Abstract
In this article we are interested in the nonlinear Schrödinger equation with non-local regional difussion
where \(f\) is a super-linear sub-critical function and \((-\Delta )_{\rho }^{\alpha }\) is a variational version of the regional laplacian, whose range of scope is a ball with radius \(\rho (x)>0\). We study the existence of a ground state and we analyze the behavior of semi-classical solutions as \(\varepsilon \rightarrow 0\).
Similar content being viewed by others
Avoid common mistakes on your manuscript.
1 Introduction
The aim of this article is to study the non-linear Schrödinger equation with non-local regional difussion
where \(0< \alpha <1\), \(\epsilon >0\), \(n\ge 2\) and \(f:\mathbb {R} \rightarrow \mathbb {R}\) is super-linear and has a sub-critical growth. The operator \((-\Delta )_{\rho }^{\alpha }\) is a variational version of the non-local regional laplacian, with range of scope determined by the positive function \(\rho \in C(\mathbb {R}^{n}, \mathbb {R}^{+})\).
Recently, a great attention has been focused on the study of problems involving the fractional Laplacian, from a pure mathematical point of view as well as from concrete applications, since this operator naturally arises in many different contexts, such as, obstacle problems, financial mathematics, phase transitions, anomalous diffusions, crystal dislocations, soft thin films, semipermeable membranes, flame propagations, conservation laws, ultra relativistic limits of quantum mechanics, quasi-geostrophic flows, minimal surfaces, materials science and water waves. The literature is too wide to attempt a reasonable list of references here, so we refer the reader to the work by Di Nezza, Patalluci and Valdinoci [4], where a more extensive bibliography and an introduction to the subject are given.
In the context of fractional quantum mechanics, the non-linear fractional Schrödinger equation has been proposed by Laskin [13, 14] as a result of expanding the Feynman path integral, from the Brownian-like to the Lévy-like quantum mechanical paths. In the last 10 years, there has been a lot of interest in the study of the fractional Schrödinger equation, see the works in [2, 5, 8, 11] and [16]. In a recent paper Felmer et al. [8] considered positive solutions of nonlinear fractional Schrödinger equation
They obtained the existence of a ground state by mountain pass argument and a comparison method devised by Rabinowitz [18] for \(\alpha = 1\). They analyzed regularity, decay and symmetry properties of these solutions. At this point it is worth mentioning that the uniqueness of the ground state with power non-linearity was proved by Frank and Lenzmann [7] in the one dimensional case, and advances in the multi-dimensional case were obtained recently by Fall and Valdinoci [6]. We also mention the work by Cheng [2], where the fractional Schrödinger equation with unbounded potential
was studied. The existence of a ground state of (1.3) is obtained by a Lagrange multiplier method and the Nehari manifold method is used to obtain standing waves with prescribed frequency.
On the other hand, research has been done in recent years for the regional fractional laplacian, where the scope of the operator is restricted to a variable region near each point. We mention the work by Guan [9] and Guan and Ma [10] where they study these operators, their relation with stochastic processes and they develop an integration by parts formula, and the work by Ishii and Nakamura [12], where the authors studied the Dirichlet problem for regional fractional laplacian modeled on the \(p\)-laplacian. These regional operators present various interesting characteristics that make them very attractive from the point of view of mathematical theory of non-local operators.
In this article we are interested in studying the non-linear Schrödinger equation when a variational version of the regional fractional laplacian is considered. We are specially interested in understanding the role of the scope function \(\rho \) on the existence of positive solution and concentration in the semi-classical limit for Eq. (1.1).
Now we make precise assumptions on \(\rho \) and \(f\). For the scope function \(\rho \) we assume \(\rho : \mathbb {R}^{n}\rightarrow \mathbb {R}^{+}\) is continuous and it satisfies the following hypotheses:
-
\((\rho _{1})\) There are numbers \(0<\rho _0<\rho _\infty \le \infty \) such that
$$\begin{aligned} \rho _{0 } \le \rho (x) < \rho _{\infty }, \quad \forall x \in \mathbb {R}^{n} \quad \text{ and }\quad \lim _{|x| \rightarrow \infty } \rho (x) =\rho _\infty . \end{aligned}$$ -
\((\rho _{2})\) In case \(\rho _\infty =\infty \) we further assume that there exists \(a\in (0,1)\) such that
$$\begin{aligned} \limsup _{|x| \rightarrow \infty } \frac{\rho (x)}{|x|}\le a. \end{aligned}$$ -
\((\rho _{3})\) For any \(x_{0}\in \mathbb {R}^{n}\), the equation
$$\begin{aligned} |x| = \rho (x+x_{0}),\quad x\in \mathbb {R}^n, \end{aligned}$$defines an \((n-1)\)-dimensional surface of class \(C^1\) in \(\mathbb {R}^{n}\).
Regarding the non-linearity \(f\) we assume that \(f:\mathbb {R}\rightarrow \mathbb {R}\) is a continuous function that satisfies the following hypotheses:
-
\((f_{1})\) \(f(t) \ge 0\) if \(t\ge 0\) and \(f(t) = 0\) if \(t \le 0\).
-
\((f_{2})\) The function \(t \rightarrow \frac{f(t)}{t}\) is increasing for \(t > 0\) and \(\lim _{t\rightarrow 0} \frac{f(t)}{t} = 0\).
-
\((f_{3})\) \(\exists \;\theta > 2\) such that \(\forall t >0\)
$$\begin{aligned} 0< \theta F(t) \le tf(t),\quad \text{ where }\quad F(t) = \int _{0}^{t}f(\xi )d\xi . \end{aligned}$$ -
\((f_{4})\) \(\exists \; C>0\) such that
$$\begin{aligned} |f(t)| \le C(1+ |t|^{p}),\;\;1 < p < \frac{n+2\alpha }{n-2\alpha }. \end{aligned}$$
Before stating our results let us introduce the main ingredients involved in our approach. We let \(H^\alpha (\mathbb {R}^{n})\) be the usual Sobolev space (see Sect. 2) equipped with the norm
Given a function \(\rho \) as above, we define
and the space
For \(u\in H_{\rho }^{\alpha }\) and \(f\) satisfying (\(f_{1}\))–(\(f_{4}\)), as we see in Sect. 2, we may define the functional
which is of class \(C^1\). We say that \(u\in H^{\alpha }(\mathbb {R}^{n})\) is a weak solution of (1.1) if \(u\) is a critical point of \(I_\rho \).
Now we are in a position to state our main existence theorem
Theorem 1.1
Assume \(0<\alpha <1\) and \(n\ge 2\). If \(f\) satisfies (\(f_{1}\))–(\(f_{4}\)) and \(\rho \) satisfies (\(\rho _{1}\))–(\(\rho _{2}\)) then (1.1) possesses at least one non-trivial weak solution. Moreover this solution satisfies \(u(x)\ge 0\) a.e. for all \(x\in \mathbb {R}^{n}.\)
We prove the existence of a weak solution of (1.1) by applying the mountain pass theorem [1] to the functional \(I_{\rho }\) defined on \(H_{\rho }^{\alpha }(\mathbb {R}^{n})\). However, since Palais-Smale sequences lose compactness in \(\mathbb {R}^{n}\), we need extra arguments based on comparison of the mountain pass critical value of \(I_{\rho }\) with that of the limiting functional \(I_{\rho _\infty }\), as devised by Rabinowitz [18]. See the work in [8], where a similar argument is used in the context of the \(\alpha \)-Laplacian.
In our second main theorem we are interested in the concentration behavior of ground states for the equation
when the positive parameter \(\epsilon \) approaches zero. The scope function \(\rho \), that describes the size of the ball of the influential region of the non-local operator, plays a key role in deciding the concentration point of ground states of the equation. Even though, at a first sight, the minimum point of \(\rho \) seems to be the concentration point, there is a non-local effect that needs to be taken in account. We define the concentration function
where the sets \(\mathcal {C}^{+}({x})\) and \(\mathcal {C}^{-}({x})\) are defined as follows
and
Here we interpret the quotient \(1/\rho _\infty ^{2\alpha }\) as zero, when \(\rho _\infty =\infty \). Now we state our second theorem
Theorem 1.2
Let \(0<\alpha <1\), \(n\ge 2\). Suppose that \(\rho \) satisfies (\(\rho _{1}\))–(\(\rho _{3}\)) and \(1 < p < \frac{n+2\alpha }{n-2\alpha }\). Then for each sequence \(\epsilon _{m} \rightarrow 0\), there exists a subsequence such that for every \(m\), there is a non-negative solution \(u_m=u_{\epsilon _m}\) of (1.7) that concentrates around a global minimum point \(x_{0}\) of \(\mathcal {H}\), as \(\epsilon _m\rightarrow 0\). In more precise terms, for every \(\delta >0\) there exists \(R>0\) and \(\epsilon _{0} >0\) such that if \(\epsilon < \epsilon _{0}\) we have
with \(C\) a constant independent of \(\delta \) and \(m\).
The proof of this theorem again uses a comparison arguments in order to obtain the concentration, but here the estimates are more delicate, see Theorem 4.1. At this point we would like to comment that regularity of weak solutions to (1.1) is not considered in this article. In [8] the authors obtained regularity, up to Hölder continuity, by a bootstrap argument together with localization technique. We think that these arguments could be adapted to this problem, but that may require a considerable amount of additional work. Regularity of the solution could, in particular, help to obtain decay estimates for the solutions and then we could study further concentration properties of Eq. (1.7).
This article is organized as follows. In Sect. 2 we present preliminaries with the main tools and the functional setting of the problem and we discuss the definition of \((-\Delta )_{\rho }^{\alpha }\) and its regional character. In Sect. 3 we prove Theorem 1.1. In Sect. 4 we provide some properties of the concentration function \(\mathcal {H}\) and we compute asymptotic values of the functional. In Sect. 5 we complete the study of the semi-classical limit, proving Theorem 1.2.
2 Preliminaries
In this section we recall some basic facts about the Sobolev space \(H^\alpha (\mathbb {R}^n)\), its relations with the space \(H^\alpha _\rho (\mathbb {R}^n)\), embeddings and compactness properties. We also discuss about the variational version of the regional fractional laplacian we consider in this paper.
Regarding the space \(H^\alpha (\mathbb {R}^n)\) we recall the following embedding theorem.
Theorem 2.1
[4] Let \(\alpha \in (0,1)\), then there exists a positive constant \(C = C(n,\alpha )\) such that
and then \(H^{\alpha }(\mathbb {R}^{n}) \hookrightarrow L^{q}(\mathbb {R}^{n})\) is continuous for all \(q \in [2, 2_{\alpha }^{*}]\). Moreover, \(H^{\alpha }(\mathbb {R}^{n}) \hookrightarrow L^{q}(\Omega )\) is compact for any bounded set \(\Omega \subset \mathbb {R}^{n}\) and for all \(q \in [2, 2_{\alpha }^{*})\), where \(2_{\alpha }^{*} = \frac{2n}{n-2\alpha }\) is the critical exponent.
For the spaces \( H^{\alpha }(\mathbb {R}^{n})\) and \( H_{\rho }^{\alpha }(\mathbb {R}^{n})\) defined in the introduction, we consider the norms \(\Vert \cdot \Vert \) and \(\Vert \cdot \Vert _{\rho }\), respectively, as defined in (1.4) and (1.5). The following result is very important in our study.
Proposition 2.1
If \(\rho \) satisfies (\(\rho _{1}\)) there exists a constant \(C = C(n, \alpha , \rho _{0})> 0\) such that
Proof
From the definition of \(\Vert \cdot \Vert \), given \(u \in H_{\rho }^{\alpha }(\mathbb {R}^{n})\) and for \(\rho _0>0\), we have
Using Fubini’s Theorem, we see that
Then, from hypothesis (\(\rho _1\)), (2.2) and (2.3) it follows that
where \(C= C(n,\alpha ,\rho _{0})\). This completes the proof. \(\square \)
Remark 2.1
By Proposition 2.1 we have that \(H_{\rho }^{\alpha }(\mathbb {R}^{n}) \hookrightarrow H^{\alpha }(\mathbb {R}^{n})\) is continuous and then, by Theorem 2.1, we have that \(H_{\rho }^{\alpha }(\mathbb {R}^{n}) \hookrightarrow L^{q}(\mathbb {R}^{n})\) is continuous for any \(q\in [2,2_{\alpha }^{*}]\) and \(H_{\rho }^{\alpha }(\mathbb {R}^{n}) \hookrightarrow L_{loc}^{q}(\mathbb {R}^{n})\) is compact for any \(q\in [2,2_{\alpha }^{*})\).
Remark 2.2
Since \(\Vert u\Vert _{\rho } \le \Vert u\Vert \), under the condition \((\rho _{1})\) Proposition 2.1 implies \(\Vert \cdot \Vert _{\rho }\) and \(\Vert \cdot \Vert \) are equivalent norms in \(H^{\alpha } (\mathbb {R}^{n})\).
The following lemma is a version of the concentration compactness principle proved by Coti Zelati and Rabinowitz [3].
Lemma 2.1
Let \(n\ge 2\). Assume that \(\{u_{k}\}\) is bounded in \(H_{\rho }^{\alpha }(\mathbb {R}^{n})\) and it satisfies
where \(R>0\). Then \(u_{k} \rightarrow 0\) in \(L^{q}(\mathbb {R}^{n})\) for \(2 < q < 2_{\alpha }^{*}\).
Proof
Let \(2 < q < 2_{\alpha }^{*}\) and consider \(\theta \in (0,1)\) such that
Then by Hölder inequality and Theorem 2.1, for every \(k\) we have
so
Taking \(\theta = \frac{2}{q}\) and covering \(\mathbb {R}^{n}\) with balls of radius \(R\), in such a way that each point of \(\mathbb {R}^{n}\) is contained in at most \(n+1\) balls, we deduce that
Then, by hypothesis, \( u_{k} \rightarrow 0,\) in \(L^{q}(\mathbb {R}^{n}).\) \(\square \)
The spaces \({H}^{\alpha }(\mathbb {R}^{n})\) and \({H}_\rho ^{\alpha }(\mathbb {R}^{n})\) are Hilbert spaces endowed with the inner products
and
respectively. Using the equivalence of the norms \(\Vert \cdot \Vert \) and \(\Vert \cdot \Vert _\rho \) proved in Proposition 2.1, we can use the Lax-Milgram representation theorem to find a unique bijective linear map \(\mathcal {M}_{\rho }: H^{\alpha }(\mathbb {R}^{n}) \rightarrow H^{\alpha }(\mathbb {R}^{n})\) such that
On the other hand, using Fourier transform we may consider an alternative way of defining the Sobolev space \({H}^{\alpha }(\mathbb {R}^{n})\) as
with inner product and norm given by
If we define the fractional laplacian as
then it can be shown that for \(0<\alpha < 1\) there exists a constant \({C}(n, \alpha )\) such that for all \(u,v \in H^{\alpha }(\mathbb {R}^{n})\),
See [4] Proposition 3.3. Then we have that
and if we define the operator \((-\Delta )_{\rho }^{\alpha }:H^{\alpha }(\mathbb {R}^{n})\rightarrow H^{-\alpha }(\mathbb {R}^{n})\) by
where \(\mathcal {I}\) is the natural injection from \(H^{\alpha }(\mathbb {R}^{n})\) to \(H^{-\alpha }(\mathbb {R}^{n})\) and \(H^{-\alpha }(\mathbb {R}^{n})\) denotes the dual space of \(H^{\alpha }(\mathbb {R}^{n})\). With these definitions we finally have that for all \(u,v\in H^{\alpha }(\mathbb {R}^{n})\)
Proposition 2.2
If \(\rho \) satisfies (\(\rho _{1}\)) and (\(\rho _{2}\)), then the operator \((-\Delta )_{\rho }^{\alpha }\) is regional, in the sense that given \(u\in H^{\alpha }(\mathbb {R}^{n})\) with compact support, there exists \(R>0\) so that for all \(\varphi \in H^{\alpha }(\mathbb {R}^{n})\) such that
then
Proof
We assume that \(\rho _\infty =\infty \) in (\(\rho _1\)) (the other case is similar). By (\(\rho _2\)) there exists \(\bar{a}\in (0,1)\) and \(R>0\) so that \(|\rho (x)|\le \bar{a} |x|\) for all \(|x|\ge R\).
If \(\varphi \in H^{\alpha }(\mathbb {R}^{n})\) is such that \(\mathrm{supp } \varphi \cap B(0, R)=\emptyset \) then
On the other hand, given \(u\in H^{\alpha }(\mathbb {R}^{n})\) with compact support, there exists \(R_0\) such that
Then we make \(R\) larger, if necessary, in order to get \(R_0< (1-\bar{a}) R\) and thus we have that for any \((x,z)\in \mathbb {R}^{n}\times B(0,\rho (x))\) such that \(x\in \mathrm{supp } \varphi \) or \(x+z\in \mathrm{supp } \varphi \) we have \(|x|\ge (1-\bar{a})R>R_0\) and then \(u(x)=u(x+z)=0\). Consequently we have that
\(\square \)
3 The ground state
In this section, our goal is to prove the existence of a ground state of Eq. (1.1), that is, a non-negative solution with lowest energy.
We start with a precise definition of the notion of solutions for Eq. (1.1).
Definition 3.1
We say that \(u\in H_{\rho }^{\alpha }(\mathbb {R}^{n})\) is a weak solution of (1.1) if
We prove the existence of weak solution of (1.1) finding a critical point of the functional \(I_{\rho }\) defined in (1.6). Using the properties of the Nemistky operators and the embeddings given in Remark 2.1, we can prove that the functional \(I_{\rho }\) is of class \(C^{1}(H_{\rho }^{\alpha }(\mathbb {R}^{n}), \mathbb {R})\) and we have
We define the Nehari manifold associated to the functional \(I_{\rho }\) as
and we observe that all non trivial solutions of (1.1) belong to \( \mathcal {N}_{\rho } \). Next, from the growth assumptions \((f_{2})\) and \((f_{4})\) it is standard to prove that, for any \(\epsilon >0\), there exists \(C_{\epsilon }\) such that
and consequently
where \(1 < p <\frac{n+2\alpha }{n-2\alpha } = 2_{\alpha }^{*} - 1\). We start our analysis with
Lemma 3.1
Assume the hypotheses (\(\rho _{1}\)) and (\(f_{1}\))–(\(f_{4}\)) hold. Then for any \(u\in H_{\rho }^{\alpha }(\mathbb {R}^{n})\setminus \{0\}\), there is a unique \(t_{u} = t(u) > 0\) such that \(t_{u}u \in \mathcal {N}_{\rho }\) and
Proof
Let \(u\in H_{\rho }^{\alpha }(\mathbb {R}^{n})\setminus \{0\}\) and consider the function \( \psi : \mathbb {R}^{+} \rightarrow \mathbb {R}\) defined as
By (3.2) we have
then \(\psi (t) > 0,\;\;\text{ for }\;\;t\;\;\text{ small }\). On the other hand, by (\(f_{3}\)) exists \(A>0\) such that \(F(t) \ge A|t|^{\theta },\;\;\forall t>0\). So
and since \(\theta > 2\), we see that \(\psi (t) < 0\) for \(t\) large. By (\(f_{2}\)), \(\psi (0) = 0\), therefore there is \(t_{u} = t(u) > 0\) such that
We see that \(\psi '(t) = 0\) is equivalent to
from where, using (\(f_{2}\)) we see that \(t_{u} >0\) such that \( t_{u}u \in \mathcal {N}_{\rho } \) is unique. \(\square \)
Now we define two critical values as follows
where \(\Gamma _{\rho }\) is given by
Under our assumptions, \(\Gamma _{\rho }\) is not empty and \(c_{\rho } >0\). We have
Lemma 3.2
Proof
We notice that \(I_{\rho }\) is bounded below on \(\mathcal {N}_{\rho }\), since by (\(f_{3}\)), \(I_{\rho }(u) >0\), for all \(u\in \mathcal {N}_{\rho }\), so that \(c_{\rho }^{*}\) is well defined. By Lemma 3.1 for any \(u\in H_{\rho }^{\alpha }(\mathbb {R}^{n})\setminus \{0\}\) there is a unique \(t_{u} = t(u) >0\) such that \( t_{u}u \in \mathcal {N}_{\rho }, \) then
On the other hand, for any \(u\in \mathcal {N}_{\rho }\), we have
so
therefore the first equality in (3.2) holds. Next we prove the other equality. We claim that for every \(\gamma \in \Gamma _{\rho }\) there exists \(t_{0} \in [0,1]\) such that \(\gamma (t_{0}) \in \mathcal {N}_{\rho }\).
In fact, by (3.1) and continuous embeddings we have
and then, for \(\gamma \in \Gamma _\rho \) we have
If we take \(r = \left( \frac{1 - \epsilon C}{C_{\epsilon } C} \right) ^{\frac{1}{p-1}}\), then we see that
On the other hand, using \((f_3)\) and since \(I_{\rho }(\gamma (1)) <0\), we have
that implies \( I'_{\rho }(\gamma (1))\gamma (1) < 0. \) Thus, by the Intermediate Value Theorem, there exists \(t_{0}\in (t_{*}, 1)\) such that \( I'_{\rho }(\gamma (t_{0}))\gamma (t_{0}) = 0\) and so \(\gamma (t_{0}) \in \mathcal {N}_{\rho }. \), completing the proof of the claim. From this result, \(\max _{t\in [0,1]} I_{\rho }(\gamma (t)) \ge I_{\rho }(\gamma (t_{0})) \ge \inf _{\mathcal {N}_{\rho }} I_{\rho }\) and then
In order to prove the other inequality we see that from (3.3), there exists \(t^{*}\) large enough such that \(I_{\rho }(t^{*} u) <0\). Now we define the curve \(\gamma _{u} : [0,1] \rightarrow H_{\rho }^{\alpha }(\mathbb {R}^{n})\) as \(\gamma _{u}(t) = t(t^{*}u)\). Then \( \gamma _{u} (0) = 0,\) \(I_{\rho }(\gamma (1)) = I_{\rho }(t^{*}u) <0\) and \(\gamma _u\) is continuous, so that \(\gamma _{u} \in \Gamma _{\rho }\). Now, by definition of \(\gamma _{u}\),
then \(c_{\rho }^{*} \ge c_{\rho }\), completing the proof. \(\square \)
Lemma 3.3
Suppose \(\{u_{k}\} \subset H_{\rho }^{\alpha }(\mathbb {R}^{n})\) and there exists \(b>0\) such that
Then either
-
(i)
\(u_{k} \rightarrow 0\) in \(H_{\rho }^{\alpha }(\mathbb {R}^{n})\), or
-
(ii)
there is a sequence \((y_{k})\in \mathbb {R}^{n}\), and \(R, \beta > 0\) such that
$$\begin{aligned} \liminf _{k\rightarrow \infty } \int _{B(y_{k}, R)} |u_{k}(x)|^{2}dx > \beta . \end{aligned}$$
Proof
By (3.9) it is standard to check, for \(k\) large enough
and then \(\{u_{k}\}\) is bounded in \(H_{\rho }^{\alpha }(\mathbb {R}^{n})\). Suppose (\(ii\)) is not satisfied, then for any \(R>0\), (2.4) holds. Consequently by Lemma 2.1
Then, noticing that
by (3.1) and the continuous embedding we have
where \(1 < p < 2_{\alpha }^{*}-1\). So
Choosing an appropriate \(C\) and using (3.9) and (3.11), we find that \(u_{k} \rightarrow 0\;\;\text{ in }\;\;H_{\rho }^{\alpha }(\mathbb {R}^{n})\), that is, (\(i\)) holds. \(\square \)
Continuing with our analysis, we consider the functional at infinity. When \(\rho _{\infty } = \infty \) then the associated limiting functional is defined as
and its Euler-Lagrange equations is given by
This problem was studied by Felmer et al. [8] and they have proved the following theorem
Theorem 3.1
[8] Under (\(f_{1}\))–(\(f_{4}\)), the functional \(I\) possesses at least one critical point with critical value \(c\), where \(c\) is defined by
and \(\Gamma \) is given by
When \(\rho _{\infty } < \infty \), then we define \(I_{\rho _{\infty }}\) as in (1.6) with \(\rho (x)\equiv \rho _\infty \). Following [8] and since the functional \(I_{\rho _{\infty }}\) is invariant under translation, we can prove that \(I_{\rho _{\infty }}\) has a critical point with critical value \(c\), given by (3.16) with \(I_{\rho _{\infty }}\) instead of \(I\). Now we prove the existence of weak solution of (1.1).
Proof of Theorem 1.1
By Lemma 3.2, for every sequence \(\{\epsilon _{k}\}\), there exists a sequence of \(\{u_{k}\}\) in \(H_{\rho }^{\alpha }(\mathbb {R}^{n})\) such that \(\Vert u_{k} \Vert _{H_{\rho }^{\alpha }} = 1\) and
As in the proof of Lemma 3.2, associated with each \(u_{k}\), there is a function \(\gamma _{k} \in \Gamma _{\rho }\) such that
Now, considering \(X = H_{\rho }^{\alpha }(\mathbb {R}^{n})\), \(K = [0,1]\), \(K_{0} = \{0,1\}\), \(M = \Gamma _{\rho }\), \(\varphi = \gamma _{k}\) and
we can use Theorem \(4.3\) of [15], to find a sequence \(\{w_{k}\}\) in \(H_{\rho }^{\alpha }(\mathbb {R}^{n})\) and \(\{\xi _{k}\} \subset [0,1]\) such that \(I_{\rho }(w_{k}) \in (c_{\rho } - \epsilon _{k}, c_{\rho } + \epsilon _{k})\),
Now, since
as in the proof of the Lemma 3.3, we show that \(\{w_{k}\}\) is bounded in \(H_{\rho }^{\alpha }(\mathbb {R}^{n})\). Moreover, up to a subsequence,
where \(w\) is weak solution of (1.1). By Lemma 3.3, there is a sequence \(\{y_{k}\} \subset \mathbb {R}^{n}\), \(\beta >0\) and \(R>0\) such that
If \(\{y_{k}\}\) contains a bounded subsequence, then (3.22) guarantees that \(w \ne 0\) and the results follows. If \(\{y_{k}\}\) is an unbounded sequence, first we prove that
in case \(\rho _{\infty } = \infty \) or \(\rho _{\infty } < \infty \), respectively. When \(\rho _\infty =\infty \) we let \(u\) be a critical point \(u\) with critical value \(c\) and for any \(y\in R^{n}\), we define \(u_{y}(x) = u(x+y)\). Then for any \(t>0\) we have
Let \(t^{*}>0\) such that \(t^{*}u_{y} \in \mathcal {N}_{\rho }\) and
consequently \( c > I_{\rho }(t^{*}u_{y}) \ge \inf _{\mathcal {N}_{\rho }} I_{\rho }(u) = c_{\rho }, \) proving the first inequality in (3.23). When \(\rho _\infty <\infty \), the other inequality is obtained similarly.
Now we may assume that, for given \(R>0\),
since the contrary implies that \(w \ne 0\) and we finish the proof.
We analyze first the case \(\rho _{\infty } = +\infty \). For this purpose we write
for \(t\ge 0\), and we estimate the second term on the right. First we see that for any \(\epsilon > 0\) and \(\bar{t}>0\), there exists \(R>0\) such that
for all \(t\in [0,\bar{t}]\). In fact, by our assumption, for any \({M} >0\), exists \(R>0\) such that, for \(|x| > R\) we have that \(\rho (x)> {M}\). From here, interchanging the order of integration and using the continuous embedding, we have
from were we conclude (3.26) choosing \(R>0\) large enough and recalling that \(\Vert u_{k}\Vert _{{\rho }} = 1\). From now on we fix \(R>0\) so that (3.24) and (3.26) hold. Next we prove that
for all \(t\in [0,\bar{t}]\). In fact, by (\(\rho _{1}\)) there exists \(\rho _{0} >0\) such that \(\rho (x) \ge \rho _{0}\) for all \(x\in \mathbb {R}^{n}\), so that
and we obtain (3.28) by (3.24). Thus, by 3.25, (3.26) and (3.28) we obtain
If we choose \(t = {t^*}\) such that \(I({t^*} u_{k}) = \max _{t \ge 0} I(t u_{k})\) then we see that \( c_{\rho } \ge c - \epsilon , \) from were we get a contradiction with (3.23) if we take \(\epsilon >0\) small enough.
Now we analyze the case \(\rho _{\infty } < +\infty \). In this case we compare the functionals \(I_{\rho }\) and \(I_{\rho _{\infty }}\) writting
By hypothesis (\(\rho _{1}\)), for any \(\epsilon >0\) there is \(R>0\) such that
Proceeding as before, for all \(t\in [0,\bar{t}]\), we obtain the estimate
where
Thus, we obtain
Choosing \(t\) appropriately and \(\epsilon \) small enough we conclude that \( c_{\rho } > c_{\rho _{\infty }}, \) contradicting (3.23). To complete the proof we only need to prove that \(u\) is non-negative. Thanks to hypothesis (\(f_1\)) we see that \(f(-u_-(x))=0\) for all \(x\in \mathbb {R}^n\), so that it is enough to prove that
Here we consider \(u_-=\max \{-u,0\}\) and \(u_+=\max \{u,0\}\) so that \(u=u_+-u_-\). An easy computation shows that for \(x,z\in \mathbb {R}^n\) we have
so that, by definition of the inner product \(\langle \cdot ,\cdot \rangle _\rho \) given in Sect. 2, we obtain (3.32) that proves that \(u_-=0\) a.e. in \(\mathbb {R}^n\). \(\square \)
4 Asymptotic values of the functional when \(\epsilon \rightarrow 0\)
In this section we make a preliminary analysis of the asymptotic behavior of the functional associated to Eq. (1.7) when \(\epsilon \rightarrow 0\). In this and next section we consider the power function \(f(s)=s^p\) to prove Theorem 1.2. For simplicity, we prefer to treat only the power function, but all the arguments can be adapted to deal with a general \(f\) satisfying the hypotheses (\(f_1\))–(\(f_4\)). We start with some basic properties of the function \(\mathcal {H}\).
Lemma 4.1
Assuming \(\rho \) satisfies \((\rho _1)\)–\((\rho _3)\), the function \(\mathcal {H}\) is continuous and
Moreover, there exists \(x_{0}\in \mathbb {R}^{n}\) such that
Proof
Hypothesis \((\rho _3)\) implies that the function \(\mathcal {H}\) is continuous in \(\mathbb {R}^n\). By definition of the sets \(\mathcal {C}^+(x)\) and \(\mathcal {C}^-(x)\), when \(\rho _\infty <\infty \), we see that
In case \(\rho _\infty =\infty \), we easily see that, for every \(M>0\) we have that \(\mathcal {C}^+(x)\subset B^c(0,M)\) if \(|x|\) is large enough. For a similar statement with \(\mathcal {C}^-(x)\), we use hypothesis \((\rho _2)\) to get that, for \(|x|\) large enough,
This implies that, for every \(M>0\) we have that \(\mathcal {C}^-(x)\subset B^c(0,M)\) if \(|x|\) is large enough. Thus we conclude that (4.1) holds.
Next we see that
where the last inequality follows from the fact that \(\mathcal {C}^{+}(x) \subset B^c(0,\rho (x))\) and \(\mathcal {C}^{+}(x)\) is a bounded set. From here and (4.1) the existence of a global minimum is a consequence of the continuity of \(\mathcal {H}\). \(\square \)
In this section we will consider a sequence of functions \(\{w_{m}\} \subset H^{\alpha }(\mathbb {R}^{n})\) such that \(\Vert w_{m} - w\Vert _{L^{2}(\mathbb {R}^{2})} \rightarrow 0\), where \(w\in H^{\alpha }(\mathbb {R}^{n})\). We will also consider sequences \(\{z_m\}\subset \mathbb {R}^n\) and \(\{\epsilon _m\}\subset \mathbb {R}\) and assume that \(\epsilon _m\rightarrow 0\) as \(m\rightarrow \infty \). We define \(\overline{\rho }_{m}\) as
and we consider the functional \(I_\rho \) defined in (1.6), with \(F(t)=|t|^{p+1}/({p+1})\) and for different scope functions \(\rho \), in particular for \({\rho _\infty }/{\epsilon _m}\) constant and \(\bar{\rho }_m\) defined in (4.3). In case \(\epsilon _m z_m\rightarrow \bar{x}\) we will also consider the functional with \({\rho (\bar{x})}/{\epsilon _m}\). Thus, in this section we will be considering the functionals
We will also consider the functional \(I\) in \(\mathbb {R}^n\) (with \(\rho \equiv \infty \)) defined in (3.14). The following theorem is a key to understand the concentration phenomenon for Eq. (1.7).
Theorem 4.1
Under hypotheses \((\rho _{1})\)–\((\rho _{3})\), we assume as above that \(w_{m}, w \in H^{\alpha }(\mathbb {R}^{n})\) are such that \(\Vert w_{m} - w\Vert _{L^{2}(\mathbb {R}^{2})} \rightarrow 0\) and \(\epsilon _m\rightarrow 0\), as \(m\rightarrow \infty \). Then we have:
-
(i)
If \(\epsilon _m z_m\rightarrow \bar{x}\) then
$$\begin{aligned} \lim _{m \rightarrow \infty } \frac{I_{\overline{\rho }_{m}}(w_{m}) - I_{\frac{\rho _{\infty }}{\epsilon _{m}}}(w_{m})}{\epsilon _{m}^{2\alpha }} = \Vert w\Vert _{L^{2}}^{2} \mathcal {H}(\overline{x})\qquad \text{ and } \end{aligned}$$(4.4) -
(ii)
If \(|\epsilon _{m}|z_{{m}} \rightarrow \infty \) then
$$\begin{aligned} \lim _{m \rightarrow \infty } \frac{I_{\overline{\rho }_{m}}(w_{m}) - I_{\frac{\rho _{\infty }}{\epsilon _{m}}}(w_{m})}{\epsilon _{m}^{2\alpha }} = 0. \end{aligned}$$(4.5)
In various stages of the proof of this theorem it will be convenient to replace the function \(w_m\) by the truncated limit \(w_{R}(x) = w(x)\chi _{B(0,R)}(x)\), where \(R>0\). We clearly have
Lemma 4.2
For all \(\delta >0\) there exist \(m(\delta )>0\) and \(R(\delta )>0\) such that
In order to prove Theorem 4.1 we first prove several lemmas under the hypotheses (i), that is
We analyze the cases \(\rho _{\infty } = +\infty \) and \(\rho _{\infty } < +\infty \) separately. It will be convenient to decompose the problem considering
For the second term of the right hand side we have
Lemma 4.3
Under assumption of Theorem 4.1 and assuming (i)
In case \(\rho _{\infty } = + \infty \), here we write \(I_{\frac{\rho _{\infty }}{\epsilon _{m}}}=I\) and \(1/\rho _\infty ^{2\alpha }=0\).
Proof
We first consider the case \(\rho _{\infty } = + \infty \). We have
If we denote by \(E_m\) the second term above and we consider \(R>0\), we have
From here, using Lemma 4.2 and the fact that \(w_R(\cdot )\) and \(w_R(\cdot +z)\) have disjoint supports if \(|z|>2R\), we obtain that \(\lim _{m\rightarrow \infty }E_m=0\), which implies the result.
In case \(\rho _{\infty } < +\infty \) we proceed similarly, noticing that we have to replace the integral over \(B^c\left( 0,\frac{\rho (\overline{x})}{\epsilon _{m}}\right) \) by an integral over \(B\left( 0,\frac{\rho _\infty }{\epsilon _{m}}\right) \setminus B\left( 0,\frac{\rho (\overline{x})}{\epsilon _{m}}\right) \) and compute accordingly. \(\square \)
Now we consider the first term in (4.6), for which it is convenient to write
Here, and in what follows, we denote by \(A(a,b)\) the annulus \(B(0,b)\setminus B(0,a)\) and we notice that \(A(a,b)=\emptyset \) when \(a\ge b\).
We start our analysis with \(\mathcal {I}_1\) and for this purpose, we first consider the second and third term in the expansion of the quadratic expression \( |w_{m}({x+z}) - w_{m}({x})|^{2}\).
Lemma 4.4
Under assumption of Theorem 4.1 and assuming (i) we have
and
Proof
The first limit is obtained using the arguments given in the proof of Lemma 4.3. To study the second limit we see that
where \(R>0\). By the continuity of \(\rho \) and the fact that \(\epsilon _m z_m\rightarrow \bar{x}\) as \(m\rightarrow \infty \), we see that
uniformly in \(B(0,R)\). From here, Lemma 4.2 and the inequality above, the result follows. \(\square \)
Next we consider the first term in the expansion of \(|w_{m}(x+z) - w_{m}(x)|^{2}\).
Lemma 4.5
Under assumption of Theorem 4.1 and assuming (i)
Proof
To start we consider
where
and \(E_{m}^{2}\) is the error term. For \(E_m^2\) we have
Next we consider \(E_m^1\) and we observe that
where
On the other hand we see that for any \((x,z)\) such that \(|x+z| < R\) we have
Therefore
where the sets \(M_m\) and \(N_m\) are defined as follows
and
Similarly, from (4.11), (4.12) and (4.13) we find that
In order to complete the analysis of \(E_m^1\) we just need to look at the limit of the integrals. We recall that, by hypothesis (\(\rho _3\)), the set defined by the equation
is an (\(n-1\))-dimensional surface and that we are assuming that \( \lim _{m\rightarrow \infty } \epsilon _m z_{m}=\bar{x}. \) So we have
Using similar arguments we obtain
and
completing the proof of the lemma. \(\square \)
Using Lemmas 4.4 and 4.5 we conclude that
In a complete analogous way we can prove that
Proof of Theorem 4.1
The proof of (i) is a consequence of (4.6), (4.8), (4.16), (4.17) and Lemma 4.3. Now we consider (ii) in case \(\rho _{\infty } = \infty \). We have
where \(R>0\). By hypothesis \((\rho _{1})\) and since \(\rho _\infty =\infty \), we have that for any \(M>0\) there is \(R>0\) and \(m_0\) large enough such that
and
Consequently
We observe that if \(|x+z|<R\) and \(|x+z_m|<R/\epsilon _m\) then \(|z|>|z_m|/2\), where we may need to make \(m\) larger. Then we further look at the integral above
On the other hand
From Lemma 4.2, (4.18)–(4.20) we can argue that (4.4) holds.
In the case \(\rho _{\infty } < \infty \), by hypothesis \((\rho _{1})\) we have that, for any \(\delta >0\) there is \(R>0\) and \(m\) large enough such that
Consequently
where
The integral in (4.21) is estimated exactly as \(E_m^1\). From here and (4.21) we can argue that (4.4) holds, completing the proof. \(\square \)
Before leaving this section we prove a lemma that will be useful later. We recall that, by Lemma 3.1, for any \(w_{m} \in H_{\rho _{m}}^{\alpha }(\mathbb {R}^{n})\), there is a unique \(t_{m} = t(w_{m}) >0\) such that \(t_{m}w_{m} \in \mathcal {N}_{\rho _{m}}\) and
Let \(\mathcal {N}\) the Nehari manifold associated to the limit problem, that is
Lemma 4.6
Assume \(w_{m} \rightarrow w\) in \(H^{\alpha }(\mathbb {R}^{n})\) and \(w \in \mathcal {N}\), then
Proof
In fact, by definition of \(t_{m}\) we have
Since \(w_{m} \rightarrow w\in \mathcal {N}\) we have that \(\Vert w_{m}\Vert ^{2} \rightarrow \Vert w\Vert ^{2}\), \(\int _{\mathbb {R}^{n}} u_{m}^{p+1}dx \rightarrow \int _{\mathbb {R}^{n}}u^{p+1}dx\) and \(w\) is non-zero. Thus \(t_{m}\) converges to \(\bar{t}\) and \(\bar{t}=1.\) \(\square \)
5 Concentration behaviour
In this section we complete our study on the concentration behavior for ground states of Eq. (1.7) and we prove Theorem 1.2. We start rescaling Eq. (1.7), for this purpose we define \(\rho _{\epsilon }(x) = \frac{1}{\epsilon } \rho (\epsilon x)\) and \((-\Delta )_{\rho _{\epsilon }}^{\alpha }\) the operator defined by (2.5) changing \(\rho \) by \(\rho _{\epsilon }\). We then consider the rescaled equation
and we see that \(u\) is a weak solution of (1.7) if and only if \(v_{\epsilon }(x) = u(\epsilon x)\) is a weak solution of (5.1). In fact, by definition (2.6) and changing variables it is easy to see that for every test function \(\varphi \)
In order to study Eqs. (1.7) and (5.1), we consider the functional \(I_{\rho _{\epsilon }}\) on the \(\epsilon \)-dependent Hilbert space \(H_{\rho _\epsilon }^{\alpha }(\mathbb {R}^{n})\) with inner product \(\langle \cdot ,\cdot \rangle _{\rho _\epsilon }\).
The functional \(I_{\rho _{\epsilon }}\) is of class \(C^1\) in \(H_{\rho _{\epsilon }}^{\alpha }(\mathbb {R}^{n})\) and the critical points of \(I_{\rho _{\epsilon }}\) are the weak solutions of (5.1). We further introduce
and the mountain pass minimax value
From Lemma 3.2 we also have
For comparison purposes we consider the functional \(I\) defined in 3.14, whose critical points are the solutions of 3.15, where \(f(u)=u^p\). We also consider the critical value 3.16 that satisfies
Now we start the proof of Theorem 1.2 with some preliminary lemmas.
Lemma 5.1
Suppose (\(\rho _{1}\)) holds. Then
Proof
Since we obviously have
for all \(u\in H_{\rho _{\epsilon }}^{\alpha }(\mathbb {R}^{n})\), then we have \( I_{\rho _{\epsilon }} (u) \le I(u) \) and therefore
Now consider \(v_\epsilon \in H^\alpha _{\rho _\epsilon }\) a solution of Eq. (5.1) with critical value \(c_{\rho _\epsilon }\), then \(c_{\rho _\epsilon }=I_{\rho _{\epsilon }} ( u_\epsilon )=\max _{t \ge 0} I_{\rho _{\epsilon }} (t u_\epsilon ) \) and \(I'_{\rho _{\epsilon }}(v_{\epsilon }) v_{\epsilon } = 0\), so that
and then
From here and using that \(c_{\rho _{\epsilon }}\) by (5.4), we see that \(\Vert v_{\epsilon }\Vert _{\rho _\epsilon }\) is bounded when \(\epsilon \rightarrow 0^{+}\). Next we see that for any \(t >0\) we have
Using hypothesis (\(\rho _{1}\)) and estimating the second term in the right side as in (3.27) and (3.29), we find
Now choosing \(t = t^{*}_\epsilon >0\), such that \(I(t^{*}_\epsilon v_\epsilon ) = \max _{t \ge 0} I(t v_\epsilon )\) we find
from where \( \liminf _{\epsilon \rightarrow 0^{+}} c_{\rho _{\epsilon }} \ge c. \) Combining with (5.4) we get (5.3). \(\square \)
Lemma 5.2
If \(v_\epsilon \) is s family solutions of (5.1) with critical value \(c_{\rho _\epsilon }\), then there exists a family \(\{y_{\epsilon }\}\) and positive constants \(R\) and \(\beta \) such that
If not, there exists a sequence \(v_{k} = v_{\epsilon _{k}}\) such that
the by Lemma 2.1, we have \( v_{k} \rightarrow 0\) in \(L^{q}(\mathbb {R}^{n})\) for any \(2 < q < 2_{\alpha }^{*}. \) However, this is impossible because by (5.5), (5.6) and Lemma 5.1
\(\square \)
Now let
then by (5.10),
To continue, we consider the rescaled scope function \(\overline{\rho }_\epsilon \), as defined in (4.3),
and then \(w_\epsilon \) satisfies the equation
Now we prove the convergence of \(w_\epsilon \) as \(\epsilon \rightarrow 0\).
Lemma 5.3
For every sequence \(\{\epsilon _m\}\) there is a subsequence, which will also be denoted by \(\{\epsilon _m\}\), so that \(w_{\epsilon _m}=w_{m} \rightarrow w\) in \(H^{\alpha }(\mathbb {R}^{n})\), when \(m\rightarrow \infty \), where \(w\) is a solution of (3.15).
Proof
From (5.6) we see that \(\{w_\epsilon \}\) is bounded, and then, for every sequence \(\{\epsilon _m\}\) there is a subsequence, we keep calling the same, so that \(w_{\epsilon _m}=w_{m} \rightharpoonup w\), which satisfies Eq. (3.15). To prove the convergence of this sequence we use its weak convergence together with (5.6) and Lemma 5.1 to get
Here \(K\) is an estimate for \(\Vert w_m\Vert \). Thus \(\Vert w_m\Vert ^2 \rightarrow \Vert w\Vert ^2\) and hence \(w_{m} \rightarrow w\) in \(H^{\alpha }(\mathbb {R}^{n})\). \(\square \)
We are now in a position to complete the proof of our second main theorem.
Proof of Theorem 1.2
We first obtain an upper bound for the critical values \(c_{\rho _{\epsilon _m}}=c_m\), for the sequence \(\{\epsilon _m\}\) given in Lemma 5.3. Next we consider the scope function
where \(x_0\) is a global minimum point of \(\mathcal {H}\), see Lemma 4.1. To continue, we consider the function \(w_m=w_{\epsilon _m}\) as given in (5.8) and let \(t_m>0\) such that \(t_m w_m\in \mathcal {N}_{\tilde{\rho }_m}\). According to Lemma 5.3, \(\{w_m\}\) converges to \(w\in \mathcal {N}\), then \(t_m\rightarrow 1\) and \(t_mw_m\rightarrow w\).
Now we apply Theorem 4.1 to obtain that
We have used part (i) of Theorem 4.1 with \(z_m=x_0/{\epsilon _m}\).
On the other hand, since \(w_{m} \in H^{\alpha }(\mathbb {R}^{n})\) is a critical point of \(I_{\bar{\rho }_m}\), we have that
We write \(y_m=y_{\epsilon _m}\). If \(\epsilon _m|y_m|\rightarrow \infty \), then we may apply part (ii) of Theorem 4.1 with \(z_m=y_m\) in 5.12 and obtain that
which contradicts (5.11). We conclude then, that \(\{\epsilon _my_m\}\) is bounded and that, for a subsequence, \(\epsilon _my_m\rightarrow \bar{x}\), for some \(\bar{x}\in \mathbb {R}^n\). Now we apply Theorem 4.1 again, but now part (i) with \(z_m=y_m\) in 5.12, and we obtain that
From (5.11) and (5.13) we finally get that
and taking the limit as \(m\rightarrow \infty \), we get
completing the proof of the theorem. \(\square \)
References
Ambrosetti, A., Rabinowitz, P.: Dual variational methods in critical points theory and applications. J. Funct. Anal. 14, 349–381 (1973)
Cheng, M.: Bound state for the fractional Schrödinger equation with undounded potential. J. Math. Phys. 53, 043507 (2012)
Coti Zelati, V., Rabinowitz, P.: Homoclinic type solutions for a semilinear elliptic pde on \(\mathbb{R}^{n}\). Commun. Pure Appl. Math. 45, 1217–1269 (1992)
Di Nezza, E., Patalluci, G., Valdinoci, E.: Hitchhiker’s guide to the fractional Sobolev spaces. Bull. Sci. Math. 136, 521–573 (2012)
Dong, J., Xu, M.: Some solutions to the space fractional Schrödinger equation using momentum representation method. J. Math. Phys. 48, 072105 (2007)
Fall, M., Valdinoci, E.: Uniqueness and nondegeneracy of positive solutions of \((-\Delta )^{\alpha }u + u = u^{p} \) in \(\mathbb{R}^{n}\) when \(\alpha \) is close to \(1\)”. arXiv:1301.4868v2 [math.AP] (2013)
Frank, R., Lenzmann, E.: Uniqueness of non-linear ground states for fractional Laplacians in \(\mathbb{R}\). Acta Math. 210(2), 261–318 (2013)
Felmer, P., Quaas, A., Tan, J.: Positive solutions of nonlinear Schrödinger equation with the fractional laplacian. Proc. R. Soc. Edinb. Sect. A Math. 142(6), 1237–1262 (2012)
Guan, Q.-Y.: Integration by parts formula for regional fractional Laplacian. Commun. Math. Phys. 266, 289–329 (2006)
Guan, Q.-Y., Ma, Z.M.: The reflected \(\alpha \)-symmetric stable processes and regional fractional Laplacian. Probab. Theory Relat. Fields 134(4), 649–694 (2006)
Guo, X., Xu, M.: Some physical applications of fractional Schrödinger equation. J. Math. Phys. 47, 082104 (2006)
Ishii, H., Nakamura, G.: A class of integral equations and approximation of p-Laplace equations. Calc. Var. 37, 485–522 (2010)
Laskin, N.: Fractional quantum mechanics and Lévy path integrals. Phys. Lett. A 268, 298–305 (2000)
Laskin, N.: Fractional Schrödinger equation. Phys. Rev. E 66, 056108 (2002)
Mawhin, J., Willen, M.: Critical point theory and Hamiltonian systems. Applied Mathematical Sciences, vol. 74. Springer, Berlin (1989)
de Oliveira, E., Costa, F., Vaz, J.: The fractional Schrödinger equation for delta potentials. J. Math. Phys. 51, 123517 (2010)
Rabinowitz, P.: Minimax method in critical point theory with applications to differential equations. In: CBMS regional conference series in mathematics, vol. 65. AMS, Providence (1986)
Rabinowitz, P.: On a class of nonlinear Schrdinguer equations. ZAMP 43, 270–291 (1992)
Wang, X.: On concentration of positive bound states of nonlinear Schrödinger equations. Commun. Math. Phys. 153, 229–244 (1993)
Wloka, J.: Partial Differential Equations. Cambridge University Press, Cambridge (1987)
Acknowledgments
P.F. was partially supported by Fondecyt Grant # 1110291, BASAL-CMM projects. C.T. was partially supported by Becas Mecesup and CMM.
Author information
Authors and Affiliations
Corresponding author
Additional information
Communicated by P. Rabinowitz.
Rights and permissions
About this article
Cite this article
Felmer, P., Torres, C. Non-linear Schrödinger equation with non-local regional diffusion. Calc. Var. 54, 75–98 (2015). https://doi.org/10.1007/s00526-014-0778-x
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00526-014-0778-x