Abstract
We consider the nonlinear fractional problem
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 ground state solution of the local problem as \(s \rightarrow 1^-\).
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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 \(V :\mathbb {R}^N \rightarrow \mathbb {R}\) and on the nonlinearity \(f :\mathbb {R}^N \times \mathbb {R} \rightarrow \mathbb {R}\). We recall that the fractional laplacian is defined as the principal value of a singular integral via the formula:
with
This formal definition needs of course a function space in which problem (1.1) becomes meaningful: we will come to this issue in Sect. 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, 13, 19].
Instead of fixing the value of the parameter \(s \in (0,1)\), we will start from the well-known identity (see [10, Proposition 4.4]):
valid for any \(u \in C_0^\infty (\mathbb {R}^N)\), and investigate the convergence properties of solutions to (1.1) as \(s \rightarrow 1^-\).
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 \(\mathbb {Z}^N\)-periodic in the space variables. Hence equations (1.1) and (1.3) are invariant under \(\mathbb {Z}^N\)-translations, and their solutions are not unique. We will prove that—up to \({\mathbb {Z}}^N\)-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.
-
(N)
\(N \ge 3\), \(1/2< s < 1\);
-
(V)
\(V \in L^\infty (\mathbb {R}^N)\) is \({\mathbb {Z}}^N\)-periodic and \(\inf _{\mathbb {R}^N} V > 0\);
-
(F1)
\(f :\mathbb {R}^N \times \mathbb {R} \rightarrow \mathbb {R}\) is a Carathéodory function, namely \(f(\cdot , u)\) is measurable for any \(u \in \mathbb {R}\) and \(f(x, \cdot )\) is continuous for a.e. \(x \in \mathbb {R}^N\). Moreover f is \({\mathbb {Z}}^N\)-periodic in \(x \in \mathbb {R}^N\) and there are numbers \(C > 0\) and \(p \in \left( 2, \frac{2N}{N-1} \right) \) such that
$$\begin{aligned} |f(x,u)| \le C (1 + |u|^{p-1}) \end{aligned}$$for \(u \in \mathbb {R}\) and a.e. \(x \in \mathbb {R}^N\).
-
(F2)
\(f(x,u)=o(u)\) as \(u \rightarrow 0\), uniformly with respect to \(x \in \mathbb {R}^N\).
-
(F3)
\(\lim _{|u| \rightarrow +\infty } \frac{F(x,u)}{u^2}=+\infty \) uniformly with respect to \(x \in \mathbb {R}^N\), where \(F(x,u)=\int _0^u f(x,s)\, ds\).
-
(F4)
The function \(\mathbb {R}{\setminus } \{0\} \ni u \mapsto f(x,u)/u\) is strictly increasing on \((-\infty , 0)\) and on \((0, \infty )\), for a.e. \(x \in \mathbb {R}^N\).
Remark 1.1
It follows from (F1) and (F2) that for every \(\varepsilon > 0\) there is \(C_\varepsilon > 0\) such that
for every \(u \in \mathbb {R}\) and a.e \(x \in \mathbb {R}^N\). Furthermore, assumption (F4) implies the validity of the inequality:
for every \(u \in \mathbb {R}\) and a.e. \(x \in \mathbb {R}^N\).
We can now state our main result.
Theorem 1.2
Suppose that assumptions (N), (V), (F1)–(F4) hold. Let \(u_s \in H^s(\mathbb {R}^N)\) be a ground state solution of problem (1.1). Then, there exists a sequence \(\{s_n\}_n \subset (1/2, 1)\), such that \(s_n \rightarrow 1\) as \(n \rightarrow + \infty \) and there exists a sequence of translations \(\{z_n\}_n\), such that \(u_{s_n}(\cdot - z_n)\) converges in \(L^2_{\mathrm {loc}}(\mathbb {R}^N)\) to a ground state solution \(u_0 \in H^1(\mathbb {R}^N)\) of the problem (1.3).
2 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 \(\mathbb {R}^N\) as
endowed with the norm:
One can show that \(C_0^\infty (\mathbb {R}^N)\) is dense in \(H^s(\mathbb {R}^N)\). For \(u \in H^s(\mathbb {R}^N)\), an equivalent norm of u is (see [14, Proposition 1.18])
More explicitly, for every \(u \in H^s(\mathbb {R}^N)\)
where
Lemma 2.1
For every \(u \in H^1(\mathbb {R}^N)\), there results
Proof
From [10, Proposition 3.6], we know that
From [10, Remark 4.3], we know that
Therefore, recalling [10, Corollary 4.2],
\(\square \)
On \(H^s (\mathbb {R}^N)\) we introduce a new norm
which is, under (V), equivalent to \(\Vert \cdot \Vert _{H^s (\mathbb {R}^N)}\). Similarly we introduce the norm on \(H^1 (\mathbb {R}^N)\) by putting
Corollary 2.2
For every \(u \in H^1(\mathbb {R}^N)\), we have
The following convergence result will be used in the sequel.
Lemma 2.3
For every \( \varphi \in C_0^\infty (\mathbb {R}^N)\), there results
Proof
We notice that
where \(C>0\) is a constant, independent of s, that depends on the definition of the Fourier transform \({{\mathcal {F}}}\). It is now easy to conclude, since the Fourier transform of a test function is a rapidly decreasing function. \(\square \)
We will need some precise information on the embedding constant for fractional Sobolev spaces.
Theorem 2.4
[9] Let \(N>2s\) and \(2_s^*=2N/(N-2s)\). Then
for every \(u \in H^s(\mathbb {R}^N)\), where \(\mathbb {S}\) denotes the N-dimensional unit sphere and \(|\mathbb {S}|\) its surface area.
The following inequality in an easy consequence of Theorem 2.4, see also [5, Lemma 2.7].
Lemma 2.5
Let \(N \ge 3\) and \(q \in [2,2N/(N-1)]\). Then there exists a constant \(C=C(N,q)>0\) such that, for every \(s \in [1/2,1]\) and every \(u \in H^s(\mathbb {R}^N)\), we have
Definition 2.6
A weak solution to problem (1.1) is a function \(u \in H^s(\mathbb {R}^N)\), such that
for every \(\varphi \in H^s(\mathbb {R}^N)\).
Weak solutions are therefore critical points of the associated energy functional \({{\mathcal {J}}}_s :H^s (\mathbb {R}^N) \rightarrow \mathbb {R}\) defined by
We recall also the definition of a weak solution in the local case.
Definition 2.7
A weak solution to problem (1.3) is a function \(u \in H^1 (\mathbb {R}^N)\) such that
for every \(\varphi \in H^1(\mathbb {R}^N)\).
For the local problem (1.3), we put \({{\mathcal {J}}}:H^1 (\mathbb {R}^N) \rightarrow \mathbb {R}\)
Recalling the notation (2.1) and (2.2), we can rewrite our functionals in the form:
3 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.
Theorem 3.1
Let \(r > 0\), \(2 \le q < \frac{2N}{N-1}\) and \(N \ge 3\). Suppose moreover that \(\{ s_n \}_n \subset (1/2, 1)\), \(u_n \in H^{s_n} (\mathbb {R}^N)\) and
where \(M > 0\) does not depend on \(s_n\). If
then \(u_n \rightarrow 0\) in \(L^p (\mathbb {R}^N)\) for all \(p \in \left( 2, \frac{2N}{N-1} \right) \).
Proof
Let \(t \in \left( q, \frac{2N}{N-1} \right) \). Then
where \(C > 0\) is independent of \(s_n\) and \(\lambda = \frac{t-q}{\frac{2N}{N-1}-q} \frac{2N}{(N-1) t}\). Choose t such that \(\lambda = \frac{2}{t}\). Then
Covering space \(\mathbb {R}^N\) by balls of radius r, in a way that each point is contained in at most \(N+1\) balls, we get
Hence \(u_{n} \rightarrow 0\) in \(L^t (\mathbb {R}^N)\). Note that
where D does not depend on \(s_n\) and n. Similarly, from Lemma 2.5, there follows that \(\{ u_n\}_n\) is bounded in \(L^{\frac{2N}{N-1}} (\mathbb {R}^N)\). From the interpolation inequality, since \(\{ u_n \}_n\) is bounded in \(L^2 (\mathbb {R}^N)\) and in \(L^{\frac{2N}{N-1}} (\mathbb {R}^N)\), we obtain \(u_n \rightarrow 0\) in \(L^p (\mathbb {R}^N)\) for all \(p \in \left( 2, \frac{2N}{N-1} \right) \). \(\square \)
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 \(s_n \rightarrow 1\), and let \(\{v_{s_n}\}_n \subset H^{s_n}(\mathbb {R}^N)\) be such that
Then, the sequence \(\{v_{s_n}\}_n\) converges, up to a subsequence, to some \(v\in H^1 (\mathbb {R}^N)\) in \(L_{\mathrm {loc}}^q(\mathbb {R}^N)\) for every \(q \in [2,2N/(N-1))\), and pointwise almost everywhere.
Proof
Note that \(H^{s_n} (\mathbb {R}^N) \subset H^{1/2} (\mathbb {R}^N)\) 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 \(n\in \mathbb {N}\), we have
Thus, \(\{ v_{s_n} \}_n\) is bounded in \(H^{1/2} (\mathbb {R}^N)\). Hence, passing to a subsequence, there exists a function v, such that \(v_{s_n} \rightharpoonup v\) in \(H^{1/2} (\mathbb {R}^N)\), \(v_{s_n} \rightarrow v\) pointwise almost everywhere, and \(v_{s_n} \rightarrow v\) in \(L^q_{\mathrm {loc}} (\mathbb {R}^N)\) for every \(q \in [2, 2N/(N-1) )\). From [7, Corollary 7], it follows that \(v \in H^1_{\mathrm {loc}} (\mathbb {R}^N)\). To complete the proof, we need to show that \(v \in H^1 (\mathbb {R}^N)\).
Let \(\widehat{v_{s_n}}\) denote the Fourier transform of \(v_{s_n}\), similarly for \({\widehat{v}}\). We may assume, without loss of generality, that \(\widehat{v_{s_n}} \rightharpoonup {\widehat{v}}\) in \(L^2 (\mathbb {R}^N)\). Note that (3.1) implies that
for some constant \(K > 0\). For \(1/2<t \le 1\), we define
First of all, we observe that
Indeed, for any \(1/2<t < 1\) we have \((1 + |\xi |^2)^{t} \le 1 + |\xi |^2\). Take \(w \in B_1\) and note that
Hence \(w \in B_t\) for any \(t < 1\). Thus
On the other hand, fix \(w \in \bigcap _{1/2< t<1} B_t\). Take any sequence \(t_n \rightarrow 1^-\) with \(t_n > 1/2\). Then, obviously
and Fatou’s lemma yields
Hence \(w \in B_1\), or
and (3.2) is proved. Fix now any \(t \in \left( 1/2, 1 \right) \) and choose \(n_0\) such that \(s_n > t\) for all \(n \ge n_0\). Then
and
Hence, \(\widehat{v_{s_n}} \in B_t\) for \(n \ge n_0\). Each \(B_t\) is a closed and convex subset in \(L^2(\mathbb {R}^N)\), and from [8, Theorem 3.7] it is also weakly closed. Hence, \({\widehat{v}} \in B_t\). Therefore, recalling (3.2),
This implies that
and \(v \in H^1 (\mathbb {R}^N)\). \(\square \)
4 Existence of ground states
It is easy to check that the energy functional \({{\mathcal {J}}}\) 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, 12, 17, 18].
Theorem 4.1
Suppose that assumptions (N), (V), (F1)–(F4) hold. Then there exists a ground state solution \(u_0 \in H^1 (\mathbb {R}^N)\) to (1.3), i.e., a critical point of the functional \({{\mathcal {J}}}\) given by (2.3), such that
where \({{\mathcal {N}}}\) is the so-called Nehari manifold
and
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
Theorem 4.2
Suppose that assumptions (N), (V), (F1)–(F4) hold and \(1/2< s < 1\). Then there exists a ground state solution \(u_s \in H^s (\mathbb {R}^N)\) to (1.1), i.e. a critical point of the functional \({{\mathcal {J}}}_s\) given by (2.3), such that
where \({{\mathcal {N}}}_s\) is the corresponding Nehari manifold
and
5 Non-local to local transition
For any \(s \in (1/2,1)\) we define
Similarly, we put also
For any \(v \in H^s (\mathbb {R}^N) {\setminus } \{0\}\) let \(t_s(v) > 0\) be the unique positive real number such that \(t_s (v) \in {{\mathcal {N}}}_s\). Then, we put \(m_s(v) := t_s (v) v\) .
Lemma 5.1
There results
Proof
Take \(u \in H^1 (\mathbb {R}^N) \subset H^s (\mathbb {R}^N)\) as a ground state solution of (1.3), in particular \(u \in {{\mathcal {N}}}\) and \({{\mathcal {J}}}(u) = c\), where \({{\mathcal {J}}}\) is given by (2.3). Consider the function \(m_s (u) \in {{\mathcal {N}}}_s\). Obviously
Hence
Recall that \(m_s(u) = t_s u\) for some real numbers \(t_s > 0\). Suppose by contradiction that \(t_s \rightarrow +\infty \) as \(s \rightarrow 1^-\). Then, in view of the Nehari identity
but the left-hand side stays bounded (see Corollary 2.2). Hence \((t_s)_s\) is bounded. Take any convergent subsequence \((t_{s_n})\) of \((t_s)\), i.e. \(t_{s_n} \rightarrow t_0\) as \(n\rightarrow +\infty \). Obviously \(t_0 \ge 0\). We will show that \(t_0 \ne 0\). Indeed, suppose that \(t_0 = 0\), i.e. \(t_{s_n} \rightarrow 0\). Then, in view of the Nehari identity
By Corollary 2.2, \(\Vert u \Vert _{s_n}^2 \rightarrow \Vert u\Vert ^2 > 0\). Hence, in view of (F2),
a contradiction. Hence \(t_0 > 0\). Again, by Corollary 2.2,
Moreover, in view of Remark 1.1,
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 \(t_s \rightarrow 1\) as \(s \rightarrow 1^-\). Repeating the same argument we see that
and the proof is completed. \(\square \)
Lemma 5.2
There exists a constant \(M > 0\), such that
for every \(s \in (1/2,1)\).
Proof
Note that \(\Vert u_s\Vert _{L^2 (\mathbb {R}^N)} + \Vert u_s\Vert _{L^\frac{2N}{N-1} (\mathbb {R}^N)} \le C \Vert u_s\Vert _{s}\), for some \(C > 0\) independent of s. So it is enough to show that \(\Vert u_s\Vert _{s} \le M\). Suppose by contradiction that
Put \(v_s := \frac{u_s}{\Vert u_s\Vert _s}\). Then \(\Vert v_s\Vert _{s} = 1\). In particular, \(\{v_s\}\) is bounded in \(L^2 (\mathbb {R}^N)\). Suppose that
Then \(v_s \rightarrow 0\) in \(L^p (\mathbb {R}^N)\). Fix any \(t > 0\). By (4.1) we obtain
From Remark 1.1 we see that
for every \(\varepsilon > 0\). Thus \(\int _{\mathbb {R}^N} F(x,tv_{s_n}) \, \mathrm{d}x \rightarrow 0\) and for any \(t > 0\)
which is a contradiction with the boundedness of \(\{{{\mathcal {J}}}_{s_n}(u_{s_n})\}_n\). Hence (5.1) does not hold, i.e. there is a sequence \(\{z_n\} \subset {\mathbb {Z}}^N\), such that
or, equivalently
From Theorem 3.2, \(v_n (\cdot - z_n) \rightarrow v_0\) in \(L^2_{\mathrm {loc}} (\mathbb {R}^N)\) and pointwise a.e., moreover \(v_0 \ne 0\). See that, for a.e. \(x \in {{\,\mathrm{supp}\,}}v_0\) we have
Thus
a contradiction. \(\square \)
Lemma 5.3
Since \(u_s \in {{\mathcal {N}}}_s\) there is (independent of s) constant \(\rho \), such that
Proof
Since \(u_s \in {\mathcal {N}}_s\), we can write by Remark 1.1
for a constant \(C>0\) independent of s. Choosing \(\varepsilon >0\) small enough, we conclude that
\(\square \)
Corollary 5.4
There exist \(u_0 \in H^1 (\mathbb {R}^N)\), a sequence \(\{z_n\}_n \subset {\mathbb {Z}}^N\) and a sequence \(\{s_n\}_n\) such that \(s_n \rightarrow 1^-\) and
for all \(\nu \in [2, 2N/(N-1))\).
Proof
From Lemma 5.2 and Theorem 3.2 we note that
for all \(\nu \in [2, 2N/(N-1))\). If \(u_0 \ne 0\), we can take \(z_n = 0\) and the proof is completed. Otherwise \(u_{s_n} \rightarrow 0\) in \(L^2_{\mathrm {loc}} (\mathbb {R}^N)\) and therefore, \(u_{s_n} (x) \rightarrow 0\) for a.e. \(x \in \mathbb {R}^N\). Assume that
Then from Theorem 3.1 we know that \(u_{s_n} \rightarrow 0\) in \(L^\nu (\mathbb {R}^N)\) for all \(\nu \in [2, 2N/(N-1))\). Then
and \(\Vert u_{s_n}\Vert _{s_n}^2 = \int _{\mathbb {R}^N} f(x, u_{s_n}) u_{s_n} \, \mathrm{d}x \rightarrow 0\), which is a contradiction with Lemma 5.3. Hence there is a sequence \(\{ z_n \} \subset {\mathbb {Z}}^N\) such that
Moreover \(\Vert u_{s_n} (\cdot - z_n)\Vert _{s_n} = \Vert u_{s_n} \Vert _{s_n}\), so that \(\Vert u_{s_n} (\cdot - z_n)\Vert _{s_n}\) is bounded (see Lemma 5.2). Hence, in view of Theorem 3.2
for some \({\tilde{u}}_0\). Moreover, in view of (5.2), \({\tilde{u}}_0 \ne 0\). \(\square \)
Lemma 5.5
The limit \(u_0 \in H^1 (\mathbb {R}^N) {\setminus } \{0\}\) is a weak solution for (1.3).
Proof
Take any test function \(\varphi \in C_0^\infty (\mathbb {R}^N)\) and note that by [20, Section 6] we have
Moreover
Hence
Obviously
Take any measurable set \(E \subset {{\,\mathrm{supp}\,}}\varphi \) and note that, taking into account Remark 1.1,
Hence, the family \(\{ f(\cdot , u_{s_n}) \varphi \}_n\) is uniformly integrable on \({{\,\mathrm{supp}\,}}\varphi \) and in view of the Vitali convergence theorem
Therefore \(u_0\) satisfies
i.e. \(u_0\) is a weak solution to (1.3). \(\square \)
Proof of Theorem 1.2
Recalling Corollary 5.4 and Lemma 5.5, it is sufficient to check that \(u_0\) is a ground state solution, i.e. \({{\mathcal {J}}}(u_0) = c\). From Lemma 5.5 it follows that \(u_0 \in H^1 (\mathbb {R}^N) {\setminus } \{0\}\) is a weak solution, so that \(u_0 \in {{\mathcal {N}}}\). Note that, from Corollary 5.4 and Fatou’s lemma,
Taking into account Lemma 5.1 we see that
Hence \(\lim _{n \rightarrow +\infty } c_{s_n}\) exists and \(\lim _{n \rightarrow +\infty } c_{s_n} = c = {{\mathcal {J}}}(u_0)\). \(\square \)
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Acknowledgements
The authors would like to thank an anonymous referee for several valuable comments helping to improve the original version of the manuscript. Bartosz Bieganowski was partially supported by the National Science Centre, Poland (Grant No. 2017/25/N/ST1/00531). Simone Secchi is member of the Gruppo Nazionale per l’Analisi Matematica, la Probabilità e le loro Applicazioni (GNAMPA) of the Istituto Nazionale di Alta Matematica (INdAM).
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Bieganowski, B., Secchi, S. Non-local to local transition for ground states of fractional Schrödinger equations on \(\mathbb {R}^N\). J. Fixed Point Theory Appl. 22, 76 (2020). https://doi.org/10.1007/s11784-020-00812-6
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DOI: https://doi.org/10.1007/s11784-020-00812-6