Non-local to local transition for ground states of fractional Schr\"{o}dinger equations on $\mathbb{R}^N$

We consider the nonlinear fractional problem \begin{align*} (-\Delta)^{s} u + V(x) u = f(x,u)&\quad \hbox{in $\mathbb{R}^N$} \end{align*} We show that ground state solutions converge (along a subsequence) in $L^2_{\mathrm{loc}} (\mathbb{R}^N)$, under suitable conditions on $f$ and $V$, to a weak solution of the local problem as $s \to 1^-$.


Introduction
The aim of this paper is to analyse the asymptotic behavior of least-energy solutions to the fractional Schrödinger problem 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 This formal definition needs of course a function space in which problem (1.1) becomes meaningful: we will come to this issue in Section 2.
Several models have appeared in recent years that involve the use of the fractional laplacian. We only mention elasticity, turbulence, porous media flow, image processing, wave propagation in heterogeneous high contrast media, and stochastic models: see [1,11,19,13].
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 subsequenceleast-energy solutions of (1.1) converge to a ground state solution to the local problem (1.3). Our result is a continuation of the previours 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.
and on (0, ∞), for a.e. x ∈ R N . Remark 1.1. It follows from (F1) and (F2) that for every ε > 0 there is C ε > 0 such that |f (x, u)| ≤ ε|u| + C ε |u| p−1 for every u ∈ R and a.e x ∈ R N . Furthermore, assumption (F4) implies the validity of the inequality for every u ∈ R and a.e. x ∈ R N .
We can now state our main result.

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 [14,10] 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 ), Proof. From [10, Proposition 3.6], we know that 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.
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.
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 P.-L. 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 Hence u n → 0 in L t (R N ). Note that Finally, we extend the locally compact embedding into Lebesgue spaces in a uniform way. Theorem 3.2. Let {s n } n be a sequence such that 1/2 < s n < 1 and s n → 1, and let {v sn } n ⊂ H sn (R N ) be such that Then the sequence {v sn } n converges, up to a subsequence, to some v ∈ H 1 (R N ) in L q loc (R N ) for every q ∈ [2, 2N/(N − 1)), and pointwise almost everywhere.
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,Proposition 1.1]. In particular, for every n ∈ N we have v sn 1/2 ≤ C v sn sn ≤ CM. (3.1) Thus {v sn } n is bounded in H 1/2 (R N ). Hence, passing to a subsequence, there exists a function v such that v sn ⇀ v in H 1/2 (R N ), v sn → v pointwise almost everywhere, and v sn → v in L q loc (R N ) for every q ∈ [2, 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 lim inf n→+∞ (1 + |ξ| 2 ) tn |w(ξ)| 2 = (1 + |ξ| 2 )|w(ξ)| 2 and Fatou's lemma yields Hence w ∈ B 1 , or and (3.2) is proved. Fix now any t ∈ (1/2, 1) and choose n 0 such that s n > t for all n ≥ n 0 . Then 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,

This implies that
and v ∈ H 1 (R N ).

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 The following existence result is well-known in the literature, and has been shown in various ways, see e.g. [4,17,18,12]. 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.  where N s is the corresponding Nehari manifold Hence lim sup

Non-local to local transition
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 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 lim sup 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 ( From Theorem 3.2, v n (· − z n ) → v 0 in L 2 loc (R N ) and pointwise a.e., moreover v 0 = 0. See that, for a.e. x ∈ supp v 0 we have |u sn (x − z n )| = u sn sn |v sn (x − z n )| → +∞. Thus 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 Corollary 5.4. There exist u 0 ∈ H 1 (R N ), a sequence {z n } n ⊂ Z N and a sequence {s n } n such that s n → 1 − and u sn (· − z n ) → u 0 = 0 in L ν loc (R N ) as n → +∞ for all ν ∈ [2, 2N/(N − 1)).
Proof. Take any test function ϕ ∈ C ∞ 0 (R N ) and note that by [20, Section 6] we have Take any measurable set E ⊂ supp ϕ and note that, taking into account Remark 1.1, E |f (x, u sn )ϕ| dx ≤ ε u sn L 2 (R N ) ϕχ E L 2 (supp ϕ) Hence the family {f (·, u sn )ϕ} n is uniformly integrable on supp ϕ and in view of the Vitali convergence theorem Therefore u 0 satisfies i.e. u 0 is a weak solution to (1.3).