- Research
- Open access
- Published:
Infinitely many positive solutions for a nonlocal problem with competing potentials
Boundary Value Problems volume 2020, Article number: 108 (2020)
Abstract
The present paper deals with a class of nonlocal problems. Under some suitable assumptions on the decay rate of the coefficients, we derive the existence of infinitely many positive solutions to the problem by applying reduction method. Comparing to the previous work, we encounter some new challenges because of competing potentials. By doing some delicate estimates for the competing potentials, we overcome the difficulties and find infinitely many positive solutions.
1 Introduction and main results
In this paper, we study the fractional Schrödinger problem
where \(0<\sigma <1\), \(n\geq 2\), \(1< p<\frac{n+2\sigma }{n-2\sigma }\) and \(A(y)\), \(B(y)\) are two radially symmetric potentials. Here the fractional Laplacian \((-\Delta )^{\sigma }\) is defined by
where P.V. stands for the Cauchy principal value and \(C_{n,\sigma }\) is a normalization constant.
Problem (1.1) has attracted considerable attention in the recent period and part of the motivation is due to looking for a standing wave \(\psi =e^{-iht}u\) of the evolution equation
since ψ solves (1.2) if and only if u solves (1.1), where i is the imaginary unit and \(h\in \mathbb {R}\). This class of Schrödinger-type equations is of particular interest in fractional quantum mechanics for the study of particles on stochastic fields modeled by Lévy processes. In recent years, there have been many investigations for the related fractional Schrödinger equation
with \(0<\sigma <1\) and \(V: \mathbb {R}^{n}\rightarrow \mathbb {R}\) is an external potential function. A complete review of the available results in this context goes beyond the aim of this paper; we refer the interested reader to [4, 5, 9, 13–17, 19–22] and the references therein.
Especially, in [19] we studied (1.1) and infinitely many nonradial positive (sign-changing) solutions were established when \(A(y)=1\) and \(B(y)\) satisfies some radial symmetry assumption by using Lyapunov–Schmidt reduction. In this paper, continuing our study in [19], we are concerned with the multiplicity of positive solutions for (1.1) in a situation in which there exist two competing potentials and even (1.1) may not have ground states.
To the best of our knowledge, not much is obtained for the existence of multiple solutions of Eq. (1.1) with competing potentials. So our purpose of this paper is to establish the existence of infinitely many nonradial positive solutions for (1.1) by constructing solutions with large number of bumps near the infinity under some assumptions for \(A(y)\), \(B(y)\) as follows:
- (A):
there are constants \(a>0\), \(m_{1}>0\), \(\theta _{1}>0\) such that
$$ A \bigl( \vert y \vert \bigr)=1+\frac{a}{ \vert y \vert ^{m_{1}}}+O \biggl(\frac{1}{ \vert y \vert ^{m_{1}+\theta _{1}}} \biggr) \quad \text{as } \vert y \vert \rightarrow +\infty ; $$- (B):
there are constants \(b\in \mathbb {R}\), \(m_{2}>0\), \(\theta _{2}>0\) such that
$$ B \bigl( \vert y \vert \bigr)=1+\frac{b}{ \vert y \vert ^{m_{2}}}+O \biggl(\frac{1}{ \vert y \vert ^{m_{2}+\theta _{2}}} \biggr)\quad \text{as } \vert y \vert \rightarrow +\infty . $$
Our main results in this paper can be stated as follows.
Theorem 1.1
Suppose that\(n\geq 2\), \(1< p<\frac{n+2\sigma }{n-2\sigma }\), \(\frac{n+2\sigma }{n+2\sigma +1}<\min \{m_{1},m_{2}\}<n+2\sigma \)and the conditions (A) and (B) hold. If\(b<0\)or\(b>0\)and\(m_{1}< m_{2}\), then problem (1.1) has infinitely many nonradial positive solutions.
To achieve our goal, we adopt a novel idea introduced in [23], by using k, the number of the bumps of the solutions, as the parameter in the construction of solutions for (1.1). In [23], the authors studied the following equation:
and applying the reduction method, they derived the existence of infinitely many solutions to (1.3) by exhibiting bumps at the vertices of the regular k-polygons for sufficiently large \(k\in \mathbb{N}\) under some suitable conditions on \(V(y)\) and p. But, in this paper, since the competing terms appear, we have to overcome many difficulties in the reduction process which involves some technical and careful computations. Furthermore, for more results on the existence of radial ground states, infinitely many bound states or nonradial solutions, higher energy bound states to (1.3), one can refer to [1–3, 6–8, 11, 12, 18] and the references therein.
In the end of this part, let us outline the main idea to prove our main results. For any integer \(k>0\), we define
where 0 is the zero vector in \(\mathbb {R}^{n-2}\), \(r\in [r_{0}k^{\frac{n+2\sigma }{n+2\sigma -m}},r_{1}k^{ \frac{n+2\sigma }{n+2\sigma -m}}]\) for some \(r_{1}>r_{0}>0\) with \(m:=\min \{m_{1},m_{2}\}\). Also we denote by \(H^{\sigma }(\mathbb {R}^{n})\) the usual Sobolev space endowed with the standard norm
Moreover, for \(y=(y',y'')\in \mathbb {R}^{2}\times \mathbb {R}^{n-2}\), set
In what follows we will use the unique ground state U of
to build up the approximate solutions for (1.1). It is well known that in [16, 17], the authors have established the uniqueness and non-degeneracy of the ground state of (1.4) with
and
Now if we define
where \(U_{y^{i}}(y)=U(y-y^{i})\), then we will prove Theorem 1.1 by verifying the following result.
Theorem 1.2
Under the assumption of Theorem 1.1, there is an integer\(k_{0}>0\), such that, for any integer\(k\geq k_{0}\), (1.1) has a solution\(u_{k}\)of the form
where\(\varphi _{r_{k}}\in H_{k}\), \(r_{k}\in [r_{0}k^{\frac{n+2\sigma }{n+2\sigma -m}},r_{1}k^{ \frac{n+2\sigma }{n+2\sigma -m}}]\)for some constants\(r_{1}>r_{0}>0\)and as\(k\rightarrow +\infty \),
This paper is organized as follows. In Sect. 2, we will carry out a reduction procedure and then study the reduced one dimensional problem to prove Theorem 1.2 in Sect. 3. Some basic estimates and an energy expansion for the functional are left to the Appendix.
2 The reduction
In the following, we always assume that \(k\in \mathbb{N}\) is a large number. Let
where \(y^{j}=(r\cos \frac{2(j-1)\pi }{k},r\sin \frac{2(j-1)\pi }{k},0)\) and
where \(r_{0}= (\frac{h_{0}(n+2\sigma )}{h_{1}m}-\alpha )^{ \frac{1}{n+2\sigma -m}}\), \(r_{1}= (\frac{h_{0}(n+2\sigma )}{h_{1}m}+\alpha )^{ \frac{1}{n+2\sigma -m}}\), \(\alpha >0\) is a small constant and \(h_{0}\), \(h_{1}\) will be given in Sect. 3.
Define
Note that the variational functional corresponding to (1.1) is
Let
We can expand \(J(\varphi )\) as follows:
where
and
In this part, we shall find a map \(\varphi (r)\) from \(S_{k}\) to \(E_{r}\) such that \(\varphi (r)\) is a critical point of \(J(\varphi )\) under the constraint \(\varphi (r)\in E_{r}\). Associated to the quadratic form \(L(\varphi )\), we define L to be a bounded linear map from \(E_{r}\) to \(E_{r}\) such that
Then we have the following lemma, which shows the invertibility of L in \(E_{r}\).
Lemma 2.1
There is a constant\(\rho >0\)independent ofk, such that, for any\(r\in S_{k}\),
Proof
Arguing by contradiction, we suppose that there are \(k\rightarrow +\infty \), \(r_{k}\in S_{k}\), and \(\varphi _{k}\in E_{r}\) such that
Set
By symmetry, we have for \(v\in E_{r}\)
In particular,
and
Let \(\tilde{\varphi }_{k}=\varphi (y+y^{1})\). Since for any \(R>0\), \(\operatorname{dist}(y^{1},\partial \varOmega _{1})=r\sin \frac{\pi }{k}\), \(B_{R}(y^{1})\subset \varOmega _{1}\). Thus
So, we may assume that there exists \(\varphi \in H^{\sigma }(\mathbb {R}^{n})\) such that, as \(k\rightarrow +\infty \),
Moreover, \(\tilde{\varphi }_{k}\) is even in \(y_{j}\), \(j=2,\ldots,n\) and
We see that φ is even in \(y_{j}\), \(j=2,\ldots,n\) and
Now, we claim that φ solves the following linearized equation in \(\mathbb {R}^{n}\):
Indeed, define
For any \(R>0\), let \(v \in C_{0}^{\infty }(B_{R}(0))\cap \widetilde{E}\) satisfying v is even in \(y_{j}\), \(j=2,\ldots,n\). Then \(v_{1}(y)=v(y-y^{1})\in C_{0}^{\infty }(B_{R}(y^{1}))\). We may identify \(v_{1}(y)\) as elements in \(E_{r}\) by redefining the values outside \(\varOmega _{1}\) with the symmetry. By using (2.2) and Lemma A.2, we can find that
But (2.5) holds for \(v=\frac{\partial U}{\partial y_{1}}\). Hence (2.5) is true for any \(v \in H^{\sigma }(\mathbb {R}^{n})\) and the claim holds. This being the nondegenerate result of U, we have \(\varphi =c\frac{\partial U}{\partial y_{1}}\) since φ is even in \(y_{j}\), \(j=2,\ldots,n\). So it follows from the orthogonal condition (2.3) that \(\varphi =0\) and thus
Due to Lemma A.2, if \(k>0\) is large enough, we have, for η satisfying \((n+2\sigma -\eta )(p-1)>n\),
So, taking \(v=\varphi _{k}\) in (2.2), one has
This shows a contradiction and our proof is finished. □
Next, we discuss the terms \(R(\varphi )\) and \(l(\varphi )\) in (2.1). We have
Lemma 2.2
There is a constant\(C>0\)independent ofk, such that
and
for\(\varphi \in E_{r}\)and\(\Vert \varphi \Vert _{\sigma }<1\).
Proof
It is clear that, for \(v_{1}, v_{2}\in E_{r}\),
and
First, if \(p\geq 2\), it follows from Lemma A.2 that \(W_{r}\) is bounded and then
and
As a result, if \(p\geq 2\), we have
and
With the same argument, if \(1< p<2\), we find
and
which completes this proof. □
Lemma 2.3
For any\(\varphi \in E_{r}\), \(r\in S_{k}\), there is a constant\(C>0\)and a small\(\epsilon >0\), independent ofk, such that
where\(m=\min \{m_{1},m_{2}\}\).
Proof
Recall that
We are in a position to discuss the terms in (2.6). Using condition (A), similar to (A.2), we compute that
With the same argument, having \(m>\frac{n+2\sigma }{n+2\sigma +1}\), we have
Finally, taking \(\eta =n+2\sigma \) in Lemma A.2, one has
Inserting (2.7)–(2.9) into (2.6), the conclusion follows. □
Proposition 2.4
There is an integer\(k_{0}>0\), such that, for each\(k\geq k_{0}\), there is a\(C^{1}\)map from\(S_{k}\)to\(H_{k}\): \(r\mapsto \varphi =\varphi (r)\), \(r= \vert y^{1} \vert \), satisfying\(\varphi (r)\in E_{r}\), and
Moreover, there exists a small constant\(\epsilon >0\), such that, for some\(C>0\), independent of k,
Proof
We will use the contraction theorem to prove it. It follows from Lemma 2.3 that \(l(\varphi )\) is a bounded linear map in \(E_{r}\). So applying the Reisz representation theorem there exists an \(l_{k}\in E_{r}\) such that
Thus, finding a critical point for \(J(\varphi )\) is equivalent to solving
By Lemma 2.1, L is invertible and then (2.11) can be rewritten as
Set
where \(\epsilon >0\) is defined in Lemma 2.3.
From Lemmas 2.2 and 2.3, we have, for \(\varphi \in E_{r}\),
On the other hand, for any \(\varphi _{1}, \varphi _{2}\in D_{k}\), we can deduce that
Therefore, T maps \(D_{k}\) to \(D_{k}\) and is a contraction map. From the contraction map theorem, there exists φ such that \(\varphi =T(\varphi )\) and
□
3 Proof of the main result
Now we are ready to prove our Theorem 1.2. Let \(\varphi _{r}:=\varphi (r)\) be the map obtained in Proposition 2.4. Define
With the same argument in [10], we can check that, if r is a critical point of \(F(r)\), then \(W_{r}+\varphi _{r}\) is a solution of (1.1).
Proof of Theorem 1.2
It follows from Propositions 2.4 and A.3 that
In the following, we only prove the case \(b>0\) and \(m_{1}< m_{2}\) since the case that \(b<0\) can be checked in similar way. If \(b>0\) and \(m_{1}< m_{2}\), then
for some \(h_{0}, h_{1}>0\).
We next consider the following maximization problem:
Suppose that (3.1) is achieved by some \(r_{k}\) in \(S_{k}\) and then we can prove that \(r_{k}\) is an interior point in \(S_{k}\) by analyzing the following problem:
By the direct computation, we find \(g(r)\) admits a maximum point
Now we claim that \(r_{k}\) is an interior point of \(S_{k}\). In fact, it is easy to see that
On the other hand,
and
Since the function \(f(t)=(\frac{1}{t})^{\frac{n+2\sigma }{n+2\sigma -m}}(t-1)\) attains its maximum at \(t_{0}=\frac{n+2\sigma }{m}\) when \(t\in [\frac{n+2\sigma }{m}-\frac{\alpha h_{1}}{h_{0}}, \frac{n+2\sigma }{m}+\frac{\alpha h_{1}}{h_{0}}]\), we have \(g(r_{0}k^{\frac{n+2\sigma }{n+2\sigma -m}})< g(r_{k})\) and \(g(r_{1}k^{\frac{n+2\sigma }{n+2\sigma -m}})< g(r_{k})\). Thus, \(r_{k}\) is an interior point of \(S_{k}\) and \(r_{k}\) is a critical point of \(F(r)\). As a result,
is a solution of (1.1). □
References
Ao, W., Wei, J.: Infinitely many positive solutions for nonlinear equations with non-symmetric potentials. Calc. Var. Partial Differ. Equ. 51, 761–798 (2014)
Bahri, A., Li, Y.: On a min–max procedure for the existence of a positive solution for certain scalar field equations in \(\mathbb{R} ^{N}\). Rev. Mat. Iberoam. 6, 1–15 (1990)
Bahri, A., Lions, P.-L.: On the existence of a positive solution of semilinear elliptic equations in unbounded domains. Ann. Inst. Henri Poincaré, Anal. Non Linéaire 14, 365–413 (1997)
Bona, J.L., Li, Y.A.: Decay and analyticity of solitary waves. J. Math. Pures Appl. 76, 377–430 (1997)
Caffarelli, L., Salsa, S., Silvestre, L.: Regularity estimates for the solution and the free boundary of the obstacle problem for the fractional Laplacian. Invent. Math. 171, 425–461 (2008)
Cerami, G.: Some nonlinear elliptic problems in unbounded domains. Milan J. Math. 74, 47–77 (2006)
Cerami, G., Passaseo, D., Solimini, S.: Infinitely many positive solutions to some scalar field equations with nonsymmetric coefficients. Commun. Pure Appl. Math. 66, 372–413 (2013)
Cerami, G., Pomponio, A.: On some scalar field equations with competing coefficients. Int. Math. Res. Not. 8, 2481–2507 (2018)
D’avila, J., del Pino, M., Dipierro, S., Valdinoci, E.: Concentration phenomena for the nonlocal Schrödinger equation with Dirichlet datum. Anal. PDE 8(5), 1165–1235 (2015)
Dávila, J., Del Pino, M., Wei, J.: Concentrating standing waves for fractional nonlinear Schrödinger equation. J. Differ. Equ. 256, 858–892 (2014)
Devillanova, G., Solimini, S.: Min–max solutions to some scalar field equations. Adv. Nonlinear Stud. 12, 173–186 (2012)
Ding, W., Ni, W.M.: On the existence of positive entire solutions of a semilinear elliptic equation. Arch. Ration. Mech. Anal. 91, 283–308 (1986)
Dipierro, S., Palatucci, G., Valdinoci, E.: Existence and symmetry results for a Schrödinger type problem involving the fractional Laplacian. Matematiche 68, 201–216 (2013)
Fall, M.M., Mahmoudi, F., Valdinoci, E.: Ground states and concentration phenomena for the fractional Schrödinger equation. Nonlinearity 28(6), 1937–1961 (2015)
Felmer, P., Quaas, A., Tan, J.: Positive solutions of nonlinear Schrödinger equation with the fractional Laplacian. Proc. R. Soc. Edinb., Sect. A 142(6), 1237–1262 (2012)
Frank, R., Lenzmann, E.: Uniqueness of non-linear ground states for fractional Laplacians in \(\mathbb{R} \). Acta Math. 210, 261–318 (2013)
Frank, R., Lenzmann, E., Silvestre, L.: Uniqueness of radial solutions for the fractional Laplacian. Commun. Pure Appl. Math. 69, 1671–1726 (2016)
Lions, P.-L.: The concentration compactness principle in the calculus of variations. The locally compactness case, part 2. Ann. Inst. Henri Poincaré, Anal. Non Linéaire 1, 223–283 (1984)
Long, W., Peng, S., Yang, J.: Infinitely many positive and sign-changing solutions for nonlinear fractional scalar field equations. Discrete Contin. Dyn. Syst. 36, 917–939 (2016)
Maris, M.: On the existence, regularity and decay of solitary waves to a generalized Benjamin–Ono equation. Nonlinear Anal. 51, 1073–1085 (2002)
Silvestre, L.: Regularity of the obstacle problem for a fractional power of the Laplace operator. Commun. Pure Appl. Math. 60, 67–112 (2007)
Sire, Y., Voldinoci, E.: Fractional Laplacian phase transitions and boundary reactions: a geometric inequality and a symmetry result. J. Funct. Anal. 256, 1842–1864 (2009)
Wei, J., Yan, S.: Infinite many positive solutions for the nonlinear Schrödinger equation in \(\mathbb{R} ^{n}\). Calc. Var. Partial Differ. Equ. 37, 423–439 (2010)
Wei, J., Yan, S.: Infinite many positive solutions for the prescribed scalar curvature problem on \(\mathbb{S}^{N}\). J. Funct. Anal. 258, 3048–3081 (2010)
Acknowledgements
The author thanks the referee’s thoughtful reading of details of the paper and nice suggestions to improve the results.
Availability of data and materials
Data sharing not applicable to this article as no data sets were generated or analyzed during the current study.
Funding
This work was partially supported by NSFC (No. 11601194).
Author information
Authors and Affiliations
Contributions
The author read and approved the final manuscript.
Corresponding author
Ethics declarations
Competing interests
The author declares that they have no competing interests.
Appendix: Energy expansion
Appendix: Energy expansion
In this section, we will give some basic estimates and the energy expansion for the approximate solutions. Recall that
and
Now we introduce the following lemmas which have been proved in [24] and [19], respectively.
Lemma A.1
For any constant\(0<\mu \leq \min \{\alpha ,\beta \}\), there is a constant\(C>0\), such that
Lemma A.2
For any\(y\in \varOmega _{1}\)and\(\eta \in (0,n+2\sigma ]\), there is a constant\(C>0\), such that
Proposition A.3
There is a small constant\(\epsilon >0\), such that
where\(d=(\frac{1}{2}-\frac{1}{p+1})\int _{\mathbb {R}^{n}}U^{p+1}\), \(d_{1}= \frac{1}{2}\int _{\mathbb {R}^{n}}U^{2}\), \(d_{2}=\frac{1}{p+1}\int _{ \mathbb {R}^{n}}U^{p+1}\)and\(q_{0}=\frac{1}{2}q_{0}'\)with\(q_{0}\), \(q_{0}'\)are some positive constants.
Proof
Using the symmetry and Lemma A.1, we have
and
On the other hand, we see that
First, we have
Second, using Lemma A.1,
where \(\tau >0\) satisfies \(n+2\sigma -\tau >n\) and we used the fact that \(\vert y-y^{j} \vert \geq \vert y-y^{1} \vert \) for \(y\in \varOmega _{1}\).
But, by Lemma A.2, we find
As a result,
and then
Now, from the symmetry, we also find
Observe that \(\vert y-y^{j} \vert \geq \vert y-y^{1} \vert \) and \(\vert y-y^{j} \vert \geq \frac{1}{2} \vert y^{j}-y^{1} \vert \) if \(y\in \varOmega _{1}\). So we have
and similarly
with \(\kappa >0\) satisfying \(\min \{\frac{p+1}{2}(n+2\sigma -\kappa ), 2(n+2\sigma -\kappa )\}>n+2 \sigma \).
Note that
By Lemma A.1, we can deduce that
since \(y\in \mathbb {R}^{n}\setminus \varOmega _{1}\), \(\vert y-y^{1} \vert \geq c\frac{r}{k}\) for some \(c>0\) and we could choose \(p(n+2\sigma )-\tau \geq n+2\sigma \).
Furthermore,
Finally,
Thus, we have proved
which, combining with (A.3), completes our proof. □
Rights and permissions
Open Access This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made. The images or other third party material in this article are included in the article’s Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article’s Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this licence, visit http://creativecommons.org/licenses/by/4.0/.
About this article
Cite this article
Yang, J. Infinitely many positive solutions for a nonlocal problem with competing potentials. Bound Value Probl 2020, 108 (2020). https://doi.org/10.1186/s13661-020-01406-4
Received:
Accepted:
Published:
DOI: https://doi.org/10.1186/s13661-020-01406-4