In this paper, we study the existence of nodal solutions of the following general Schödinger-Kirchhoff type problem:
where a,b > 0, N ≥ 3, g : R → R+ is an even differential function and
g''(s) ≥ 0 for all s ≥ 0, h : R → R 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 replacement, we first prove that (KP1) is equivalent to the following problem:
whereand G-1 is the inverse of G. Next, we prove that (KP2) is equivalent to the following system with respect to :
For every integer k > 0, 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 N ≥ 4 and b is sufficiently large.
In this paper, we consider the existence of nodal solutions for the following generalized quasilinear Schödinger-Kirchhoff type problem:
where , . is an even differential function and for all , is an odd differential function, the potential is positive in . It is necessary pointing out that we only consider the potential .
These equations are related to the quasilinear Schödinger (QS) equations
where , is a given potential, is a real constant and are suitable functions. Set , where and u is a real function, (QS) can be reduce to the corresponding equation of elliptic type (see ):
The form of the above equation has derived as models of several physical phenomena corresponding to various types of . For instance, the case 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
. In the case models the self-channeling of a high-power ultra
short laser in matter, the propagation of a high-irradiance laser in a plasma creates an optional index depending nonlinearity on the light intensity and leads to new interesting nonlinear wave equation (see   ). For more physical motivations and more references dealing with applications, we can refer to     and references therein.
If we take , Equation (1) can derive the corresponding equations with nonlocal term. For instance, If we set , i.e., , we get the superfluid film equation with nonlocal term in plasma physics:
If we set , i.e., , we get the equation:
which is related to the model of the self-channeling of high-power ultrashort laser in matter.
Denote . We observe that the natural variational functional corresponding to Equation (1)
may be not well defined in . Moreover, the set is not linear space. To overcome this difficulty, we make a change of variable constructed by Shen and Wang in , as .
Then we get
Under suitable assumptions on g and h, we conclude that J is well defined in and .
If u is a nontrivial solution of (1), then it should satisfy
for all . Let , (2) is equivalent to
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     ). To begin with, Equation (3) can be derived as a nonlocal model for the vibrating string. It is related to the stationary analogue of equation
proposed by Kirchhoff in  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,
is the mass density, 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 , 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   , we provide a new viewpoint motivated by  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 , but also give information about the case that . We achieve our purpose by solving the following system respect to :
Recall that a node of a radial solution of (1) is a radius such that with . The main purpose in this paper is to prove the equivalent of (3) and (4).
Proposition 1. Problem (3) has least one radial solution if and only if system (4) has at least one solution such that w is radial. Moreover, w and v have the same number of nodes.
Proposition 2. Problem (1) has least one radial solution 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:
(h1) as .
(h2) There exists and such that for all .
(h3) There exists such that for any , there holds , where .
(h4) is an odd function, for , is an even positive function and for all , for all .
Under the assumptions (h1)-(h2), g and h possess many important properties. Readers can find them in .
Applying Proposition 1 and 2, we can prove the following theorems.
Theorem 3. Suppose that (h1)-(h4) are satisfied. Then for any integer the following holds.
i) If , (3) has a pair and of radial solutions for any .
ii) If , there exists with such that (3) has a pair and of radial solutions for any and , has no nontrivial solution for any and .
iii) If , there exists , with such that (3) has two pair and of radial solutions if , has a radial solution if , and has no nontrivial solution if .
Moreover, each solution obtained in i)-iii) has exactly k nodes .
Theorem 4. Suppose that (h1)-(h4) are satisfied. Then for any integer the following holds.
i) If , (1) has a pair and of radial solutions for any .
ii) If , there exists with such that (1) has a pair and of radial solutions for any and , has no nontrivial solution for any and .
iii) If , there exists , with such that (1) has two pair and of radial solutions if , has a radial solution if , and has no nontrivial solution if .
Moreover, each solution obtained in i)-iii) has exactly k nodes .
2. 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.
Proposition 5. Suppose that (h1)-(h4) are satisfied. Then for any integer the following holds.
i) If , (4) has a pair and of radial solutions for any .
ii) If , there exists with such that (4) has a pair and of radial solutions for any and , has no nontrivial solution for any and .
iii) If , there exists , with such that (4) has two pair and of radial solutions if , has a radial solution if , and has no nontrivial solution if .
Moreover, the first component of every solution obtained in i)-iii) is radially symmetric and has exactly k nodes .
Clearly, is a trivial solution of (4). Here a solution of (4) is called a nontrivial one if .
In order to obtain conclusion of Proposition 5, we first establish the result about existence of nodal solutions for the following nonlinear elliptic equation:
Lemma 1. Suppose that (h1)-(h4) are satisfied. Then for any integer , there exists a pair and of radial solutions of (5) with , having exactly k nodes .
Proof. This theorem was proved in , here we omit the detail.
Let be the set of solutions of (5) and define
According to Theorem 2.1, we obtain that and is well defined.
Lemma 2. Suppose that (h1), (h2) and (h4) are satisfied. Then it holds .
Proof. For any , one has the Pohozaev identity
From assumptions (h1), (h2) and (h4), we obtain ,
For any , there exists such that
Obviously, this implies that for some . Applying Sobolev embedding theorem, we conclude , which establish that there exists a constant such that . Thus .
Proof of Proposition 5. This result was proved in ( , Proposition 2.1). However, it plays a key role in this paper and for the sake of completeness and convenience ro reader, we here give the detail. For any , define a function in given by
Case 1: . Theorem 5 implies that for any integer , problem (5) has a pair and of radial solutions with exactly k nodes . Since , we conclude
which implies that for any . Noting that for all , there exists such that . Thus is a pair of solutions of (3) and conclusion i) holds.
Case 2: . In the case, we have
for all . For any integer , let be a pair of radial solutions of (5) with exactly k nodes , , and , where is given in Lemma 2. Then . Let and in , on . Then , are nontrivial solutions of (5), which implies that and hence as . For any and , taking , then one has and . Thus solve (2). However, for any and , we have
for all , where is defined in (6). This establishes that (4) has no nontrivial solution if and .
Case 3: . For each , we see that and hence has a unique maximum point
Let be a pair of radial solution of (5) with exactly k nodes. It is easy to check that for , and
Define , where
From Case 2, we have as . Clearly, it holds if , which implies that there exists and such that and if .
Moreover, it is easy to verify that either or if .
where is given in (6). For any with , we conclude that
for any . This implies that (4) has no nontrivial solutions if and iii) holds.
3. Proofs of Proposition 1 and 2
Proof of Proposition 1. If (4) has a solution , then one has
at least in a weak sense, and
Make a change of variable, as . Then, we conclude that
which implies that w is a solution of (3).
If (3) has a solution , then one has
at least in a weak sense.
Letting and , then we have
which implies that is a solution of (4). Moreover, it is evident that v and w have the same radial symmetry and sign.
Proof of Proposition 2. If (3) has a solution , then one has
at least in a weak sense. Letting and combining the assumptions on g and h, then we conclude that
which implies that is a solution of (1). Moreover, by the definition of , it is obvious that u and v have the same radial symmetry and sign.
By the proof of Proposition 1 and 2, we establish problem (1) has least one radial solution if and only if system (4) has at least one solution such that w is radial. And w and v have the same number of nodes. Proving the existence of nodal solutions of problem (1) is equivalent to solve problem (4). In Section 2, we show the existence and nonexistence of solutions of (4). Combining the above statements the conclusions of Theorem 3 follow directly from Proposition 1 and 5. Similarly, the conclusions of Theorem 4 follow directly from Proposition 1, 2 and 5.
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