existence of nodal solutions for the following nonlinear elliptic equation: Proof. This theorem was proved in [19], here we omit the detail. Let  be the set of solutions of (5) and define According to Theorem 2.  , .
, there exists k a µ ± > such that is a pair of solutions of (3) and conclusion i) holds.
In the case, we have  . For any integer 0 k > , let k v ± be a pair of radial solutions of (5) with exactly k nodes Let k v ± be a pair of radial solution of (5) with exactly k nodes. It is easy to check that Moreover, it is easy to verify that either where β is given in (6). For any , for any µ + ∈  . This implies that (4) has no nontrivial solutions if

Proofs of Proposition 1 and 2
at least in a weak sense, and . Then, we conclude that which implies that w is a solution of (3).
If (3)    is a solution of (4). Moreover, it is evident that v and w have the same radial symmetry and sign.

Conclusion
By the proof of Proposition 1 and 2, we establish problem (1)