Blow-up phenomena and asymptotic profiles passing from $H^1$-critical to super-critical quasilinear Schr\"{o}dinger equations

We study the asymptotic profile, as $\hbar\rightarrow 0$, of positive solutions to $$-\hbar^2\Delta u+V(x)u-\hbar^{2+\gamma}u\Delta u^2=K(x)|u|^{p-2}u,\ \ x\in \mathbb{R}^N $$ where $\gamma\geq 0$ is a parameter with relevant physical interpretations, $V$ and $K$ are given potentials and $N\geq 5$. We investigate the concentrating behavior of solutions when $\gamma>0$ and, differently form the case $\gamma=0$ where the leading potential is $V$, the concentration is here localized by the source potential $K$. Moreover, surprisingly for $\gamma>0$ we find a different concentration behavior of solutions in the case $p=\frac{2N}{N-2}$ and when $\frac{2N}{N-2}


Introduction
We are concerned with blow-up phenomena for positive solutions to the following class of quasilinear Schrödinger equations (1.1) where > 0 is the adimensionalized Planck constant, γ ∈ R is a parameter which is relevant in several applications in Physics for which we refer to [20,21], and which we assume here to be positive, V and K are given potentials, for the moment real continuous functions, and the nonlinearity in the range 2N N −2 p < 4N N −2 .We refer to [6,7,24,27] and references therein for the Physics context of (1.1).The existence of nontrivial solutions, in particular ground states for (1.1) has been intensively studied in recent years throughout a very extensive literature, among which let us mention [22,23,26].Though it is not possible to give exhaustive references on the subject, let us recall a few results which are strictly related to our problem.For semi-classical states of (1.1), namely γ = 0 and → 0, assume 2 < p < 4N N −2 , N 3, K(x) ≡ 1, and that V : R N → R is Hölder continuous and satisfying the following conditions: 0 < V 0 < inf x∈R N V (x) and there is a bounded open set Λ such that 0 < a := inf x∈Λ V (x) < min x∈∂Λ V (x).Then, the existence of localized solutions concentrating near Ω := {x ∈ Λ : V (x) = a} has been obtained in [10,18] and, by scaling properties, as → 0 the limit equation turns out to be the following quasilinear autonomous Schrödinger equation (1.2) − ∆u + au − u∆u 2 = |u| p−2 u, x ∈ R N .
We refer to [14,15,19,31,33] for related results.Notice that the scaling invariance of (1.1) breaks down as soon as γ > 0.Recently in [11], it has been proved that in this context both the cases γ = 0 and γ > 0 have similar concentration behavior.However, the limit equation for γ > 0 is different form the case γ = 0 and turns out to be the following semilinear Schrödinger equation As we are going to see, this fact will play a crucial role in studying the blow-up profile of solutions to (1.1).Indeed, loosely speaking one expects solutions can be localized along suitable normalized truncations and translations of ground states to the limit equation (1.2) or (1.3).Here the situation is completely different from the case K ≡ 1 and γ = 0, as a proper normalized, translated and rescaled solution will concentrate around critical points of the potential K.
It is well known from [3,5,30] that for the nonautonomous semilinear Schrödinger equation retains important information for the concentrating behavior of solutions.Remarkably, for our problem (1.1) the external Schrödinger potential V does not play any role in the blow-up phenomenon which is governed by the source potential K.
Another interesting phenomenon addressed in this paper is the different concentrating behavior which occurs passing form critical to supercritical nonlinearities in (1.1).This is due to the fact that the limit equation, as → 0, for (1.1) changes passing from Surprisingly, in the critical case the limit equation turns out to be the zero mass semilinear Schrödinger equation.This fact to the best of our knowledge has not been observed before.In order to state our main results, set For simplicity set κ = y) by V ε (y), K ε (y), respectively.Thus, equation (1.4) can be written in the following form (1.5) We assume the potential V and K satisfying the following conditions: Set M := {x ∈ O : K(x) = m}.
Our main results are the following: Then, for sufficiently small ε > 0, there exists a positive solution v ε of (1.5).
The solution v ε obtained in Theorem 1.1 is actually uniformly bounded with respect to ε.As a consequence we will obtain the blow-up profile of solutions to the original equation (1.1).In Section 2, we prove some preliminary results, in particular we deal with the zero mass case and prove that the equation ), as → 0. Blow-up phenomena for the autonomous version of problem (1.1) (namely, V (x) = λ > 0 and K(x) ≡ 1) have been studied in [1], where in order to get the asymptotic profile of the solution, uniform estimates of the rescaled ground state and energy estimates were established.However, their method can not be applied to deal with the non-autonomous problem (1.1).In [12], the Lyapunov-Schmidt reduction method has been used to deal with the problem (1.7) , as |x| → +∞, the authors proved that for ε sufficiently small, problem (1.7) has a positive fast decaying solution provided 2N N −2 < p < 4N N −2 , N 3. Surprisingly, the limit equation for (1.1) changes again when p = 2N N −2 .Precisely, let Note that λ, ζ → 0 as → 0.
The solution to (1.8) is closely related to the (unique) solution of the following zero mass mean field limit equation [4] (1.9) It is well known since [29] that equation (1.9) possesses an explicit one parameter family of solutions given by , µ > 0.
Throughout this paper, C will denote a positive constant whose exact value may change from line to line without affecting the overall result.

Preliminaries
In this Section we collect a few results, which we will use in the sequel, on the following zero mass equation (2.1) . Uniqueness and non-degeneracy of positive solutions oto (2.1) have been completely solved in [2], see also [12].For convenience of the reader we recall below a few results we need in the sequel.The energy functional related to equation (2.1) is given by and it is well defined in the set Theorem 2.1 (Theorem 1.1 in [2] or Theorem 1.1 in [12]).Equation (2.1) has a unique positive radial solution which belongs to D 1,2 (R N ).In particular, the ground state of (2.1) is unique up to translations.
Lemma 2.2 (Lemma 2.1, [32]).Let g(s) = √ 1 + 2s 2 and G(t) = t 0 g(s)ds.Then, G(t) is an odd smooth function as well as the inverse function G −1 (t).Moreover, the following properties hold: Next consider the following semilinear elliptic equation, which in some sense is the dual problem of (2.1): ( The energy functional corresponding to (2.2) is defined by Solutions to v of (2.2) satisfy the following Pohozaev identity Moreover, the ground state has a mountain pass characterization, namely Theorem 2.3 (Propositions 2.6 and 3.2 in [2]).The following properties hold: 2) has a unique fast decay positive radial solution v(r), namely ) be a positive radially decreasing solution of (2.2).Then there exists C > 0 such that for sufficiently large r.Here is the fundamental solution of −∆ on R N .In particular we have that

Proof of Theorems 1.1 and 1.2
We next consider the following quasilinear Schrödinger equation where g(s) = √ 1 + 2s 2 .Direct calculations show that (3.1) is equivalent to (1.5).The energy functional corresponding to (3.1) is given by Note that J ε is not even well defined in H 1 (R N ).However, it is well known since [13,22] that a suitable dual approach, hidden in change of variables, turns the energy functional to be smooth and well defined in a proper function space setting, see [25], and also [10] for an Orlicz space approach.Here, the change of variables u = G −1 (v), yields the following smooth energy The Euler-Lagrange equation associated to P ε is the following which implies that u is a weak solution of (3.1).Therefore, in order to find nontrivial solutions to (3.1), we are renconducted to find nontrivial solutions of (3.4).Since we are concerned with positive solutions, we actually consider the following truncated energy functional However, in order to avoid cumbersome notations, hereafter we write v in place of v + in the last integral, when this does not yield confusion.Set where where τ > 0 has to be determined later on.By inspection Γ ε ∈ C 1 (H 1 (R N ), R).The functional Q ε will act as a penalization to force the concentration phenomena to occur inside O.This type of penalization was introduced in [8,9].Let U be the unique fast decay positive radial solution of (2.2).Without loss of generality, we may assume U(0) = max U(x) and that 0 ∈ M. Set U t (x) := U( x t ) for t > 0. By (2.3), there exists t 0 > 1 such that Choose a positive number β < dist(M,R N \O) 100 and a cut-off funtion ϕ(x) ∈ C ∞ 0 (R N , [0, 1]) such that ϕ(x) = 1 for |x| β and ϕ(x) = 0 for |x| 2β.Set ϕ ε (x) := ϕ(εx) and then define where We aim at finding a solution of (1.5 ) near the set Next we borrow some ideas from [11], however, here the situation is quite different in particular for the decaying behavior of the ground state solution of the limit equation and the different concentrating behavior of the solution.
Note that any v ∈ E R ε can be regarded as an element of In what follows, for small d > 0, let v n ∈ X d εn ∩ E Rn εn with ε n → 0 and R n → +∞ be such that lim From the definition of X d εn , we can find a sequence {y n } ⊂ M β such that This implies that {v n } is bounded in H 1 (R N ).Since M β is compact, we may assume, up to a subsequence, that y n → y 0 ∈ M β .Lemma 3.4.
Proof.Suppose by contradiction that there exist R > 0 and a sequence {z n } ⊂ {z ∈ R N : ).Then, by (3.16), we get ṽ2 n dx > 0, (3.17) which yields ṽ ≡ 0. Let φ ∈ C ∞ 0 (R N ), then for large n, we have φ( where By the Lebesgue dominated convergence theorem, we get (3.20) Combine (3.18), (3.19) and (3.20), to have the following which implies that ṽ is a positive solution of the following equation Recall that in the right hand side of the above equality, ṽ is actually ṽ+ .Thus, by the maximum principle, we get ṽ > 0. Because of K(z 0 ) m, we get C K(z 0 ) C m .Choosing R > 0 sufficiently large, by Pohozaev's identity we obtain dx.
This fact together with Lions' concentration-compactness lemma give η n → 0 in L q (R N ), q ∈ (2, 2 * ).So, we obtain lim Then, let us prove the following Therefore, by Lemma 2.2−(iv) − (v) and G −1 (0) = 0, for large n, we deduce that In what follows we use the following notation: Proof.From (3.15), we get Similarly, we get For n large enough, we have v n,2 H 1 (R N ) Cd for small d > 0. On the other hand, by Lemma 2.2-(i) − (ii), we get This concludes the proof of Lemma 3.6.

Denote the usual norm in
Lemma 3.7.For small d > 0, there exist a sequence {z n } ⊂ R N and y 0 ∈ M with ε n → 0 and R n → +∞ satisfying, up to a subsequence, the following: ).Thus, up to a subsequence if necessary, we may assume w n ⇀ w in H 1 (R N ), w n → w in L q loc (R N ), q ∈ [2, 2 * ), w n → w a.e. in R N .From (3.15), for given R > 0, as n is large enough we get Thus, we have φ εn,Rn and analogously to the proof of (3.19) and (3.20), w is a positive solution of the following equation We have which is impossible.Thus, up to a subsequence, we may assume As in the proof of (3.30), we have By the maximum principle v 1 > 0. which yields lim n→∞ v n,2 * εn,Rn = 0 and the Lemma is proved.
Let d ∈ (0, d 0 ) such that Lemmas 3.4, 3.5, 3.6 and 3.7 hold and define Lemma 3.8.For any d ∈ (0, d 0 ), there exist positive constants δ d , R d and ε d such that By contradiction, we assume that for some d ∈ (0, d 0 ), there exists up to a subsequence.Thus, for large n, εn , which contradicts the fact v n ∈ X d 0 εn \ X d εn .Lemma 3.9.For any given δ > 0, there exist small positive constants ε 1 and Similarly to the proof of Lemma 3.1, for small ε > 0, we have On the other hand, for v ∈ X d ε , by choosing d small enough, we have The result follows from (3.35) and (3.36).Lemma 3.10.For sufficiently small ε > 0 and large R > 0, there exists a sequence The proof is similar to [17,18].For reader's convenience, let us give a detailed proof.By contradiction, for small ε > 0 and large R > 0, there exists On the other hand, by Lemma 3.8, there exists δ > 0 independent of ε ∈ (0, ε 0 ) and R > R 0 such that Thus, there exists a pseudo-gradient vector field We choose two positive Lipschitz continuous functions ζ R ε and ξ satisfying (3.37) Then the initial value problem For the properties of F R ε , we refer to e.g.[18,28].Let η ε (s) = W ε,st 0 = ϕ ε U st 0 , s ∈ [0, 1] as before.Then, for the small d 1 > 0, there exists some µ > 0 such that if In this case one of the following alternatives holds: Thus, lim t→+∞ Γ ε (η R ε (s, t)) = −∞, which contradicts to Lemma 3.9.So, we have that (b) holds.For any fixed s with |st 0 − 1| µ, we find t s | > σ for some σ > 0 dependent of d 0 and d 1 .Thus, by Remark 3.3, we get Lemma 3.11.For sufficiently small ε > 0, there exists a critical point Proof.By Lemma 3.10, there exists ε 0 and R 0 > 0 such that there exists a sequence .By applying standard Moser's iteration (see [16]), {v R ε } is bounded in L q loc (R N ) uniformly on R R 0 and ε ∈ (0, ε 0 ) for any q < ∞.Moreover, for any y ∈ R N , we have v R ε L q (B 3 (y)) C v R ε L p 2 (B 4 (y)) .(3.41)By Theorem 8.17 [16] and (3.41), we have sup In particular this implies that v R ε stays bounded in L ∞ (R N ).Since v R ε ε and {Γ ε (v R ε )} are bounded, we get {Q ε (v R ε )} is uniformly bounded on R R 0 and ε ∈ (0, ε 0 ).So, we have for any R R 0 and ε ∈ (0, ε 0 ).Thus, for |x| (3.44) Since v ε ∈ X d 0 ε , by Moser's iteration [16], {v ε } is uniformly bounded in L ∞ (R N ) for small ε > 0. By Lemma 3.4, for small d > 0, there exist a sequence {z ε } ⊂ R N and y 0 ∈ M satisfying