Concentration Phenomena for Fractional Elliptic Equations Involving Exponential Critical Growth
-
Claudianor O. Alves
and
Olímpio H. Miyagaki
Abstract
In this paper, we deal with the singular perturbed fractional elliptic problem
1 Introduction
In this paper, we are concerned with existence and concentration of positive solutions for the following singular perturbed fractional elliptic problem:
where ε is a small positive parameter, the potential V is bounded away from zero, the non-linearity
where
for functions ϕ in the Schwartz class. Also, for sufficiently smooth u,
and, by [18, Propostion 3.6],
Here
endowed with the norm
We suppose that the potential
V is locally Hölder continuous and there exists
There exists a bounded interval
The function
where
There exist constants
where
with
and
where
We are interested in a bound state solution of (Pϵ) (solution with finite energy), when f has the maximal growth, which allows us to treat problem (Pϵ) variationally in the fractional Sobolev space
There exists
for all
In view of (1.1), we say that f has exponential critical growth at
1.1 Statement of the Main Result
The following theorem contains our main result.
Assume (V1), (V2), (AR), and (f1)–(f4) hold. Then there exists
Theorem 1.2 may be considered as the extension of the main result for the Laplacian in [20] to the case of the square root of the Laplacian. The proof is made combining the Ozawa inequality [26] with the Del Pino and Felmer truncation argument [17] and a recent approach developed by Alves and Miyagaki [4]. In [16, 27, 29, 12], existence results in non-local situation were established, while in [21, 28, 13, 14], concentration phenomena were proved imposing a global condition in V. Recently, Felmer and Torres [23] studied a new class of the problem involving a non-linear Schrödinger equation with non-local regional diffusion, which generalizes, for instance, the fractional Laplacian operator. In [31], Zhang, do Ó and Squassina treated some problems involving fractional Schrödinger–Poisson systems.
(i) We recall that condition (AR) imposes some superquadratic growth condition on the non-linearity F.
(ii) Condition (f4) appeared first in [11], then for instance in [2, 20]. For the non-local situation it was used, e.g., in [19].
(iii) Critical growth of Trudinger–Moser type was used in [15] and also in [20, 1, 2]. The Ozawa inequality to discuss the non-local problem in bounded and unbounded domain was used in [25] and [19], respectively.
(iv) Notice that, if
Throughout the paper, unless explicitly stated, the symbol C will denote a generic positive constant, which may vary from line to line.
1.2 Outline
The sequel of the paper is organized as follows. Section 2 contains some technical results, which are crucial tools to prove our main theorem. In Section 3, we adapt a method due to L. Caffarelli and L. Silvestre to obtain a local realization of the fractional Laplacian via a Dirichlet-to-Neumann operator. As a consequence of this argument we transform our non-local problem (Pϵ) into one local problem (LPϵ) defined on the upper half plane. Using variational techniques combined with Del Pino and Felmer’s truncation argument, we prove Theorem 1.2 in Section 4.
2 Preliminary Results
In this section we collect preliminary facts for future reference. First of all, let us set the standard notations to be used in the paper. We denote the upper half-space in
where
Moreover, we denote by
Since the potential V is bounded from above and below, it is easy to see that
Using the above definition, we see that if
Since
are compact for any bounded measurable set
Our first lemma is an important Trudinger–Moser inequality on
Let
In particular, if
Using the above lemma, we are able to prove some technical lemmas. The first of them is crucial in the study of the behavior of Palais–Smale sequences.
Let
Then, there exist
Proof.
Using (2.2), there are
Fix
for every
for some positive constant
Let
as
Proof.
By (f1), for each
from where it follows that
Setting
we can fix
which is an immediate consequence of Lemma 2.2. Therefore, up to subsequence, we derive that
Since
On the other hand, we know that
Gathering the above limits with (2.6), we get (2.3):
Limits (2.4) and (2.5) follow with the same type of arguments. ∎
The next lemma is a Lions type result, which is crucial in our approach. Since it follows with the same arguments found in [2, Proposition 2.3], we will omit its proof.
Let
then
3 Caffarelli and Silvestre’s Method
First of all, using the change of variable
Hereafter, to get a solution for (P’ϵ), we will use a version of the Caffarelli–Silvestre extension [10] due to Frank and Lenzmann [24] defined on the whole real line. In both papers, a local interpretation of the fractional Laplacian given in
is called
To get a solution for the non-local problem (P’ϵ), we will study the existence of solutions for the local problem defined on the upper half plane:
where
since if w is a solution for (LPϵ), the function
Associated with (LPϵ), we have
which is
We would like to point out that u is a solution of (P’ϵ) if, and only if,
In what follows, we will not work directly with functional
Hereafter, we will use the same approach explored in [17], modifying the non-linearity of a suitable way. The idea is the following:
First of all, without loss of generality, we will assume that
We recall that in assumption (f1) we imposed that
where
and
where Λ was given in (V2) and
Using the above functions, we will study the existence of positive solutions for the following problem:
where
We recall from [10] that, to get a solution for problem (AP), it is enough to study the existence of solutions for the following problem:
because if w is a solution of (AP’), the function
Here, we would like to point out that if
where
Associated with (AP’), we have the energy functional
where
Using the definition of g, it follows that
and
From assumption (3.2), we deduce
and
The functional
there are
there is
Proof.
By (3.1)–(3.4), there exist
From the above inequality, we deduce that there are
showing (i).
To prove (ii), fix
From (f3), we know that there are
Using the above inequality, we derive
Thereby, (ii) follows with
In what follows, we denote by
The existence of
The minimax level
Proof.
Consider
By characterization of
Consequently, by (f4),
which implies that
Hence, by (f4),
Hereafter, we will assume that k is large enough such that
The next lemma establishes an important relation between
The numbers
Hence, there is
Proof.
By (V1),
Next, fix
By definition of
Using standard arguments as those in [20], we can prove
Thus,
showing (3.5). Inequality (3.6) is an immediate consequence of (3.5) and Lemma 3.2. ∎
Let
Proof.
Gathering
from where it follows that
and thus (3.9). ∎
For
Proof.
Let
By Lemma 3.4,
Moreover, by Corollary 2.3,
as
Using the above limits, it is possible to prove that v is a critical point for
Considering
On the other hand, using the limit
Since
and
Therefore,
Now, recalling that
Hence,
and
The last limit combined with the definition of function M gives
Gathering this limit with (3.10), we deduce that
Since
showing that
For
Proof.
By Lemma 3.3, there is
Decreasing
Proof.
First of all, we recall that since
To show (3.14), it is enough to prove that, for any sequence
does not hold for any
implying that
For any
Moreover, there is
Proof.
We begin the proof showing that
Since
Then,
From the definition of g, we see that
Recalling that
Therefore, there is
Using the characterization of
As
it follows that
Recalling that
By change of variable, we also have
Using Ekeland’s variational principle, we can assume that
A direct computation shows that
Thus, in what follows, we can assume that for some subsequence, there is
On the other hand, by Fatou’s lemma we obtain
which implies
The above equalities combined with Fatou’s lemma, up to a subsequence, give
Recalling that
showing (3.16).
Using the last limit, we are able to prove (3.17). First, we prove the following fact:
Indeed, if the claim does not hold, there is
Consequently, there is
From definition of
Note that
and so,
Therefore,
implying that
where
and
which contradicts (3.18). This proves Claim 3.9.
By Claim 3.9, there are a subsequence of
Indeed, from definition of
Then, by (3.16),
Hence, there is
where
If
Hence,
from where it follows that
As
Moreover, by (V2), we have
Let
and
As an immediate consequence, the sequence
Proof.
In what follows, for each
with
adapting the same approach explored in [3, Lemma 4.1] and using the fact that
Now, the limit (3.19) is obtained by interpolation on the
In what follows, we denote by
Since
we have that
or equivalently
where
Denoting
and that there is
Motivated by some results found in [7] (see also [22]), which hold for the whole line, we deduce that
where
There is
Using the above information, we are able to prove the following result.
The sequence
Proof.
Given
By (K2), we have that, for all
On the other hand,
Fix
Since
and
we deduce that there are
The same approach can be used to prove that for each
Hence, increasing R, if necessary, we must have
Since δ is arbitrary, the proof is finished. ∎
There is
Hence,
Proof.
By Lemma 3.8, we know that
Hence,
or equivalently
Now, by Lemma 3.12, there is
from where it follows that
On the other hand, we have that
Thus, there is
implying that
finishing the proof. ∎
4 Proof of Theorem 1.2
By Theorem 3.6, we know that problem (AP) has a non-negative solution
that is,
is a solution for original problem (Pϵ).
If
In what follows, setting
Now, we claim that
Indeed, by Lemma 3.12, we know that
Therefore,
Hence,
implying that
showing that (4.1) holds.
Funding statement: The authors were supported by INCTMAT/CNPq/Brazil. C. O. Alves was supported by CNPq/Brazil Proc. 304036/2013-7. J. M. do Ó was supported by CNPq/Brazil Proc. 407099/2013-1. O. H. Miyagaki was supported by CNPq/Brazil Proc. 304015/2014-8 and CAPES/Brazil Proc. 2531/14-3.
References
[1] Alves C. O., Existence of positive solution for a nonlinear elliptic equation with saddle-like potential and nonlinearity with exponential critical growth in
[2] Alves C. O., do Ó J. M. B. and Miyagaki O. H., On nonlinear perturbations of a periodic elliptic problem in
[3] Alves C. O. and Figueiredo G. M., Multiplicity of Positive Solutions for a class of quasilinear problems in
[4] Alves C. O. and Miyagaki O. H., Existence and concentration of solution for a class of fractional elliptic equation in
[5] Ambrosetti A. and Rabinowitz P. H., Dual variational methods in critical point theory and applications, J. Funct. Anal. 14 (1973), 349–381. 10.1016/0022-1236(73)90051-7Search in Google Scholar
[6] Barrios B., Colorado E., de Pablo A. and Sánchez U., On some critical problems for the fractional Laplacian operator, J. Differential Equations 252 (2012), 613–6162. 10.1016/j.jde.2012.02.023Search in Google Scholar
[7] Blumenthal R. M. and Getoor R. K., Some theorems on stable processes, Trans. Amer. Math. Soc. 95 (1960), 263–273. 10.1090/S0002-9947-1960-0119247-6Search in Google Scholar
[8] Brändle C., Colorado E. and Sánchez U., A concave-convex elliptic problem involving the fractional Laplacian, Proc. Roy. Soc. Edinburgh Sect. A 143 (2013), 39–71. 10.1017/S0308210511000175Search in Google Scholar
[9] Cabré X. and Sire Y., Nonlinear equations for fractional Laplacians. I: Regularity, maximum principles, and Hamiltonian estimates, Ann. Inst. H. Poincaré Anal. Non Linéaire 31 (2014), 23–53. 10.1016/j.anihpc.2013.02.001Search in Google Scholar
[10] Caffarelli L. and Silvestre L., An extension problems related to the fractional Laplacian, Comm. Partial Differential Equation 32 (2007), 1245–1260. 10.1080/03605300600987306Search in Google Scholar
[11] Cao D. M., Nontrivial solution of semilinear elliptic equation with critical exponent in
[12] Chang X. and Wang Z. Q., Nodal and multiple solutions of nonlinear problems involving the fractional Laplacian, J. Differential Equations 256 (2014), 2965–2992. 10.1016/j.jde.2014.01.027Search in Google Scholar
[13] Chen G. and Zheng Y., Concentration phenomena for fractional nonlinear Schrödinger equations, Comm. Pure Appl. Anal. 13 (2014), 2359–2376. 10.3934/cpaa.2014.13.2359Search in Google Scholar
[14] Dávila J., del Pino M. and Wei J. C., Concentrating standing waves for the fractional nonlinear Schrödinger equation, J. Differential Equations 256 (2014), 858–892. 10.1016/j.jde.2013.10.006Search in Google Scholar
[15] de Figueiredo D. G., Miyagaki O. H. and Ruf B., Elliptic equations in
[16] de Souza M. and Araújo Y. L., On nonlinear perturbations of a periodic fractional Schrödinger equation with critical exponential growth, Math. Nachr. 289 (2016), no. 5–6, 610–625. 10.1002/mana.201500120Search in Google Scholar
[17] del Pino M. and Felmer P. L., Local mountain pass for semilinear elliptic problems in unbounded domains, Calc. Var. Partial Differential Equations 4 (1996), 121–137. 10.1007/BF01189950Search in Google Scholar
[18] Di Nezza E., Palatucci G. and Valdinoci E., Hitchhiker’s guide to the fractional Sobolev spaces, Bull. Sci. Math. 136 (2012), 521–573. 10.1016/j.bulsci.2011.12.004Search in Google Scholar
[19] do Ó J. M., Miyagaki O. H. and Squassina M., Nonautonomous fractional problems with exponential growth, NoDEA Nonlinear Differential Equations Appl. 22 (2015), no. 5, 1395–1410. 10.1007/s00030-015-0327-0Search in Google Scholar
[20] do Ó J. M. B. and Souto M. A. S., On a class of nonlinear Schodinger equations in
[21] Fall M. M., Mahmoudi F. and Valdinoci E., Ground states and concentration phenomena for the fractional Schrödinger equation, Nonlinearity 28 (2015), no. 6, 1937–1961. 10.1088/0951-7715/28/6/1937Search in Google Scholar
[22] Felmer P., Quass A. and Tan J., Positive solutions of nonlinear Schrödinger equation with the fractional Laplacian, Proc. Roy. Soc. Edinburgh A 142 (2012), 1237–1262. 10.1017/S0308210511000746Search in Google Scholar
[23] Felmer P. and Torres C., Non-linear Schrödinger equation with non-local regional diffusion, Calc. Var. Partial Differential Equations 54 (2015), no. 1, 75–98. 10.1007/s00526-014-0778-xSearch in Google Scholar
[24] Frank R. and Lenzmann E., Uniqueness of non-linear ground states for fractional Laplacians in
[25] Iannizzotto A. and Squassina M.,
[26] Ozawa T., On critical cases of Sobolev’s inequalities, J. Funct. Anal. 127 (1995), 259–269. 10.1006/jfan.1995.1012Search in Google Scholar
[27] Secchi S., Ground state solutions for nonlinear fractional Schrödinger equations in
[28] Shang X. and Zhang J., Concentrating solutions of nonlinear fractional Schrödinger equation with potentials, J. Differential Equations 258 (2015), 1106–1128. 10.1016/j.jde.2014.10.012Search in Google Scholar
[29] Shang X., Zhang J. and Yang Y., On fractional Schödinger equation in
[30] Willem M., Minimax Theorems, Progr. Nonlinear Differential Equations Appl. 24, Birkhäuser, Boston, 1996. 10.1007/978-1-4612-4146-1Search in Google Scholar
[31] Zhang J., do Ó J. M. and Squassina M., Fractional Schrödinger–Poisson systems with a general subcritical or critical nonlinearity, Adv. Nonlinear Stud. 16 (2016), no. 1, 15–30. 10.1515/ans-2015-5024Search in Google Scholar
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This work is licensed under the Creative Commons Attribution 4.0 International License.
Articles in the same Issue
- Frontmatter
- Positive Answer to Berestycki’s Open Problem on the Unit Ball
- Existence and Multiplicity of Nontrivial Solutions to Quasilinear Elliptic Equations
- A Singular Elliptic System with Higher Order Terms of p-Laplacian Type
- Torsion Functions and the Cheeger Problem: A Fractional Approach
- On a Biharmonic Equation with Steep Potential Well and Indefinite Potential
- On Invariant Tori with Prescribed Frequency in Hamiltonian Systems
- Nonlinear Nonhomogeneous Robin Problems with Superlinear Reaction Term
- Existence and Multiplicity of Periodic Solutions for Dirichlet–Neumann Boundary Value Problem of a Variable Coefficient Wave Equation
- A Generalized Approach for the Modeling of Goodwin-Type Cycles
- Some Remarks on Profile Decomposition Theorems
- Spacelike Graphs of Prescribed Mean Curvature in the Steady State Space
- Critical Nonlocal Systems with Concave-Convex Powers
- Concentration Phenomena for Fractional Elliptic Equations Involving Exponential Critical Growth
- Optimal Functional Inequalities for Fractional Operators on the Sphere and Applications
Articles in the same Issue
- Frontmatter
- Positive Answer to Berestycki’s Open Problem on the Unit Ball
- Existence and Multiplicity of Nontrivial Solutions to Quasilinear Elliptic Equations
- A Singular Elliptic System with Higher Order Terms of p-Laplacian Type
- Torsion Functions and the Cheeger Problem: A Fractional Approach
- On a Biharmonic Equation with Steep Potential Well and Indefinite Potential
- On Invariant Tori with Prescribed Frequency in Hamiltonian Systems
- Nonlinear Nonhomogeneous Robin Problems with Superlinear Reaction Term
- Existence and Multiplicity of Periodic Solutions for Dirichlet–Neumann Boundary Value Problem of a Variable Coefficient Wave Equation
- A Generalized Approach for the Modeling of Goodwin-Type Cycles
- Some Remarks on Profile Decomposition Theorems
- Spacelike Graphs of Prescribed Mean Curvature in the Steady State Space
- Critical Nonlocal Systems with Concave-Convex Powers
- Concentration Phenomena for Fractional Elliptic Equations Involving Exponential Critical Growth
- Optimal Functional Inequalities for Fractional Operators on the Sphere and Applications
