Nodal Solutions for a Quasilinear Schrödinger Equation with Critical Nonlinearity and Non-Square Diffusion

This paper is concerned with a type of quasilinear Schrodinger equations of the form 
 
\begin{eqnarray} 
-\Delta u+V(x)u-p\Delta(|u|^{2p})|u|^{2p-2}u=\lambda|u|^{q-2}u+|u|^{2p2^{*}-2}u, 
\end{eqnarray} 
 
where $\lambda>0, N\ge3, 4p < q < 2p2^*, 2^*=\frac{2N}{N-2}, 1< p < +\infty$. For any given $k \ge 0$, by using a change of variables and Nehari minimization, we obtain a sign-changing minimizer with $k$ nodes.

Such types of equations have been derived as models of several physical phenomena and have been the subject of extensive study in recent years. For example, solutions to (1.1) are related to the solitary wave solutions for quasilinear Schrödinger equations of the form i∂ t z = −∆z + W (x)z − f (|z| 2 )z − κ∆h(|z| 2 )h (|z| 2 )z, (1.2) where z : R × R N → C, W : R N → R is a given potential, κ is a real constant and f, h : R + → R are suitable functions. Quasilinear equation (1.2) appears naturally in mathematical physics and have been derived as modes of several physical phenomena corresponding to various types of h. The case h(s) = s modes the time evolution of the condensate wave function in super-fluid film ( [21,23]). This equation has been called the superfluid film equation in fluid mechanics by Kurihara [21]. In the case h(s) = (1 + s) 1 2 , problem (1.2) modes the self-channeling of a high-power ultra short laser in matter (see [7,10,14,34]).
Let z(x, t) = exp(−iEt)u(x) in (1.2), we find that u(x) solves the following elliptic equation Considering the case h(s) = s, problem (1.3) becomes u → 0, as |x| → ∞, (1.4) In this case, q ∈ [4, 22 * ) if we take f (s 2 )s ∼ |s| q−2 s as s → ∞, is called subcritical exponent in sprit of [27]. This case was studied extensively in the past several years. We can refer to the references [12,16,24,25,26,27,28,33] where positive or sign-changing solutions were obtained by using the variational argument. The case q = 22 * correspond to a critical growth for (1.4). In fact, it was shown in [27] by using a variational identity that (1.4) has no positive solution in H 1 (R N ) with u 2 |∇u| 2 ∈ L 1 (R N ) if f (u 2 )u = |u| q−2 u, q ≥ 22 * and V satisfies ∇v(x) · x ≥ 0 in R N . As pointed by Liu et al. in [26], the critical case for (1.4) is very interesting. Concerning this case, Moanemi in [30] considered the related singularly perturbed problem and obtained a positive radial solution in the radially symmetric case. Later on, an existence result of positive solutions was given by João Marcos et al. in [6] via Mountain-Pass lemma. Recently, Liu et al. considered the existence of positive solutions for general quasilinear elliptic equations in [22] by perturbation method. Y. Deng, S. Peng and J. Wang in [17] studied (1.4) for the case of critical growth, i.e. f (u 2 )u = λ|u| q−2 u + |u| 22 * −2 u, where λ > 0, 4 < q < 22 * . They constructed infinitely many sign-changing solutions for (1.4) by the Nehari method.
It is worth pointing out that for the related semilinear equations for κ = 0 , the existence of radial sign-changing solutions has been explored thoroughly, we refer the readers to [3,9,13,15,38] and the references therein. However, there seems to be little progress on the existence and nonexistence of nontrival solution for the case when h(s) = s p (1 < p < +∞). In this case, problem and q ∈ [4p, 2p2 * ) if we take f (s 2 )s ∼ |s| q−2 s as s → ∞, is called subcritical exponent and 2p2 * is the correspond critical exponent. Comparing with (1.4), the quasilinear term κp∆(|u| 2p )|u| 2p−2 u for p > 1 in problem (1.5) is called the nonsquare diffusion; meanwhile, the quasilinear term κ∆(u 2 )u in (1.4) is called the square diffusion.
In this paper, we focus on the case when h(s) = s p (1 < p < +∞), κ = 1 and f (s 2 )s = λ|s| q−2 s + |s| 2p2 * −2 s, i.e. problem (1.1). We will construct infinitely many solutions for problem (1.1) with critical growth by the Nehari method. To this end, we introduce some definitions and notations.
For any given set K ⊂ R N , we denote ( K |u(x)| s dx) 1 s =: |u| s,K and we write K instead of K dx. We use " → " and " " to denote the strong and weak convergence in the related function space respectively. C will denote a positive constant unless specified. We denote by u + and u − the functions defined by u + (x) = max{u(x), 0} and u − (x) = max{−u(x), 0}. For a radial domain Ω, we set which is equivalent to the usual norm in H 1 (R N ) by (V 1 ).
The weak form of the equation , which is formally the variational formulation of the following functional: where K(s) = λ q |s| q + 1 2p2 * |s| 2p2 * , k(s) = K (s) = λ|s| q−2 s + |s| 2p2 * −2 s. We point out that we cannot apply directly variational method here because I(u) is not well defined in H 1 (R N ). To overcome this difficulty, we generalize the argument developed by Liu-Wang-Wang in [26] and make the change of variables v = f −1 (u), where f is defined by the following ODE After the change of variables, I(u) can be reduced to the following functional which is well defined on the usual Sobolev space H 1 (R N ) under suitable assumptions on the potential V (x) and the nonlinearity K(s). Moreover, the nontrival critical points of the functional J correspond precisely to the nontrival weak solutions of the following equation For the convenience, we rewrite equation (1.6) in the following form: and the corresponding variational functional is and We can easily verify that problem (1.1) is equivalent to problem (1.7) and the nontrival critical points of J(v) are the nontrival solutions of problem (1.7). In the following, we construct infinitely many k − node solutions for problem (1.7). For any k ∈ {0, 1, 2, · · · }, v ± is said to be a pair of k-node solution of (1.7) if v ± is a radical solution with the following properties: (ii) v ± possess exactly k nodes r i with 0 < r 1 < r 2 < · · · < r k < +∞, and v ± (r i ) = 0, i = 1, 2, · · · , k.
To find nontrival critical points of J, difficulties lie in two aspects. The first difficulty is caused by the usual lack of compactness since the problem involve critical exponent and are dealt with in the whole R N . To overcome this difficulty, we should work out a threshold value of energy under which a (PS) sequence is precompact. The second difficulty lies in a new phenomenon in which the nonlinear term g(x, s) only satisfies lim s→∞ g(x, s) |s| 2 * −1 = 0 instead of the usual subcritical condition g(x, s) = o(|s| t ) (2 < t < 2 * ) at infinity. Further more, as we will see later, the functions G(x, v), g(x, v)v given by (1.8), (1.9) and 1 2 g(x, v)v − G(x, v) may change sign. These two new phenomena cause two more difficulties. On one hand, the usual Ambrosetti-Rabinowitz condition is not satisfied. On the other hand, the usual argument to verify that the functional corresponding to (1.7) satisfies the (PS) condition cannot be employed directly. Hence, we need to analyze the exact asymptotic behavior of g(x, s) and should apply more delicate analysis to the functional corresponding to (1.7).
To construct nodal solutions for (1.7), we will look for a minimizer of a constrained minimization problem in a special space in which each function changes sign k (k ∈ {0, 1, 2, · · · }) times and then verify that the minimizer is smooth and indeed a solution to (1.7) by analyzing the least energy related to the minimizer. We mention here that the main method to prove our theorem was essentially introduced by Bartsch and Willem in [3], Cao and Zhu in [9] and G. Cerami, S. Solimini and M. Struwe in [11].
It should be mentioned that Shen and Wang [35] studied the problem (1.3) for general h(s) and an existence result for positive solution was obtained when f (s 2 )s is subcritical. Some interesting results about the existence and nonexistence of solutions for quasilinear elliptic problems can be found in [31,18,36,4,19].
The paper is organized as follows: in Section 2, we will provide some useful lemmas. The existence of positive solution is proved in Section 3 and Theorem 1.1 is proved in Section 4 respectively.
2. Some preliminary lemmas. In this Section, we give some properties of f, g and G which are defined in the introduction. (1) f is uniquely defined C ∞ function and invertible.
(9) There exists a positive constant C such that Thus, there exist positive constants C 1 and C 2 such that Proof. Points (1)-(4) are immediate. It's easy to obtain (5) using L'Hospital rule.
To establish the first inequality of (6), we need to show that, for all t ≥ 0, In this aim we study the function g : Thus the first inequality is proved. The second one is derived in a similar way.

YINBIN DENG, YI LI AND XIUJUAN YAN
This together with the fact that f is odd proves (7). Estimate (8) follows directly from the definition of f and we can easily get (9) combining (4) and (5). The Lemma is proved.
Lemma 2.2. There exist two positive constants α and β such that Proof. From Lemma 2.1(9), there exists a, b > 0 such that which gives that On the other hand, if t ≤ 1, we have that and hence which implies that Our Lemma now follows from (2.1) and (2.2).
Lemma 2.3. G(x, s) and g(x, s) satisfy the following properties: Proof. In fact, we must analyze the terms On the other hand, from Lemma 2.1 (5), the term Combining (2.4)-(2.6), we have (G 2 ). Similarly we can prove (G 4 ).
Lemma 2.1 (7) suggest that the functions G(x, v), g(x, v)v and 1 2 g(x, v)v − G(x, v) may be sign-changing. As we can see later, this fact causes more new obstacle when one tries to obtain nontrival solutions. To overcome those obstacle, we need to analyze the properties of f (t) for t large.

YINBIN DENG, YI LI AND XIUJUAN YAN
Denote , we obtain that Thus Remark 2.5. Using above Lemma we can prove that 1 2 In fact, by Lemma 2.4 and noting (a − b) α ≥ a α − αa α−1 b for all 0 ≤ b ≤ a and α ≥ 1, we have 3. The existence of positive solutions. In this section, we prove the existence of positive solution for problem (1.7) which is equivalent to problem (1.1), by using Mountain-Pass Lemma due to Ambrosetti-Rabinowitz [1]. Using Lemmas in Section 2 and proceeding as done in [6] or [17] we can verify that the functional J exhibits the Mountain-Pass geometry.
By Lemma 3.1, Lemma 3.3 and Mountain-Pass lemma, we can easily verify the following lemma:
Now we are ready to prove the following result. Thus we have By Lemmas 2.3 and Lemma 3.2, for any δ > 0 there exists a constant C > 0 such that On the other hand, from (3.4) we have Next, we are going to estimate J(t ψ ). From Lemma 3.2 and (3.3), we have In the following, we estimate Bρ G(x, t ψ ) and G(x, t ψ ).

By Lemma 2.1 (3) and Remark 2.5, we have for small that
On the other hand, denote ω = B 2ρ \Bρ .
Remark 3.6. By the same argument, we can consider the similar problem on a bounded domain The following corollary hold:

4.
The existence of sign-changing solutions. In this Section, we prove that the existence of node solutions for problem (1.7) by Nehari technique.
Let Ω be one of the following three types of domains:  J Ω (v).
By a standard argument, we can obtain the following compact lemma on the annular region (see [5]).
can be achieved by a positive function w * which is a positive solution of the following problem under the assumption of Theorem 1.1, where g(x, v) is defined by (1.8).
Proof. Firstly, we provec can be attained. Using Lemma 2.1, Lemma 2.3 and proceeding as done in [6] or [17] we can verify that the functional J Ω exhibits the Mountain-Pass geometry: (i) there exist α, ρ > 0 such that J Ω (v) ≥ α for all v = ρ; (ii) there exists w ∈ H 1 0 (Ω), such that w > ρ and J Ω (w) < 0. By Lemma 4.1, we deduce that, for any v ∈ M (Ω), In order to show thatc is attained by w * which is the solution of (4.2). We distinguish two cases: Case 1: Ω = B R (0 < R ≤ +∞). As done in the proof of Theorem 3.5, we can prove that c * is the critical value by Mountain-Pass lemma under the assumptions of Theorem 1.1. Thus there exists Combining (4.3) and (4.4), we obtainc is the critical value and J Ω (w * ) =c, J Ω (w * ) = 0. This shows that w * is the weak solution of (4.2). Case 2: Ω = {x ∈ R N |0 < R 1 ≤ |x| ≤ R 2 ≤ +∞}. By Lemma 4.2 and Mountain-Pass lemma, we can deduce that c * is the critical value of J Ω . Repeating the the argument of Case 1, we can conclude thatc can be attained by w * and w * is a week solution of (4.2).
Note that M ± k = ∅, k = 1, 2, · · ·. In the following we will always refer M k and we will drop the "+". For M − k , everything could be done exactly in the same way. Set Let h(v) be the functional defined in H 1 (R N ) by under the assumption of Theorem 1.1.
Proof. Let c k be attained by v, r 1 is the first node of v.

YINBIN DENG, YI LI AND XIUJUAN YAN
Thus from Lemma 3.2, (3.13) and above inequality we have Next we repeat the last part in the proof of Theorem 3.5 to obtain that I 1 < 0 and hence f or small . The proof of this Lemma is almost a repetition of the proof of Lemma 4.5 in [17], so we omit the detail here. Now, we are ready to prove the main result.
As a consequence, we complete the proof.