Quasilinear Schrödinger equations with superlinear terms describing the Heisenberg ferromagnetic spin chain

In this paper, we consider a model problem arising from a classical planar Heisenberg ferromagnetic spin chain:


Introduction
This paper is concerned with the existence of standing-wave solutions for quasilinear Schrödinger equations of the form where W (x) is a given potential, κ is a real constant, and ρ, l are real functions of essentially pure power forms.Quasilinear equations of the form (1.1) appear more naturally in mathematical physics and have been derived as models of several physical phenomena corresponding to various types of l.For instance, the case of l(s) = s is used for the superfluid film equation in plasma physics [1].In the case l(s) = (1 + s) 1 2 , (1.1) models the self-channeling of a high-power ultrashort-wavelength laser in matter [2].If l(s) = (1s) 1 2 , (1.1) also appears in the theory of the Heisenberg ferromagnetic spin chain.We refer to [3][4][5][6] and their references for more details on this subject.
Here, our special interest is in the existence of standing-wave solutions, that is, solutions of type φ(x, t) = exp(iFt)u(x), where F ∈ R and u > 0 is a real function.It is well known that φ satisfies (1.1) if and only if the function u solves the following equation of the elliptic type: where V (x) = W (x) + F is the new potential function.If we let l(s) = (1s) 2 and V (x) = λ + ε , we obtain the equation which originally appears in the Heisenberg ferromagnetic spin chain.In the mathematical literature, few results are known on (1.3).In a one-dimensional space, Brüll et al. [7] studied the ground states u for (1.3) with lim |x|→∞ u(x) = 0.For a higher-dimensional space, in [4], Takeno and Homma constructed the expression of the solution to boundary value problems for second-order nonlinear ordinary differential equations.More recently, Wang in [8] considered the following quasilinear Schrödinger equation: - He generalized the result given in [7] to a three-dimensional space.
The main objective of the present paper is to study the following quasilinear Schrödinger equation To the best of our knowledge, up to now, there are no results for (1.5) on R N for the superlinear case.
We observe that the critical point of the functional solves the Euler-Lagrange equation (1.5).From the variational point of view, there exist two difficulties to overcome for this functional (1.6).One is that the functional is not well defined in H 1 (R N ).The other is how to guarantee the positiveness of the principle part.
In order to overcome these two difficulties, we will focus on the following functional: where κ > 0 is a constant.Obviously, if u κ is a critical point of I 0 (u), then u κ solves the equation For the solution u κ of (1.8), we rescale u κ = κ -1 2 u.Then, u satisfies (1.5).Furthermore, according to [9], (1.8) can be reformulated as the following problems of the form: where g(t) = 1 -κt 2 1-κt 2 .It is obvious that g(t) is a singular function.Now, to avoid the singularity, by using the cutoff technique introduced in [8], we continuously extend the domain of the function g(t) to all of [0, +∞).More precisely, we consider the function where θ > 5+ √ 17 2 .Clearly, g κ (t) ∈ C 1 ([0, +∞), [0, +∞)) and g κ (t) decreases in [0, +∞).Substituting this form for g(t) in (1.9), we obtain the following Schrödinger equation: and the critical point of the functional satisfies the equation (1.11).
Here, the previously defined g κ (t) is obviously bounded satisfying 0 < a 1 ≤ g κ (t) ≤ 1, where . Hence, the functional I κ (u) is regular and nonsmooth.For the existence and the L ∞ estimate of the critical point of the functional (1.12), we follow the ideas shown in [9,10] and make the change of variables: (1.13) Thus, by using the change of variable (1.13), the nonsmooth functional I κ (u) can be transformed into a smooth functional and the quasilinear problem (1.11) is reformulated as a semilinear equation Consequently, in order to find the nontrivial solutions of (1.11), it suffices to show the existence of the nontrivial solutions of (1.15).We also observe that if v κ is a critical point of the functional J κ (v), then u κ = G -1 κ (v κ ) is a solution of the problem (1.11).Hence, in this way we only need to discuss the existence of the critical point v κ of the smooth functional J κ (v) by the critical-point theory.In what follows, we assume that cκ p-2 2 = 1 only as a convenience.If we can prove that the critical point u κ of the functional (1.12) satisfies .16)then this function u κ is good for what we want since g κ (u) = g(u) = 1 -κt 2 1-κt 2 under this situation.That is, in this case, the functional (1.12) is exactly the functional (1.7) and thus u κ is a weak solution of equations (1.8) and (1.9).Then, the function is the solution of (1.5).
Based on the description in the previous paragraph, the key step is to construct the estimate of |v κ | ∞ .Then, we can achieve the expression of c 0 by the inequality (1.16) such that, if c > c 0 , (1.16) holds and so u = c -1 p-2 u κ solves the equation (1.5).To this aim, according to the arguments in [11], we first obtain the H 1 estimate of v κ .Then, combining this H 1 estimate, we construct the L ∞ estimate |v κ | ∞ .We must point out explicitly that, instead of the Morse iteration method used in [11], we use the method of converting integral inequalities into differential inequalities, which can be found in Lemma 5.1 on p. 71 in Ladyzhenskaya and Ural'tseva [12] and is used to study the L ∞ estimate of the nonlinear elliptic equations on bounded domains, to construct the estimate of |v κ | ∞ .Moreover, all the constants in this estimate are well known.
Throughout this paper, we assume the potential ≤ V ∞ and we make use of the following notations: Let X be the completion of the space C ∞ 0 (R N ) with respect to the norm . By (V 1 ) and (V 2 ), X is equivalent to H 1 (R N ).The symbols |u| q and |u| ∞ are used for the norm of the space L q (R N ) with 2 ≤ q < +∞ and q = ∞, respectively.
The corresponding result is as follows: , (1.18) Then, for c > c 0 , the quasilinear problem (1.5) admits a solution u under the conditions (V 1 ) and (V 2 ).
Furthermore, we obtain a Pohozaev identity for this class of quasilinear equations, which is used to prove the nonexistence results of solution for (1.5), while we justify that p = 2 * is the critical exponent for equation (1.5).

The modified problem
In this section, we consider the following equation If we rescale the solution of (2.1) In what follows, we will establish a positive solution of (2.1).To this aim, we first introduce g κ (t) defined in (1.10) and focus on the following Schrödinger equation: We will prove that there exists a positive solution u κ for (2.3) with |u κ | ≤ 1 √ θκ .Direct calculation shows that u κ is indeed a solution of (2.1) and thus √ κu κ is a solution of (2.2).It is well known that (2.3) is the Euler-Lagrange equation associated with the energy functional Thus, by using the change of variable (1.13) and recalling our assumption cκ p-2 2 = 1, the nonsmooth functional I κ (u) can be transformed into a smooth functional and the quasilinear problem (2.3) is reformulated as a semilinear equation Therefore, in order to find the positive solution of (2.3), it suffices to study the solutions of (2.6) via the mountain-pass theorem.Thus, we need the following lemma to show some properties of the inverse function G -1 κ (t).
Proof This lemma is mainly from [8], here the proof is provided to readers only as a convenience.By the definition of g κ (t) and L'Hospital's rule, properties (1)-( 3) are obvious.By (1), for t > 0, we have θ 1 t ≤ G κ (t) ≤ g κ (0)t, which implies (4).Now, we prove the property (5).If t < 1 √ θκ , we have In the following lemma, we establish the geometric hypotheses of the mountain-pass theorem.
In consequence of Lemma 2.2, we can apply the mountain-pass theorem without the (PS)-condition found in [13] to obtain a (PS) c κ sequence {v n }, where c κ is the well-known mountain-pass level associated with the function J κ , that is, (2.7) Moreover, for any φ ∈ H 1 (R N ), we have (2.9) On the other hand, using Lemma 2.1-( 1) and ( 4), we have Combining (2.9) and (2.10), we see that (2.11) Therefore, combining (2.7), (2.8), and (2.11), we infer the inequality which shows the boundedness of {v n }.
Since {v n } is a bounded sequence in H 1 (R N ), there exist v κ ∈ H 1 (R N ) and a subsequence of {v n }, still denoted by itself, such that and Proof To begin with, we first prove that v κ is a weak solution.To this aim, we must prove that By [13], there is z ∈ L q (B 2R (0)) such that Consequently, Moreover, by Lemma 2.1-( 1) and (4), Hence, by the Lebesgue Dominate Theorem, we have The same type of arguments shows the limits below and Now, the above limits combined with Recalling that R is arbitrary and Now, we will show that v κ ≡ 0. To this aim, we suppose that v κ = 0 and claim that in this case {v n } is also a Palais-Smale sequence for functional J κ,∞ : On the other hand, we know that (2.17) Next, we claim that for all R > 0, the vanishing cannot occur.Suppose, by contradiction, that (2.19) occurs, then by Lions' compactness lemma [14], v n → 0 in L q (R N ) for any q ∈ (2, 2 * ).Jointly with Lemma 2.1, we derive that v n dx = 0.
Moreover, using the limits below we also have Therefore, which is a contradiction, since c κ ≥ a 0 > 0.
The last limit, together with Fatous' Lemma, lead to which shows that J κ,∞ (ṽ κ ) ≤ c κ .Now, following the arguments given in [15], if we define and γ (t) = ṽκ,t (x), we achieve and J κ,∞ (γ (L)) < 0 for sufficiently large L > 1.Then, by the definition of c κ , there holds which is a contradiction.Thereby, v κ is a nontrivial critical point for J κ .Moreover, repeating the same type of arguments explored in (2.21), we have that J κ (v κ ) ≤ c κ .

L ∞ estimate
This section is mainly to show the L ∞ estimate of the function v κ = G κ (u κ ) obtained in Proposition 2.1.To this aim, we need the following fact first to show the H 1 estimate of v κ .

Lemma 3.1 The solution v
Proof As v κ is a critical point of J κ , it follows that Then, by Lemma 2.1-(4), From now on, we consider the functional and we denote c ∞ the mountain-pass level associated with J ∞ , which is independent of κ.
Since J κ (v) ≤ J ∞ (v), we deduce that c κ ≤ c ∞ .Consequently, by Lemma 3.1, the solution v κ must satisfy the estimate Proof For any φ ∈ H 1 (R N ), the solution v κ of (2.6) satisfies By taking φ = (v κl) + as a test function in (3.2) with l > 0, applying Lemma 2.1-( 1) and (4), we have Combining the Sobolev inequality and the Minkowski inequality, we have Moreover, by the Hölder inequality, we have If we take Consequently, combining (3.4) and (3.6), we conclude, if l > l 0 , that Thus, jointly with we finally have Proof Inspired by Lemma 5.1 of [12], we consider the function For this function, we have -f (l) = |A l |.Therefore, (3.8) can be rewritten as If we integrate this inequality with respect to l from l 0 to l max := |v κ | ∞ , we obtain Moreover, jointly with (3.5), recalling that l and then, by (3.8), where b 1 = 1 + 1 a + 2 *p.
Combining Lemma 2.1-( 4) and (3.1), we infer that Now, to ensure that p-2 u κ is the solution of (1.5).Thus, we complete the proof.
Then, if F(x, u, ∇u), x • F x (x, u, ∇u), and F r (x, u, ∇u) • ∇u ∈ L 1 (R N ), there holds the following identity We omit the proof of this lemma, since it can be mainly found in [16].
(5.3) Thus, the integrands in (5.2) can be expressed as

4 Proof of Theorem 1.1
Proof of Theorem 1.1 A direct consequence of Proposition 2.1 and Lemma 3.3 is that v κ = G κ (u κ ) solves(1.15)andhas the estimate