A Unified Approach to Singularly Perturbed Quasilinear Schrödinger Equations

In this paper we present a unified approach to investigate existence and concentration of positive solutions for the following class of quasilinear Schrödinger equations, -ε2Δu+V(x)u∓ε2+γuΔu2=h(u),x∈RN,\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$-\varepsilon^2\Delta u+V(x)u\mp\varepsilon^{2+\gamma}u\Delta u^2=h(u),\ \ x\in \mathbb{R}^N, $$\end{document} where N⩾3,ε>0,V(x)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$N\geqslant3, \varepsilon > 0, V(x)$$\end{document} is a positive external potential,h is a real function with subcritical or critical growth. The problem is quite sensitive to the sign changing of the quasilinear term as well as to the presence of the parameter γ>0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\gamma>0$$\end{document}. Nevertheless, by means of perturbation type techniques, we establish the existence of a positive solution uε,γ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$u_{\varepsilon,\gamma}$$\end{document} concentrating, as ε→0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\varepsilon\rightarrow 0$$\end{document}, around minima points of the potential.


Introduction
In this paper, we consider the following class of quasilinear Schrödinger equations, the literature in the context of plasma physics and the continuum limit of discrete molecular structures. In particular, κ turns out to be small and negative when (1.2) represents the weak nonlocal limit of some general nonlocal models, whereas κ has no prescribed sign in plasma physics, see for instance [6,7,20,26,28] for more details on the Physics involved. A well-known standard tool in the investigation of the elliptic side of Schrödinger equations, is the ansatz z(t, x) = exp(−iEt/ε)u(x) where E ∈ R and u is a real function, by which equation (1.2) turns into the following quasilinear elliptic equation, where V (x) = W (x) − E and h(u) = f (|u| 2 )u. On the one hand, the study of the existence of ground states (minimal energy solutions) and bound states (finite energy) solutions to (1.3) with ε = 1 has received considerable attention in recent years, see [12,23,24,27,30,34] for arbitrary fixed κ > 0 and [4,32] for κ < 0 and |κ| small. On the other hand, al lot of work has been done on the existence of semi-classical states for (1.3), namely κ = ε 2 and ε → 0 + . Assume V : R N → R is Hölder continuous and satisfies the following condition (V ): and there is a bounded open set O such that With the assumption (V ), the existence of a localized solution concentrating around M := {x ∈ O : V (x) = m} has been studied under various conditions on the nonlinearity h. When N 3, we refer to [11,16,36] for the subcritical case and [18,33] for the critical case. The case of dimension N = 2 has been addressed in [13,14]. Recently, S. Adachi et al. in [1][2][3] have studied asymptotic properties of the ground state of the quasilinear Schrödinger equation −∆u + λu − κu∆u 2 = |u| p−1 u, x ∈ R N .
As κ → 0 + , they have proved that the ground state u κ convergences to a ground state solution of a limit equation for 2 < p < 2 * whence it blows up, in a suitable sense, for p > 2 * . Therefore, it is a legitimate and somewhat interesting question what happens when the two parameters ε and κ in (1.3) tend to zero at different speed rates. The purpose of this paper is to present a unified approach to study the asymptotic properties of the solutions to (1.3) for κ = ∓ε 2+γ with γ > 0 and ε → 0 + . Let us stress the fact that both signs for κ, which is assumed to be small in absolute value, turn out to be physically relevant as developed in [31]. We will show that beyond the expected concentration phenomenon, the limit equations for the case γ > 0 and γ = 0 turn out to be different when κ = ε 2+γ . Moreover, to the best of our knowledge, no result is known in the case κ = −ε 2+γ . Observe that (1.1) is the Euler-Lagrange equation associated to the energy functional From the variational point of view, the first difficulty to handle is to find a proper function space setting where (1.4) is well defined. For κ = ε 2+γ , this difficulty can be overcome by minimization on spheres (mass constraint) or on the Nehari manifold, see [22,24]. Another important tool for this type of equations is to perform a suitable change of variable [23,29] which reduces (1.1) to a semilinear elliptic equation for which variational methods are available, see [12]. Unfortunately, the methods which work in the case κ = ε 2+γ can not be directly adapted to deal with the case κ = −ε 2+γ . Indeed, here the principle part of the corresponding energy functional does change sign, see [4,32]. As we look for positive solutions, let us assume h(t) = 0 for t 0 and we require h ∈ C(R + , R) to enjoy the following conditions, which were introduced by Byeon and Jeanjean in [10], see also [9]: Our main result is the following: Theorem 1.1 (Subcritical case). Let γ > 0 and assume that (V ) and (h 1 )-(h 3 ) hold. Then, for sufficiently small ε > 0, there exists a positive solution u ε,γ of (1.1) satisfying the following: (1) There exists a local maximum point x ε of u ε,γ such that lim ε→0 dist(x ε , M) = 0 and u ε,γ (ε · −εx ε ) converges locally uniformly to a positive ground state of Critical case. Assume in place of (h 2 ) the following critical growth, The above results extend to cover the critical case provided we further assume In [16], similar results to those of Theorem 1.1 are obtained for the equation (1.3) with γ = 0, namely κ = ε 2 and p ∈ (1, 2(2 * ) − 1) in (h 2 ). The limit equation for In our case, where κ = ε 2+γ with γ > 0, the situation is different and we can not cover the range p ∈ (2 * − 1, 2(2 * ) − 1) as the solution can blow up. Closely related problems were considered in [4], where the authors study the following equation, and assuming V is a trapping potential, they have proved the existence of a positive solution for 2 < p < 2 * − 1 and α > 0 small enough. Moreover, in [32] the authors prove that if V ∈ C 1 (R N , [0, +∞)) is such that x·∇V (x)+2V (x) 0 for all x ∈ R N , then the equation (1.6) has only trivial C 2 solutions for any p 2 * − 1 and any α > 0. Finally, let us point out that difficulties in (1.3) with κ = −ε 2 are due to the fact that it is still an open problem if there exists a solution to equation (1.6) for all α > 0. The outline of the paper is as follows: in Section 2, we modify (1.1) by introducing an auxiliary function. In Section 3, we prove the existence of a positive solution for the modified problem. Then, Sections 4 and 5 are devoted to prove Theorem 1.1 respectively in the subcritical and critical case where we use a truncation approach.
Notation. Without loss of generality, we may assume 0 ∈ M. For any set Ω ⊂ R N and ε, δ > 0, we define Ω ε := {x ∈ R N : εx ∈ Ω} and Ω δ := {x ∈ R N : dist(x, Ω) δ}. Let We denote by C a positive constants whose exact value may change from line to line without affecting the overall result.

Vol.88 (2020)
A Unified Approach to Quasilinear Schrödinger Equations 511 A Unified Approach to Quasilinear Schrödinger Equations 5 Note that (2.1) formally is the Euler-Lagrange equation associated to the energy functional However, classical variational methods can not be directly applied since I ε,γ is not well defined on the Hilbert space E ε because of the presence of four homogeneous term w 2 |∇w| 2 .
Remark 2.1. The cut-off function η(t) can be constructed as follows. Let then G ε,γ,± (t) is an odd and smooth function as well as the inverse function G −1 ε,γ,± (t). For any fixed γ > 0 and small ε > 0, next we establish a few properties of G −1 ε,γ,± (t). Lemma 2.2. The following properties hold: By definition of g ε,γ,± (t) and G ε,γ,± (t), (iii) follows. Let us prove (iv): for t > 0 we have Then, for small ε > 0, by (iii) we have Next we consider the following modified quasilinear Schrödinger equation, Direct calculations show that (2.4) turns into (2.1) when g ε,γ,± (t) = √ 1 ± ε γ t 2 and the energy functional related to (2.4) is given bȳ which is well defined on E ε , though it is still non-smooth. As in [29], we exploit the change of variable w = G −1 ε,γ,± (v) to replace the functionalĪ ε,γ,± by the following smooth functional: which implies that w is a weak solution of (2.4).
Define the functional where The functional Q ε will have a penalization effect which forces concentration phenomena to occur inside O. This type of penalization was first introduced in [8]. Clearly, Γ ε,γ,± ∈ C 1 (E ε , R).

By Lemma 2.2-(iii) we also have
Let us estimate P ε,γ,+ and P ε,γ,− separately: L m (G −1 ε,γ,+ (η ε (ts 0 )) C m . So, by (3.9) and (3.10), we get So, by (3.9) and (3.11), we get Combine (i) and (ii) to have lim inf ε→0 C ε,γ,± C m which completes the proof.  Next we borrow some ideas from Byeon and Jeanjean [10] in order to prove the existence of critical points for the modified problem. This method has been used to deal with the case γ = 0 in [16]. Here the situation is quite different as the limit equation is different from the case γ = 0 and we set the problem in a Sobolev space setting instead of the Orlicz space setting used in [16]. We set the problem in the space Note that any v ∈ E R ε can be regarded as an element in E ε by defining v = 0 on R N \ B R/ε (0), so that we may assume the two norms · ε and · ε,R coincide. Define the level set Γ c ε,γ,± := {u ∈ E R ε : Γ ε,γ,± c} and Let v n ∈ X d εn ∩ E Rn εn with ε n → 0 and R n → +∞ such that From the definition of X d εn , we can find {U n } ⊂ S m and a sequence of points Since S m and M β are compact, there exist U ∈ S m and y 0 ∈ M β such that U n → U in H 1 (R N ), U n (x) → U (x) a.e. in R N and y n → y 0 . It is easy to check that for provided n is large enough and in particular {v n } stays bounded in E ε .  Let ε n z n → z 0 ∈ {z ∈ R N : 1 2 β |z − y 0 | 3β}, as n → ∞. Setṽ n := v n (· + z n ) and, up to a subsequence, we may assumeṽ n ṽ in H 1 (R N ) andṽ n →ṽ in L p loc (R N ), p ∈ [2, 2 * ). Then, (3.14) yields Indeed, by Lemma 2.2-(iv), one has G −1 εn,γ,± (ṽ n ) −ṽ n → 0, a.e. in R N . Sinceṽ n →ṽ, a.e. in R N , we get G −1 εn,γ,± (ṽ n ) →ṽ, a.e. in R N . Moreover, g εn,γ,± (G −1 εn,γ,± (ṽ n )) → 1, a.e. in R N . Thus, (3.17) and (3.18) are consequences of the Lebesgue dominated convergence theorem. Combine (3.16)-(3.18) to obtain so thatṽ turns out to be a nontrivial solution to the following equation, Next choose R > 0 sufficiently large and apply Pohozaev's identity to get, on the one hand, on the other hand, for large n, it follows from (3.13) that Note that lim n→∞ |z n − yn εn | = +∞ since 1 2εn β |z n − yn εn | 3β εn . Thus, for n large, we get which is a contradiction for small d > 0. This completes the proof of Lemma 3.4.
Lemma 3.7. For d > 0 sufficiently small, there exist a sequence {z n } ⊂ R N , y 0 ∈ M, and U 0 ∈ S m satisfying, up to a subsequence, the following: Proof. Let w n := v n,1 (· + yn εn ). Then, by (3.13), we get {w n } is bounded in H 1 (R N ). Thus, up to a subsequence if necessary, we may assume w n w in H 1 (R N ), w n → w in L p loc (R N ), p ∈ [2, 2 * ) and w n → w a.e. in R N . From (3.13), for given R > 0, when n large, we get Thus, we have which yields w = 0. Now let φ ∈ C ∞ 0 (R N ) and note that w n (x) = v n,1 (x + yn εn ) = v n (x + yn εn ) for x ∈ supp(φ) and large n. Moreover, supp(w n (x)) ⊂ {x ∈ R N : |ε n x| 2β} ⊂ O. Thus, from Γ εn,γ,± (v n ), φ = o n (1) φ and analogous to the proof of (3.17) and (3.18), w is a nontrivial solution of equation (3.26) By the maximum principle, w > 0.
We have |ε n z n − y n | 1 2 β. In fact, if |ε n z n − y n | 1 2 β, by (3.14), we have which is impossible. Thus, up to a subsequence, we may assume that ε n z n → z 0 ∈ {z ∈ R N : |z| 2β}. Assume that v n,1 (· + z n + yn εn ) v 1 in H 1 (R N ). Analogously to the proof of (3.26), we have By the maximum principle, v 1 > 0.