Boundary Singularities of Solutions to Semilinear Fractional Equations
-
Phuoc-Tai Nguyen
Abstract
We prove the existence of a solution of
1 Introduction
Let
where
We denote by
Definition 1.1.
We say that a function
The s-boundary trace of u is denoted by
Let
We denote by
there exist
Definition 1.2.
Let
The boundary value problem with measure data for semilinear elliptic equations
was first studied by Gmira and Véron in [18], and then the typical model, i.e. problem (1.3) with
where
To our knowledge, few papers concerning boundary singularity problems for nonlinear fractional elliptic equations have been published in the literature. The earliest works in this direction are the papers [17, 10] by Felmer et al., which deal with the existence, nonexistence and asymptotic behavior of large solutions for equations involving fractional Laplacians. Afterwards, Abatangelo [1] presented a suitable setting for the study of fractional Laplacian equations in a measure framework and provided a fairly comprehensive description of large solutions which improve the results in [17, 10]. Recently, Chen, Alhomedan, Hajaiej and Markowich [9] investigated semilinear elliptic equations involving measures concentrated on the boundary by employing an approximate method.
In the present paper, we aim to establish the existence and uniqueness of weak solutions of (1.1). To this end, we develop a theory for linear equations associated to (1.1):
An existence and uniqueness result for (1.4) is stated in the following proposition.
Proposition 1.3.
Assume
Moreover, there exists a positive constant
This proposition allows us to study the semilinear equation (1.1). We first deal with the case of
Theorem 1.4.
Assume
Existence and uniqueness: For every
(1.7)(1.8)Monotonicity: The mapping
Remark.
The restriction
We reveal that, in measures framework, because of the interplay between the nonlocal operator
This yields substantial new difficulties and leads to disclose new types of results. The new aspects are both on the technical side and on the one of the new phenomena observed.
Theorem 1.5.
Assume
Stability: Assume that
If μ is a Dirac mass concentrated at a point on
Theorem 1.6.
Under the assumption of Theorem 1.5, let
Then
We next assume that
By Theorem 1.5,
with
the unit sphere in
where
Theorem 1.7.
Let
As a consequence of this result, we obtain the behavior of
Theorem 1.8.
Assume
If
(1.14)If
If Ω is a bounded
(1.15)locally uniformly on
Assume
The main ingredients of the present study are the following: estimates on the Green kernel and Martin kernel, theory for linear fractional equations in connection with the notion s-boundary trace as mentioned above, similarity transformation and the study of equation (1.13).
The paper is organized as follows: In Section 2, we present important properties of s-boundary trace, and prove Proposition 1.3. Theorems 1.4–1.6 and 1.8 are obtained in Section 3. Finally, in Appendix A, we discuss separable solutions of (1.12) and demonstrate Theorem 1.7.
2 Linear Problems
Throughout the present paper, we denote by
2.1 s-Harmonic Functions
We first recall the definition of s-harmonic functions (see [3, p. 46], [4, p. 230] and [6, p. 20]). Denote by
Denote by
where
For each Borel set
Definition 2.1.
Let u be a Borel measurable function in
We say that u is singular s-harmonic in Ω if u is s-harmonic and
Put
The following result follows from [5, Corollary 3.10 and Theorem 3.12] and [6, p. 20] (see also [20]).
Proposition 2.2.
Let
u is s -harmonic in Ω if and only if
u is singular s -harmonic in Ω if and only if u is s-harmonic in Ω and
2.2 Green Kernel, Poisson Kernel and Martin Kernel
In what follows, the notation
Denote by
where
Proposition 2.3.
There holds
(2.1)The similarity constant in the above estimate depends only on Ω and s.
Denote by
Put
Chen and Véron obtained the following estimate for the Green operator [11, Proposition 2.3 and Proposition 2.6].
Lemma 2.4.
Assume
There exists a constant
(2.3)Assume
Let
The relation between
Proposition 2.5.
There holds
The similarity constant in the above estimate depends only on Ω and s.
Denote by
Fix a reference point
By [15, Theorem 3.6], the Martin boundary of Ω can be identified with the Euclidean boundary
The next result [4, 15, 19] is important in the study of s-harmonic functions, which give a unique presentation of s-harmonic functions in terms of the Martin kernel.
Proposition 2.6.
The mapping
There holds
(2.4)The similarity constant in the above estimate depends only on Ω and s.
For every
If u is a nonnegative s -harmonic function in Ω , then there exists a unique
Lemma 2.7.
There exists a constant
(2.5)If
Proof.
(i) Using (2.4) and an argument similar to the one in the proof of [2, Theorem 2.5], we obtain (2.5).
(ii) By combining the fact that
2.3 Boundary Trace
We recall that, for
The following geometric property of
Proposition 2.8.
There exists
For every point
The mappings
Proposition 2.9.
Assume
Proof.
For
which implies
Note that for
In order to prove (2.6), we may assume that the coordinates are placed so that
Hence if
Therefore, for
By combining estimates (2.7) and (2.8), we obtain the second estimate in (2.6). The first estimate in (2.6) follows from (2.4). ∎
As a consequence, we get the following estimates.
Corollary 2.10.
Assume
with c being as in (2.6).
Proposition 2.11.
Assume
Moreover,
Proof.
Without loss of generality, we may assume that
Note that if
where the last inequality holds since
Combining the above estimates, we obtain (2.9).
Next we demonstrate (2.10). Given
Thus the choice of
Put
which yields
On the other hand, due to (2.9),
Lemma 2.12.
Assume
If
Moreover, if
Proof.
By the assumption, there exists a nonnegative Radon measure τ on Ω such that
We first prove that
By [1, p. 5547], there is a positive constant
where the similarity constant depends only on Ω and s. This implies
We define
For any
Moreover, the solution can be written as
By the maximum principle [1, Lemma 3.9],
Letting
We next show that there exists a measure
Definition 2.13.
A function u possesses an s-boundary trace on
The s-boundary trace of u is denoted by
Remark.
(i) The notation of s-boundary trace is well defined. Indeed, suppose that μ and
By Kato’s inequality [8, Theorem 1.2],
(ii) It is clear that for every
(iii) This kind of boundary trace was first introduced by Nguyen and Marcus [21] in order to investigate semilinear elliptic equations with Hardy potential. In the present paper, we prove that it is still an effective tools in the study of nonlocal fractional elliptic equations.
2.4 Weak Solutions of Linear Problems
Definition 2.14.
Let
Proof of Proposition 1.3.
The uniqueness follows from [11, Proposition 2.4]. Let u be as in (1.5). By [11],
This implies (2.16), and therefore u is the unique solution of (1.4). Since
3 Nonlinear Problems
In this section, we study the nonlinear problem (1.1). The definition of weak solutions of (1.1) is given in Definition 1.2.
3.1 Subcritical Absorption
Proof of Theorem 1.4.
We prove this theorem in several steps. Monotonicity. Let
Similarly,
Therefore,
By Kato’s inequality, the assumption
Therefore,
Since
Existence.
Step 1: Assume that
Then
By [12, Proposition 3.1], there exists a unique weak solution v of (3.1). This means
Put
Step 2: Assume that
Let
Then
Hence
The uniqueness follows from the monotonicity.
Step 3: Assume that
Let
Moreover, for any
it follows that
This together with (3.6) implies
Put
By step 2, the sequences
The uniqueness follows from the monotonicity. ∎
Define
This space is endowed with the norm
We say that a sequence
Proof of Theorem 1.5.
Monotonicity. The monotonicity can be proved by using an argument similar to the one in the proof of Theorem 1.4.
Existence. Let
Let
This, together with (2.3), (2.5) and (3.9), implies that
We have
From this
follows. We infer from (3.7) and the estimate
This implies that
Next we prove that
We take an arbitrary Borel set
On one hand, we have
From (3.10), we infer
By assumption (1.9), there exists a sequence
From assumption (1.9), we see that the right-hand side of (3.13) tends to 0 as
This means
By Vitali’s convergence theorem, we deduce that, up to a subsequence,
Stability. Assume
Proposition 3.1.
Assume f is a continuous nondecreasing function on
Proof.
By (2.1),
Hence,
Put
Therefore, for every
Hence,
We next estimate
Therefore, by (1.9),
Finally, we estimate
Put
If
By combining (3.18) and (3.19), we obtain
3.2 Power Absorption
In this subsection, we assume that
For any
If u is a solution of (1.14) in Ω, then
By Corollary A.9, the function
where
Lemma 3.2.
Assume
Proof.
Let
Put
Then
Moreover,
Put
We will compare
Step 1: We show that
where
it follows that, for any
Therefore, if we choose
Next we see that there exists
Consequently,
Therefore, if we choose
By combining (3.23) and (3.24), for
Step 2: We show that
Then
This contradiction implies that
Step 3: End of proof. From step 2 we deduce that
We note that
This implies
Put
If
By combining (3.25) and (3.26), we get
If
Lemma 3.3.
Let
Proof.
We use an argument similar to the one in the proof of Proposition 3.1. It is easy to see that for every
Let
Now, by proceeding as in the proof of Proposition 3.1, we deduce easily that there is a positive constant
and
Proposition 3.4.
Assume
Proof.
We first claim that for any
Indeed, by (2.4),
Since
By proceeding as in step 2 of the proof of Lemma 3.2, we deduce that
Consequently,
Next we prove the lower bound in (3.30). By (2.4) and Lemma 3.3, for any
For
where
By choosing
Since
Proposition 3.5.
Assume
Proof.
For any
Case 1:
and take
Here the first estimate holds since
Case 2:
and take
Here the third estimate holds since
Case 3:
and take
Here the third estimate holds since
Case 4:
and take
Here the inequality holds since
Since
By putting
Proposition 3.6.
Assume
Proof.
The proposition can be obtained by adapting the argument in the proof of [9, Theorem 1.2]. Let
Then
which implies
Fix
By [13, Lemma 2.2],
Next put
where
Therefore,
Let
We can choose k large enough so that the function
is a subsolution of (3.37). By [13, Lemma 2.2], we obtain
Put
Then we derive from (3.38) that
By combining (3.35), (3.36) and (3.39), we deduce that
This implies
Theorem 3.7.
Assume
Moreover, the map
Proof.
We divide the proof into two steps. Step 1: Existence. For
Then
Since
From (3.40) we deduce that
where c depends only on N, s and the
Step 2: Uniqueness. Suppose u and
Take
By an argument similar to the one in step 2, we can show that
Proof of Theorem 1.8.
We have two cases. (i) Case 1:
Since
where the first inequality holds in B and the second inequality holds in Ω.
Let
with
Step 1:
Therefore,
where
This implies (3.30).
Step 2:
and
When
Furthermore, there also holds for
Letting
Again this implies that
Step 3: End of the proof. From (3.42) and (3.44) there holds
Since the left-hand side and the right-hand side of (3.45) converge to the same function
and this convergence holds in any compact subset of Ω. Taking
(ii) Case 2:
Funding statement: The first author is supported by Fondecyt Grant 3160207. The second author is supported by collaboration programs ECOS C14E08.
A Separable Solutions
A.1 Separable s-Harmonic Functions
We denote by
Hence
with
where
where
whenever this integral is defined. We will see in the next two lemmas that the role of the exponent
Lemma A.1.
If
Then the following statements hold:
Proof.
Since
By the change of variable
where
the claim follows.∎
As a byproduct of (A.1) we have the following monotonicity formula.
Lemma A.2.
If
In the next result we analyze the behavior of
Lemma A.3.
Assume
If
(A.2)If
Proof.
First, notice that the quantity
is uniformly bounded with respect to
as
Hence,
If
for some
and if
as
Proposition A.4.
Assume
Furthermore,
Proof.
By Lemma A.3, for any
if
The above calculations justify the name of the fractional Laplace–Beltrami operator given to
Lemma A.5.
Assume
where
Proof.
If
Then we put
This implies
Since
Lemma A.6.
Under the assumption of Lemma A.5, there holds
where
Proof.
There holds, by the Cauchy–Schwarz inequality,
Since by invariance by rotation we have
we derive (A.4).∎
We denote the upper hemisphere of the unit sphere in
Proposition A.7.
Let
Furthermore, the mapping
Proof.
We first notice that
for any
where
Since by the Poincaré inequality [16] there holds
we obtain that the right-hand side of (A.6) is bounded from above by
Next we use the expansion estimates in Lemma A.5 to obtain that
where
Therefore,
Finally, we obtain
We consider the following bilinear form in
Then
and
By the Riesz theorem, for any
We set
is a compact positive operator (here we use the fact that
The function
Using (A.1), Lemma A.2 and the monotone convergence theorem, we derive that the mapping
is increasing and continuous. This implies that
when
we derive from the Poincaré inequality [16] and (A.4) that
Since
Hence if
This means that the function
satisfies
The uniqueness of the positive eigenfunction implies that this function is
A.2 The Nonlinear Problem
A.2.1 Separable Solutions in ℝ N
If we look for separable positive solutions of
under the form
Proposition A.8.
Assume
If
Proof.
If
Since
Then
Next, if
Using invariance by rotation of the integral term on
Hence
Corollary A.9.
Assume
A.2.2 Separable Solutions in ℝ + N
If we consider separable solutions
Proof of Theorem 1.7.
We prove this in three steps. Step 1: Nonexistence. Assume that such a solution
Hence,
If
Step 2: Existence. Consider the following functional with domain
Because of Lemma A.6,
This implies that
Step 3: Uniqueness.
(i) Existence of a maximal solution. By [27], any solution ω is smooth. Hence, at its maximum
This implies that
Furthermore, this sequence
(ii) Existence of a minimal solution. From Theorem 3.7 follows that
Hence
if
(iii) End of the uniqueness proof. By combining (A.10) and (A.11), we infer that there exists
Assume
is a positive supersolution (by convexity) of (1.13). Moreover,
is a positive subsolution of (1.13) smaller than
References
[1] N. Abatangelo, Large S-harmonic functions and boundary blow-up solutions for the fractional Laplacian, Discrete Contin. Dyn. Syst. 35 (2015), no. 12, 5555–5607. 10.3934/dcds.2015.35.5555Suche in Google Scholar
[2] M.-F. Bidaut-Véron and L. Vivier, An elliptic semilinear equation with source term involving boundary measures: The subcritical case, Rev. Mat. Iberoam. 16 (2000), no. 3, 477–513. 10.4171/RMI/281Suche in Google Scholar
[3] K. Bogdan, The boundary Harnack principle for the fractional Laplacian, Studia Math. 123 (1997), no. 1, 43–80. 10.4064/sm-123-1-43-80Suche in Google Scholar
[4] K. Bogdan, Representation of α-harmonic functions in Lipschitz domains, Hiroshima Math. J. 29 (1999), no. 2, 227–243. 10.32917/hmj/1206125005Suche in Google Scholar
[5] K. Bogdan and T. Byczkowski, Potential theory for the α-stable Schrödinger operator on bounded Lipschitz domains, Studia Math. 133 (1999), no. 1, 53–92. 10.4064/sm-133-1-53-92Suche in Google Scholar
[6] K. Bogdan, T. Byczkowski, T. Kulczycki, M. Ryznar, R. Song and Z. Vondraček, Potential Analysis of Stable Processes and its Extensions, Lecture Notes in Math. 1980, Springer, Berlin, 2009. 10.1007/978-3-642-02141-1Suche in Google Scholar
[7] K. Bogdan, T. Kulczycki and M. Kwaśnicki, Estimates and structure of α-harmonic functions, Probab. Theory Related Fields 140 (2008), no. 3–4, 345–381. 10.1007/s00440-007-0067-0Suche in Google Scholar
[8] L. A. Caffarelli and Y. Sire, On some pointwise inequalities involving nonlocal operators, Harmonic Analysis, Partial Differential Equations and Applications, Appl. Numer. Harmon. Anal., Birkhäuser/Springer, Cham (2017), 1–18. 10.1007/978-3-319-52742-0_1Suche in Google Scholar
[9] H. Chen, S. Alhomedan, H. Hajaiej and P. Markowich, Complete study of the existence and uniqueness of solutions for semilinear elliptic equations involving measures concentrated on boundary, Complex Var. Elliptic Equ. 62 (2017), no. 12, 1687–1729. 10.1080/17476933.2016.1278441Suche in Google Scholar
[10] H. Chen, P. Felmer and A. Quaas, Large solutions to elliptic equations involving fractional Laplacian, Ann. Inst. H. Poincaré Anal. Non Linéaire 32 (2015), no. 6, 1199–1228. 10.1016/j.anihpc.2014.08.001Suche in Google Scholar
[11] H. Chen and L. Véron, Semilinear fractional elliptic equations involving measures, J. Differential Equations 257 (2014), no. 5, 1457–1486. 10.1016/j.jde.2014.05.012Suche in Google Scholar
[12] H. Chen and L. Véron, Weakly and strongly singular solutions of semilinear fractional elliptic equations, Asymptot. Anal. 88 (2014), no. 3, 165–184. 10.3233/ASY-141216Suche in Google Scholar
[13] H. Chen and J. Yang, Semilinear fractional elliptic equations with measures in unbounded domain, Nonlinear Anal. 145 (2016), 118–142. 10.1016/j.na.2016.08.003Suche in Google Scholar
[14] Z.-Q. Chen and R. Song, Estimates on Green functions and Poisson kernels for symmetric stable processes, Math. Ann. 312 (1998), no. 3, 465–501. 10.1007/s002080050232Suche in Google Scholar
[15] Z.-Q. Chen and R. Song, Martin boundary and integral representation for harmonic functions of symmetric stable processes, J. Funct. Anal. 159 (1998), no. 1, 267–294. 10.1006/jfan.1998.3304Suche in Google Scholar
[16] E. Di Nezza, G. Palatucci and E. Valdinoci, Hitchhiker’s guide to the fractional Sobolev spaces, Bull. Sci. Math. 136 (2012), no. 5, 521–573. 10.1016/j.bulsci.2011.12.004Suche in Google Scholar
[17] P. Felmer and A. Quaas, Boundary blow up solutions for fractional elliptic equations, Asymptot. Anal. 78 (2012), no. 3, 123–144. 10.3233/ASY-2011-1081Suche in Google Scholar
[18] A. Gmira and L. Véron, Boundary singularities of solutions of some nonlinear elliptic equations, Duke Math. J. 64 (1991), no. 2, 271–324. 10.1215/S0012-7094-91-06414-8Suche in Google Scholar
[19] P. Graczyk, T. Jakubowski and T. Luks, Martin representation and relative Fatou theorem for fractional Laplacian with a gradient perturbation, Positivity 17 (2013), no. 4, 1043–1070. 10.1007/s11117-012-0220-6Suche in Google Scholar
[20] M. Kwaśnicki, Ten equivalent definitions of the fractional Laplace operator, Fract. Calc. Appl. Anal. 20 (2017), no. 1, 7–51. 10.1515/fca-2017-0002Suche in Google Scholar
[21] M. Marcus and P.-T. Nguyen, Moderate solutions of semilinear elliptic equations with Hardy potential, Ann. Inst. H. Poincaré Anal. Non Linéaire 34 (2017), no. 1, 69–88. 10.1016/j.anihpc.2015.10.001Suche in Google Scholar
[22] M. Marcus and L. Véron, The boundary trace of positive solutions of semilinear elliptic equations: The subcritical case, Arch. Ration. Mech. Anal. 144 (1998), no. 3, 201–231. 10.1007/s002050050116Suche in Google Scholar
[23] M. Marcus and L. Veron, The boundary trace of positive solutions of semilinear elliptic equations: The supercritical case, J. Math. Pures Appl. (9) 77 (1998), no. 5, 481–524. 10.1016/S0021-7824(98)80028-7Suche in Google Scholar
[24] M. Marcus and L. Veron, Removable singularities and boundary traces, J. Math. Pures Appl. (9) 80 (2001), no. 9, 879–900. 10.1016/S0021-7824(01)01209-0Suche in Google Scholar
[25] M. Marcus and L. Véron, The boundary trace and generalized boundary value problem for semilinear elliptic equations with coercive absorption, Comm. Pure Appl. Math. 56 (2003), no. 6, 689–731. 10.1002/cpa.3037Suche in Google Scholar
[26] M. Marcus and L. Véron, Nonlinear Second Order Elliptic Equations Involving Measures, De Gruyter Ser. Nonlinear Anal. Appl. 21, De Gruyter, Berlin, 2014. 10.1515/9783110305319Suche in Google Scholar
[27] X. Ros-Oton and J. Serra, The Dirichlet problem for the fractional Laplacian: Regularity up to the boundary, J. Math. Pures Appl. (9) 101 (2014), no. 3, 275–302. 10.1016/j.matpur.2013.06.003Suche in Google Scholar
© 2018 Walter de Gruyter GmbH, Berlin/Boston
This work is licensed under the Creative Commons Attribution 4.0 International License.
Artikel in diesem Heft
- Frontmatter
- Solving Becker’s Problem on Periodic Solutions of Parabolic Evolution Equations
- Global Dynamics of Generalized Logistic Equations
- Boundary Singularities of Solutions to Semilinear Fractional Equations
- On Parabolic Variational Inequalities with Multivalued Terms and Convex Functionals
- Refined Boundary Behavior of the Unique Convex Solution to a Singular Dirichlet Problem for the Monge–Ampère Equation
- Symmetric and Asymmetric Solutions of p-Laplace Elliptic Equations in Hollow Domains
- Optimal Design Problems for the First p-Fractional Eigenvalue with Mixed Boundary Conditions
- Boundedness of Solutions to a Parabolic-Elliptic Keller–Segel Equation in ℝ2 with Critical Mass
- On a Class of Quasilinear Elliptic Equations with Degenerate Coerciveness and Measure Data
- The Bahri–Coron Theorem for Fractional Yamabe-Type Problems
- Radial Nonlinear Elliptic Problems with Singular or Vanishing Potentials
Artikel in diesem Heft
- Frontmatter
- Solving Becker’s Problem on Periodic Solutions of Parabolic Evolution Equations
- Global Dynamics of Generalized Logistic Equations
- Boundary Singularities of Solutions to Semilinear Fractional Equations
- On Parabolic Variational Inequalities with Multivalued Terms and Convex Functionals
- Refined Boundary Behavior of the Unique Convex Solution to a Singular Dirichlet Problem for the Monge–Ampère Equation
- Symmetric and Asymmetric Solutions of p-Laplace Elliptic Equations in Hollow Domains
- Optimal Design Problems for the First p-Fractional Eigenvalue with Mixed Boundary Conditions
- Boundedness of Solutions to a Parabolic-Elliptic Keller–Segel Equation in ℝ2 with Critical Mass
- On a Class of Quasilinear Elliptic Equations with Degenerate Coerciveness and Measure Data
- The Bahri–Coron Theorem for Fractional Yamabe-Type Problems
- Radial Nonlinear Elliptic Problems with Singular or Vanishing Potentials
