Asymptotic behavior of multiple solutions for quasilinear Schrödinger equations

. This paper establishes the multiplicity of solutions for a class of quasilinear Schrödinger elliptic equations:


Introduction
This paper deals with multiplicity and asymptotic behavior of solitary wave solutions for quasilinear Schrödinger equations of the form where z : R 3 × R → C, W : R 3 → R is a given potential, γ is a real constant and l, ρ are real functions.Quasilinear equations of the form (1.1) have been established in the past in several areas of physics with different types of ρ.For example, the case ρ(t) = t was used in [18] for the superfluid film equation in plasma physics; the case ρ(t) = (1 + t) 1/2 was considered for the self-channeling of a high-power ultrashort laser in matter, see [11] and [12].These types of equations also appear in fluid mechanics [19], in the theory of Heidelberg ferromagnetism and magnus [20], in dissipative quantum mechanics [17] and in condensed matter theory [27].
We now consider the case of the superfluid film equation in plasma physics, namely ρ(t) = t.If we look for standing waves, that is, solutions of the form z(t, x) := exp(−iEt)u(x) with E > 0, we are lead to investigate the following elliptic equation with V(x) = W(x) − E and f : R 3 × R → R given by f (x, t) := l(x, |t| 2 )t is a new nonlinear term.Later on, we shall pose precisely the hypotheses on V and f .Taking γ = 0, the equation (1.2) is a semilinear case, scholars have obtained a large number of existence and multiplicity results based on variational methods, see e.g.[10,14,21,22].When γ > 0, the first existence of positive solutions is proved by Poppenberg, Schmitt and Wang in [28] with a constrained minimization argument.While a general existence result for (1.1) is due to Liu et al. in [25] through using of a change of variable to reformulate the quasilinear problem (1.2) to a semilinear one in an Orlicz space framework.Colin and Jeanjean in [13] used the same method of changing variables, but the classical Sobolev space H 1 (R N ) was chosen.We refer the readers to [5,26,31,33,34] for more results.Recently, in [23], by using perturbation methods, Liu et al. proved the existence of nodal solutions for the general quasilinear problem in bounded domains.
In the above references mentioned, the γ in the quasilinear problem (1.2) was assumed to be a fixed constant.While, the constant γ represents several physical effect and is assumed to be small in some situation.This indicates the importance of the study of the asymptotic behavior of ground states as γ → 0 + .But, asymptotic behavior of solutions for quasilinear Schrödinger equations is much less studied.In [1], Adachi et al. considered the problem for N = 3, λ > 0, γ > 0 and f (x, s) = |s| p−2 s (4 < p < 6): They showed the ground states u γ of (1.3) satisfies u γ → u 0 in H 2 (R 3 ) ∩ C 2 (R 3 ) as γ → 0 + , where u 0 is a unique ground state of Then, in [34], Wang and Shen proved the asymptotic behavior of positive solutions for (1.3) when p ∈ (2, 4), which complemented the result given by Adachi et al. in [1].By applying the blow-up analysis and the variational methods, in [2][3][4] Adachi et al. obtained the precise asymptotic behavior of ground states when N ≥ 3 and the nonlinear term has H 1 -critical growth or H 1 -supercritical growth.However, the work in the literature always assumed that V(x) ≡ λ > 0 and studied the asymptotic behavior of one ground state solution for (1.4).We are interested in the problem that whether or not we can find the multiplicity of solutions for (1.4) with some suitable potential conditions.Furthermore, as γ → 0 + , whether these solutions have any asymptotic behavior.Specifically, the main purpose of the present paper is to solve the following three problems: We have the multiplicity of solutions for (1.4) in unbounded domains, which complements the results given by Liu et al. in [23].
(Q 2 ) We obtain the asymptotic properties of solutions for (1.4) under some suitable potential conditions.Our result, in the sense that we do not need the restrictive conditions V(x) ≡ λ > 0, improves the one obtained in [1].
(Q 3 ) All the papers mentioned above only studied the asymptotic behavior of a positive ground state solution for (1.4).In this paper, we explore the asymptotic behavior of multiple solutions for quasilinear Schrödinger equations.More precisely, we can obtain the asymptotic behavior of sign-changing solution for (1.4).
For this purpose, we consider the multiplicity and asymptotic behavior of solutions for the following one-parameter family of elliptic equations with general nonlinearities: where γ > 0 and V(x) ∈ C(R 3 , R) satisfying: (V 1 ) : For any M, r > 0, there is a ball B r (y) centered at y with radius r such that Remark 1.1.The condition (V 1 ) was firstly introduced by Bartsch, Pankov and Wang [8] to guarantee the compactness of embeddings of the work space.The limit of condition (V 1 ) can be replaced by one of the following simpler conditions: For the continuous nonlinearity f , we suppose that it satisfies the following conditions: ( f 1 ) : there exist a constant C and p ∈ (4, 6) such that where F(x, t) = t 0 f (x, s)ds.Note that (1.4) is the Euler-Lagrange equation associated to the natural energy functional: which is not well defined in H 1 (R 3 ).Due to this fact, the usual variational methods can not be applied directly.This difficulty makes problem like (1.4) interesting and challenging.Inspired by the work of Shen [29], we first establish the existence of signed solutions for a modified quasilinear Schrödinger equation where In what follows, instead of using the dual method, we search the existence of signchanging solutions for the problem (1.4) via the perturbation method and invariant sets of descending flow.
For asymptotic behavior of solutions for the problem (1.4), arguments we apply are rather standard.Using a bootstrap argument, we obtain the uniform boundedness of L ∞ -norm of u γ .Then we apply the uniform estimates for the energies to show the strong convergence in ) will be defined in Section 2), this is a key problem to the study.Next, we give our main results.Theorem 1.2.Assume that (V 0 ), (V 1 ), and ( f 1 )-( f 3 ) hold.Then, for fixed γ ∈ (0, 1], the problem (1.4) has at least three solutions: a positive solution u γ,1 , a negative solution u γ,2 and a sign-changing solution u γ,3 .
Remark 1.4.In order to prove the existence of a sign-changing solution, we need a restriction p > 4 because of the degeneracy of the quasilinear term.Moreover we require that p is H 1 -subcritical to prove the L ∞ -norm of the solutions of (1.5) are uniformly bounded.Since 4 < 2N N−2 if and only if N < 4. Hence we only show the asymptotic behavior of multiple solutions for the quasilinear Schrödinger for N = 3.This paper is organized as follows.In Section 2, we describe the variational framework associated with the problem (1.4).We give the proofs of existence of signed and sign-changing solutions in Sections 3-4, respectively.Section 5 is devoted to the study of asymptotic behavior of solutions.
In what follows, C and C i (i = 1, 2, . . . ) denote positive generic constants.In this paper, the norms of L s (R N )(s ≥ 1) is denoted by | • | s .

The modified problem
Let and the norm ∥u∥ 2 . From [9], we know that under the assumptions (V 0 ) and (V 1 ), the embedding Note that (1.4) is the Euler-Lagrange equation associated to the natural energy functional: ). Inspired by [13,29,30], we consider the following quasilinear Schrödinger equation: Here we choose g γ (t) : R → R given by It follows that g γ (t) It is well known that G γ (t) is an odd function and inverse function G −1 γ (t) exists.Moreover, we summarize some properties of G −1 γ (t) as follows.Lemma 2.1 ([30]).
(1) lim t→0 (2) lim t→+∞ (5) 0 ≤ s g γ (s) g ′ γ (s) ≤ 1, for all s ∈ R; (6) there exists a positive constant C independent of γ such that (7) there exists θ > 4 such that In what follows, taking the change variable we observe that the functional I γ (u) can be written of the following way From Lemma 2.1 and conditions (V 0 ), (V 1 ) and ( f 1 )-( f 3 ), we obtain the functional for all φ ∈ H 1 V (R 3 ).Moreover, the critical points of the functional J γ correspond to the weak solutions of the following equation

The existence of signed solutions
In this section we fix 1 ≥ γ > 0. Let u + = max{u, 0} and u − = min{u, 0}.Set and and we have where we need sufficiently small δ > 0 and the Sobolev inequality.Thus, it implies On the other hand, the condition ( f 3 ) implies that For w ∈ C ∞ 0 (R 3 ) with supp(w) = B 1 and w(x) ≥ 0, As a consequence of Lemma 3.1 and the Ambrosetti-Rabinowitz Mountain Pass Theorem, for the constant ) be a Palais-Smale sequence.Then Therefore, by (3.1)-(3.3)and Lemma 2.1-( 7), we have Next, we will prove that there exists a constant C > 0 such that Otherwise, there exists a sequence {v Hence, by (3.4), [30], we get a contradiction.This Proof.First, we show that the sequence {v n } possesses a convergent subsequence in H 1 V (R 3 ).Indeed, by the boundedness of {v n } and the compactness of embedding 3 .
. By standard regular arguments, the weak limit v of {v n } is a critical point of ) and v can be shown to be positive critical point of J γ by applying the maximum principle in [16].Hence, u = G −1 γ (v) is a positive weak solution of (1.4).By the similar argument, we know that the equation (1.4) also has a negative weak solution.
The next two results establish the uniform boundedness of H 1 V -norm of v γ .This important estimate will be used in Section 5.
Proof.Let v γ be a critical point of J + γ .Similar with Lemma 3.2, we get the following estimates Similar with Lemma 3.1, we have where we need sufficiently small δ > 0 and the Sobolev inequality.Thus, if v ∈ ∂Σ ρ , take ρ small enough, it implies that J + γ (v) ≥ Cρ 2 := m 1 , where m 1 does not depend on γ.Note that where the constant T > 0. For T large enough, we have Due to h(t) ∈ Γ, then we get where m 2 does not depend on γ.

The existence of sign-changing solutions
The goal of this section is to consider the existence of sign-changing solutions.To do this, we define the work space E as follows where which endowed with the norm and W 1,4 (R 3 ) endowed with the norm The norm of E is denoted by ) is compact (see [9]).Applying the interpolation inequality, we obtain that the embedding from In what follows, we formally formulate (1.4) in variational structure as follows Notice that I γ is an ill-behaved functional in H 1 V (R 3 ).To avoid this difficulty, in the sequel, for each µ, γ > 0 fixed, let us consider the perturbation functional I µ,γ : E → R associated with (1.4) given by By deducing as in [15] (see also [23]), it is normal to verify that I µ,γ ∈ C 1 (E, R) and for each φ ∈ E, we get (4.4) In the following, we prove a compactness condition for I µ,γ .
Proof.Let {u n } ⊂ E be a (PS) sequence for I µ,γ , that is {u n } satisfies: By a standard argument, we can prove that every bounded (PS) sequence {u n } ⊂ E of I µ,γ possesses a convergent subsequence, cf.[15].This completes the proof.
In the following, we would like to construct a descending flow guaranteeing existence of desired invariant sets for the functional I µ,γ .For this purpose, we introduce an auxiliary operator A where and C 0 > 0 large enough.It is normal to verify that J µ,γ ∈ C 1 (E, R) and for all ω ∈ E we have Clearly, we notice that the following two statements are equivalent: u is a fixed point of A and u is a critical point of I µ,γ .
From conditions ( f 1 ), ( f 2 ) and the Sobolev embeddings theorem, for any δ > 0, there exists C δ , such that This deduces Therefore, the functional Φ µ,γ is coercive.We can see that the functional Φ µ,γ is bounded from below and maps bounded sets into bounded sets.In the following, we shall prove that the functional Φ µ,γ is also strictly convex.In fact, since where θ t = tv + (1 − t)ω (t ∈ (0, 1)).By Young's inequality, for any δ > 0, there exists Taking δ = 3µ 2 and choosing C 0 > From the above analysis, we obtain that the functional Φ µ,γ is coercive, bounded below, weakly lower semicontinuous and strictly convex.Thus, the functional Φ µ,γ admits a unique minimizer v = A(u).Moreover, the operator A maps bounded sets into bounded sets.
Next, we will verify the continuity of the operator A on E. To prove this, let Furthermore, by the similar estimates of (4.6), for C 0 large enough, we get Then, combining (4.7) with (4.8), we have Since K ∈ C 1 (E, R) and u n → u strongly in E, we get that v n → v strongly in E and the operator A is continuous.
Next, we shall verify ( 1) and (2) as follows.By (4.5), we get Furthermore, we have the following estimates where In fact, there hold Using similar methods, we can also estimate other terms in (4.10).Hence For (2), by the similar estimates of (4.6), set φ = u − v, we have ).
In order to prove (3), we consider Hence, for any δ > 0, there exists C δ , such that ).
Taking δ > 0 small enough, by direct calculation, we obtain the following estimates Combining (4.11) and Lemma 4.3-( 1), we can obtain , we obtain Consider a positive cone P in E defined by P := {u ∈ E : u ≥ 0 a.e. on x ∈ R 3 }.For an arbitrary ε > 0, let

Lemma 4.4.
There exists ε 0 > 0 such that for all ε ∈ (0, ε 0 ), then Proof.Since the proofs of the two conclusions are similar, we just give the proof of A(∂P (4.12) Next, we will give the estimates of both sides of above equality.On one hand, we have On the other hand, by Young inequality, we obtain , for any δ > 0. (4.14) Fix δ = V 0 and choose ε 0 such that C δ ( ε 0 S ) 4 ≤ S 2 .For 0 < ε < ε 0 and u ∈ P + ε , we have By (4.13)-(4.15),we get .
From the above analysis, we know that A is merely continuous.But A itself is not applicable to construct a descending flow for I µ,γ , and we have to construct a locally Lipschitz continuous operator B which inherits the main properties of A.
To apply Theorem 4.7 to obtain one sign-changing critical point of I µ,γ , we take Then we need to prove the following crucial lemma.
Proof.The proof is similar to many existing literature (see [25,32]).For the readers' convenience, here we give the details.
To verify (3), for any u ∈ Σ, from the conditions ( f 1 ) and ( f 2 ) and the definition of Σ, for all δ > 0, there exists C δ > 0, such that On the other hand, by the condition ( f 3 ), we have F(x, t) ≥ C|t| θ for all x ∈ R 3 .For any u ∈ φ 0 (∂ 0 χ), then which together with (4.17) implies that for R large enough and ε small enough, we obtain sup u∈φ 0 (∂ 0 χ) Hence, by Theorem 4.7, I µ,γ has at least one critical point u in E \ (P + ε ∪ P − ε ).The next result establishes an important estimate associated with critical values.Lemma 4.10.Assume 0 < µ < 1 and 0 < γ < 1.Then there exists a positive constant m 3 (independent on µ and γ), such that where u µ,γ is a sign-changing critical point of I µ,γ .
Proof.For fixed 0 < µ < 1 and 0 < γ < 1, take a path φ 1,1 (s, t) , where the constant T > R (R is defined in the proof of Lemma 4.9).A simple computation ensures that φ 1,1 (0, s) By the similar estimates of (4.18), taking T sufficiently large, we obtain where C 1 > 0 is large enough.On the other hand, for ε small enough, we have here choose C 1 large enough, such that 0 < C 2 < C 1 .Then estimates (4.19) and (4.20) ensure that max This implies where where m 3 is independent on γ and µ.
Finally, the existence of a sign-changing critical point to the original functional I γ is based on the following convergence result for the perturbation functional I µ,γ .Proposition 4.11 ([23]).Let µ i → 0 and {u i } ⊂ E be a sequence of critical points of I µ i ,γ satisfying I ′ µ i ,γ (u i ) = 0 and I µ i ,γ (u i ) ≤ C for some C independent of i.Then as i → ∞, up to a subsequence Lemma 4.12.Assume 0 < γ < 1.Then there exist a positive constant m 3 and a sign-changing critical point u γ of I γ , such that where m 3 is independent on γ.
Proof.From Lemma 4.9 and Lemma 4.10, it permits to apply the Proposition 4.11.Therefore, there exists a critical point u γ of I γ such that ).In the following, we will show that u γ is a sign-changing critical point of I γ .To this end, we need estimate u γ+ ̸ = 0 as follows.Consider ⟨I ′ µ i ,γ (u i ), u i+ ⟩ = 0, it follows from Sobolev inequality and the conditions where δ > 0 small enough.This implies |u i+ | 6 ≥ C > 0. Recall that u i+ → u γ+ strongly in L 6 (R 3 ).Therefore, we see that u γ+ ̸ = 0.By the same argument we can prove that u γ− ̸ = 0. Hence we obtain u γ is a sign-changing critical point of I γ .
Moreover, by Lemma 4.10, we obtain where m 3 is independent on γ and µ.
Having this in mind, taken µ → 0, from the Proposition 4.11 we have where u γ is sign-changing critical point of I γ .
Before concluding this section, we would like to complete the proof of Theorem 1.2.
Proof of Theorem 1.2.From Lemma 3.3 and Lemma 4.12, the problem (1.4) has at least three solutions: a positive solution u γ,1 , a negative solution u γ,2 and a sign-changing solution u γ,3 .

Asymptotic behavior of solutions
In this section, our goal is to study the asymptotic behavior of u γ = G −1 (v γ ).Having this in mind, we are going to show the L ∞ estimates of the critical points of J γ .
Proof.The result can be proved similarly to [5,14] but we give a proof for the convenience of the readers.In what follows, for simplicity, we denote v γ by v.
Set T > 0, and denote Choosing φ = |v T | 2(η−1) v in (5.1),where η > 1 to be determined later, we get Combining the fact that the second term in the left side of the above equation is nonnegative and Lemma 2.1-( 4), we obtain Taking δ small enough in (5.2), we have On the other hand, using the Sobolev inequality, we have where we used that (a + b) 2 ≤ 2(a 2 + b 2 ) and η 2 ≥ (η − 1) 2 + 1.By (5.3), the Hölder inequality and the Sobolev embedding theorem, , By conditions ( f 1 ) and ( f 2 ), the Lebesgue dominated theorem and the fact that u γ → u 0 strongly in L p (R 3 ), we get In what follows, define the following functional: F(x, u)dx.
Proof of Theorem 1.3.From Lemma 3.3, we know that for all γ ∈ (0, 1], there exists a positive critical point u γ,1 .Then, by Lemma 5.2, we obtain u γ,1 → u 1 strongly in H 1 V (R 3 ) as γ → 0 + , where u 1 is critical point of I.Note that at this stage, we do not know whether u 1 ̸ = 0. To this end, by Lemma 3.5, we know that 0 < m 1 ≤ I + γ (u γ,1 ) and so, by u γ,1 → u 1 strongly in H 1 V (R 3 ) as γ → 0 + , I + γ (u 1 ) ≥ m 1 > 0. Consequently, u 1 ̸ = 0, then u 1 can be shown to be positive critical point of I + γ by applying the maximum principle in [16], that is, u 1 is a positive solution of (1.6).Similarly, we can show u 2 is a negative solution of problem (1.6).