Non-local to local transition for ground states of fractional Schrödinger equations on RN\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathbb {R}^N$$\end{document}

We consider the nonlinear fractional problem (-Δ)su+V(x)u=f(x,u)inRN\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} (-\Delta )^{s} u + V(x) u = f(x,u)&\quad \hbox {in } \mathbb {R}^N \end{aligned}$$\end{document}We show that ground state solutions converge (along a subsequence) in Lloc2(RN)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$L^2_{\mathrm {loc}} (\mathbb {R}^N)$$\end{document}, under suitable conditions on f and V, to a ground state solution of the local problem as s→1-\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$s \rightarrow 1^-$$\end{document}.


Introduction
The aim of this paper is to analyse the asymptotic behavior of least-energy solutions to the fractional Schrödinger problem: (x, u) in R N u ∈ H s (R N ), (1.1) under suitable assumptions on the scalar potential V : R N → R and on the nonlinearity f : R N ×R → R. We recall that the fractional laplacian is defined as the principal value of a singular integral via the formula: u(x) − u(y) |x − y| N +2s dy with 1 C(N, s) = flow, image processing, wave propagation in heterogeneous high contrast media, and stochastic models: see [1,11,13,19]. Instead of fixing the value of the parameter s ∈ (0, 1), we will start from the well-known identity (see [10,Proposition 4.4]): valid for any u ∈ C ∞ 0 (R N ), and investigate the convergence properties of solutions to (1.1) as s → 1 − .
In view of (1.2), it is somehow natural to conjecture that solutions to (1.1) converge to solutions of the problem: We do not know if this conjecture is indeed correct with this degree of generality.
In this paper, we will always assume that both V and f are Z N -periodic in the space variables. Hence equations (1.1) and (1.3) are invariant under Z N -translations, and their solutions are not unique. We will prove that-up to Z N -translations and along a subsequence-least-energy solutions of (1.1) converge to a ground state solution to the local problem (1.3). Our result is a continuation of the previous paper [5], in which we consider the equation on a bounded domain and extend the very recent analysis of Biccari et al. (see [2]) in the linear case for the Poisson problem to the semilinear case. See also [6].
We collect our assumptions.
Carathéodory function, namely f (·, u) is measurable for any u ∈ R and f (x, ·) is continuous for a.e. x ∈ R N . Moreover f is Z N -periodic in x ∈ R N and there are numbers C > 0 and p ∈ 2, 2N and on (0, ∞), for a.e. x ∈ R N .

The variational setting
In this section we collect the basic tools from the theory of fractional Sobolev spaces we will need to prove our results. For a thorough discussion, we refer to [10,14] and to the references therein. For 0 < s < 1, we define a Sobolev space on R N as endowed with the norm: , an equivalent norm of u is (see [14,Proposition 1.18 More explicitly, for every u ∈ H s (R N ) From [10,Remark 4.3], we know that Therefore, recalling [10,Corollary 4.2], On H s (R N ) we introduce a new norm Similarly we introduce the norm on H 1 (R N ) by putting The following convergence result will be used in the sequel.
Proof. We notice that where C > 0 is a constant, independent of s, that depends on the definition of the Fourier transform F. It is now easy to conclude, since the Fourier transform of a test function is a rapidly decreasing function.
We will need some precise information on the embedding constant for fractional Sobolev spaces.
Vol. 22 (2020) Non-local to local transition Page 5 of 15 76 The following inequality in an easy consequence of Theorem 2.4, see also [5,Lemma 2.7].
Weak solutions are therefore critical points of the associated energy functional J s : We recall also the definition of a weak solution in the local case.
For the local problem (1.3), we put J : Recalling the notation (2.1) and (2.2), we can rewrite our functionals in the form:

Uniform Lions' concentration-compactness principle
Since the summability exponent of our space is not fixed, we need a "uniform" version of a celebrated result by Lions.
Covering space R N by balls of radius r, in a way that each point is contained in at most N + 1 balls, we get where D does not depend on s n and n. Similarly, from Lemma 2.5, there fol- Finally, we extend the locally compact embedding into Lebesgue spaces in a uniform way. Proof. Note that H sn (R N ) ⊂ H 1/2 (R N ) and where C > 0 does not depend on s n (and therefore also on n): see for instance [14, 2N/(N − 1)). From [7,Corollary 7], it follows that v ∈ H 1 loc (R N ). To complete the proof, we need to show that v ∈ H 1 (R N ).
Let v sn denote the Fourier transform of v sn , similarly for v. We may assume, without loss of generality, that v sn v in L 2 (R N ). Note that (3.1) implies that First of all, we observe that Indeed, for any 1/2 < t < 1 we have (1 + |ξ| 2 ) t ≤ 1 + |ξ| 2 . Take w ∈ B 1 and note that Hence w ∈ B t for any t < 1. Thus On the other hand, fix w ∈ 1/2<t<1 B t . Take any sequence t n → 1 − with t n > 1/2. Then, obviously and Fatou's lemma yields Hence w ∈ B 1 , or Hence, v sn ∈ B t for n ≥ n 0 . Each B t is a closed and convex subset in L 2 (R N ), and from [8, Theorem 3.7] it is also weakly closed. Hence, v ∈ B t . Therefore,

Existence of ground states
It is easy to check that the energy functional J has the mountain-pass geometry. In particular, there is radius r > 0, such that inf u =r J (u) > 0.
The following existence result is well-known in the literature, and has been shown in various ways, see e.g. [4,12,17,18].
where N is the so-called Nehari manifold The same methods can be applied also in the nonlocal case, and the following existence result can be shown, see e.g. [3,15,16]. In what follows, r s > 0 is the radius chosen so that inf u s=rs J s (u) > 0. J s (γ(t)), (4.1) where N s is the corresponding Nehari manifold and

Non-local to local transition
For any s ∈ (1/2, 1) we define Similarly, we put also Hence lim sup Recall that m s (u) = t s u for some real numbers t s > 0. Suppose by contradiction that t s → +∞ as s → 1 − . Then, in view of the Nehari identity F (x, t s u) t 2 s u 2 u 2 dx → +∞, but the left-hand side stays bounded (see Corollary 2.2). Hence (t s ) s is bounded. Take any convergent subsequence (t sn ) of (t s ), i.e. t sn → t 0 as 76 Page 10 of 15 B. Bieganowski and S. Secchi JFPTA n → +∞. Obviously t 0 ≥ 0. We will show that t 0 = 0. Indeed, suppose that t 0 = 0, i.e. t sn → 0. Then, in view of the Nehari identity By Corollary 2.2, u 2 sn → u 2 > 0. Hence, in view of (F2), a contradiction. Hence t 0 > 0. Again, by Corollary 2.2, t 2 sn u 2 sn → t 2 0 u 2 as n → +∞. Moreover, in view of Remark 1.1, |f (x, t sn u)t sn u| ≤ εt 2 sn |u| 2 + C ε t p sn |u| p ≤ C(|u| 2 + |u| p ) for some constant C > 0, independent of n. In view of the Lebesgue's convergence theorem, Thus, the limit t 0 satisfies Taking the Nehari identity into account we see that t 0 = 1. Hence t s → 1 as s → 1 − . Repeating the same argument we see that and the proof is completed.
Proof. Note that u s L 2 (R N ) + u s L Then v s → 0 in L p (R N ). Fix any t > 0. By (4.1) we obtain From Remark 1.1 we see that for every ε > 0. Thus R N F (x, tv sn ) dx → 0 and for any t > 0 which is a contradiction with the boundedness of {J sn (u sn )} n . Hence ( and pointwise a.e., moreover v 0 = 0. See that, for a.e. x ∈ supp v 0 we have a contradiction.
Proof. Since u s ∈ N s , we can write by Remark 1.1 for a constant C > 0 independent of s. Choosing ε > 0 small enough, we conclude that  2N/(N − 1)).
Proof. From Lemma 5.2 and Theorem 3.2 we note that 2N/(N − 1)). If u 0 = 0, we can take z n = 0 and the proof is completed. Otherwise u sn → 0 in L 2 loc (R N ) and therefore, u sn (x) → 0 for a.e. x ∈ R N . Assume that sup y∈R N B(y, 1) |u sn | 2 dx → 0.