December  2018, 38(12): 5963-5991. doi: 10.3934/dcds.2018131

Attainability of the fractional hardy constant with nonlocal mixed boundary conditions: Applications

1. 

Laboratoire d'Analyse Nonlinéaire et Mathématiques Appliquées, Université Abou Bakr Belkaïd, Tlemcen, Tlemcen 13000, Algeria

2. 

Département de Mathématiques, Université Ibn Khaldoun, Tiaret, Tiaret 14000, Algeria

3. 

Departamento de Matemáticas, Universidad Autonoma de Madrid, 28049 Madrid, Spain

* Corresponding author

To Rafa de la Llave in his 60th birthday, with our best wishes.

Received  September 2017 Revised  January 2018 Published  April 2018

Fund Project: This work is partially supported by research grants MTM2013-40846-P and MTM2016-80474-P, MINECO, Spain.

The first goal of this paper is to study necessary and sufficient conditions to obtain the attainability of the fractional Hardy inequality
$\Lambda_{N}\equiv \Lambda_{N}(\Omega): = \inf\limits_{\{\varphi\in \mathbb{E}^{s}(\Omega, D), \varphi \ne0\}}\dfrac{\frac{a_{d, s}}{2}\displaystyle\int_{\mathbb R^d}\int_{\mathbb R^d}\dfrac{|\varphi(x)-\phi(y)|^{2}}{|x-y|^{d+2s}}dx dy}{\displaystyle\int_{\Omega}\frac{\varphi^2}{|x|^{2s}}\, dx}, $
where
$\Omega$
is a bounded domain of
$\mathbb R^d$
,
$0<s<1$
,
$D\subset \mathbb R^d\setminus \Omega$
a nonempty open set,
$N = (\mathbb R^d\setminus \Omega)\setminus\overline{D}$
and
$\mathbb{E}^{s}(\Omega, D) = \{ u \in H^s(\mathbb R^d):\, u = 0 \text{ in } D\}.$
The second aim of the paper is to study the mixed Dirichlet-Neumann boundary problem associated to the minimization problem and related properties; precisely, to study semilinear elliptic problem for the fractional Laplacian, that is,
${P_\lambda } \equiv \left\{ {\begin{array}{*{20}{l}}{{{\left( { - \Delta } \right)}^s}u\;\;\; = \;\;\;\lambda \frac{u}{{|x{|^{2s}}}} + {u^p}}&{{\rm{in}}\;\Omega ,}\\{\;\;\;\;\;\;\;\;\;u\;\;\; > \;\;\;0}&{{\rm{in}}\;\Omega ,}\\{\;\;\;\;\;\;{{\cal B}_s}u\;\;\;: = \;\;u{\chi _D} + {{\cal N}_s}u{\chi _N} = 0}&{{\rm{in}}\;{{\mathbb {R}}^d}\backslash \Omega ,}\end{array}} \right.$
with
$N$
and
$D$
open sets in
$\mathbb R^{d}\backslash\Omega$
such that
$N \cap D = \emptyset$
and
$\overline{N}\cup \overline{D} = \mathbb R^{d}\backslash\Omega$
,
$d>2s$
,
$\lambda> 0$
and
$<p\le 2_s^*-1_s^* = \frac{2d}{d-2s}$
. We emphasize that the nonlinear term can be critical.
The operators
$(-\Delta)^s $
, fractional Laplacian, and
$\mathcal{N}_{s}$
, nonlocal Neumann condition, are defined below in (7) and (8) respectively.
Citation: Boumediene Abdellaoui, Ahmed Attar, Abdelrazek Dieb, Ireneo Peral. Attainability of the fractional hardy constant with nonlocal mixed boundary conditions: Applications. Discrete and Continuous Dynamical Systems, 2018, 38 (12) : 5963-5991. doi: 10.3934/dcds.2018131
References:
[1]

B. Abdellaoui and R. Bentifour, Caffarelli-Kohn-Nirenberg Type Inequalities of Fractional Order and Applications, J. Funct. Anal., 272 (2017), 3998-4029.  doi: 10.1016/j.jfa.2017.02.007.

[2]

B. AbdellaouiA. Dieb and E. Valdinoci, A nonlocal concave-convex problem with nonlocal mixed boundary data, Comm. on Pure Appl. Analysis, 17 (2018), 1103-1120.  doi: 10.3934/cpaa.2018053.

[3]

B. AbdellaouiE. Colorado and I. Peral, Some remarks on elliptic equations with singular potential and mixed boundary conditions, Advanced Nonlinear Studies, 4 (2004), 503-533. 

[4]

B. AbdellaouiE. Colorado and I. Peral, Effect on the boundary conditions in the behaviour of the optimal constant of some Caffarelli-Kohn-Nirenberg inequalities. Application to some doubly critical nonlinear elliptic problems, Advances in Differential Equations, 11 (2006), 667-720. 

[5]

B. AbdellaouiM. MedinaI. Peral and A. Primo, A note on the effect of the Hardy potential in some Calderon-Zygmund properties for the fractional Laplacian, J. Differential Equations, 260 (2016), 8160-8206.  doi: 10.1016/j.jde.2016.02.016.

[6]

B. AbdellaouiM. MedinaI. Peral and A. Primo, Optimal results for the fractional heat equation involving the Hardy potential, Nonlinear Anal., 140 (2016), 166-207.  doi: 10.1016/j.na.2016.03.013.

[7]

B. AbdellaouiI. Peral and A. Primo, A remark on the fractional Hardy inequality with a remainder term, C. R. Acad. Sci. Paris, Ser. I, 352 (2014), 299-303.  doi: 10.1016/j.crma.2014.02.003.

[8]

B. Barrios and M. Medina, Strong maximum principles for fractional elliptic and parabolic problems with mixed boundary conditions. arXiv: 1607.01505.

[9]

W. Beckner, Pitt's inequality and the uncertainty principle, Proceedings of the American Mathematical Society, 123 (1995), 1897-1905. 

[10]

H. Brézis and E. Lieb, A relation between pointwise convergence of functions and convergence of functionals, Proc. Amer. Math. Soc., 88 (1983), 486-490. 

[11]

H. Brézis and L. Nirenberg, Positive solutions of nonlinear elliptic equations involving critical Sobolev exponents, Comm. Pure Appl. Math., 36 (1983), 437-477.  doi: 10.1002/cpa.3160360405.

[12]

C. Bucur and E. Valdinoci, Nonlocal Diffusion and Applications, Lecture Notes of the Unione Matematica Italiana, 20. Springer; Unione Matematica Italiana, Bologna, 2016.

[13]

X. Cabré and J. Tan, Positive solutions of nonlinear problems involving the square root of the Laplacian, Adv. Math., 224 (2010), 2052-2093.  doi: 10.1016/j.aim.2010.01.025.

[14]

L. Caffarelli and L. Silvestre, An extension problem related to the fractional Laplacian, Comm. Partial Differential Equations., 32 (2007), 1245-1260.  doi: 10.1080/03605300600987306.

[15]

E. Di NezzaG. Palatucci and E. Valdinoci, Hitchhiker's guide to the fractional Sobolev spaces, Bull. Sci. Math., 136 (2012), 521-573.  doi: 10.1016/j.bulsci.2011.12.004.

[16]

S. Dipierro, L. Montoro, I. Peral and D. Sciunzi, Qualitative properties of positive solutions to nonlocal critical problems involving the Hardy-Leray potential, Calc. Var. Partial Differential Equations, 55 (2016), Art. 99, 29 pp.

[17]

S. DipierroX. Ros-Oton and E. Valdinoci, Nonlocal problems with Neumann boundary conditions, Rev. Mat. Iberoam, 33 (2017), 377-416.  doi: 10.4171/RMI/942.

[18]

I. Ekeland, On the variational principle, J. Math. Anal. Appl., 47 (1974), 324-353.  doi: 10.1016/0022-247X(74)90025-0.

[19]

F. Ferrari and I. Verbitsky, Radial fractional Laplace operators and Hessian inequalities, J. Differential Equations, 253 (2012), 244-272.  doi: 10.1016/j.jde.2012.03.024.

[20]

R. L. Frank and R. Seiringer, Non-linear ground state representations and sharp Hardy inequalities, Journal of Functional Analysis, 255 (2008), 3407-3430.  doi: 10.1016/j.jfa.2008.05.015.

[21]

R. FrankE. H. Lieb and R. Seiringer, Hardy-Lieb-Thirring inequalities for fractional Schrödinger operators, J. Amer. Math. So, 21 (2008), 925-950. 

[22]

G. Grubb, Local and nonlocal boundary conditions for $mu$-transmission and fractional order elliptic pseudodifferential operators, Anal. PDE, 7 (2014), 1649-1682.  doi: 10.2140/apde.2014.7.1649.

[23]

I. W. Herbst, Spectral theory of the operator $(p^2+m^2)^{1/2}-Ze^2/r$, Commun. math. Phys., 53 (1977), 285-294. 

[24]

N. S. Landkof, Foundations of Modern Potential Theory, Translated from the Russian by A. P. Doohovskoy, Die Grundlehren der mathematischen Wissenschaften, Band 180, Springer-Verlag, New York-Heidelberg, 1972.

[25]

T. Leonori, M. Medina, I. Peral, A. Primo and F. Soria, Principal eigenvalue of mixed problem for the fractional Laplacian: Moving the boundary conditions, J. Differential Equations, 265 (2018), no. 2, 593-619. doi: 10.1016/j.jde.2018.03.001.

[26]

T. LeonoriI. PeralA. Primo and F. Soria, Basic estimates for solutions of a class of nonlocal elliptic and parabolic equations, Discrete Contin. Dyn. Syst., 35 (2015), 6031-6068.  doi: 10.3934/dcds.2015.35.6031.

[27]

G. Molica Bisci, V. Radulescu and R. Servadei, Variational Methods for Nonlocal Fractional Problems, Encyclopedia of Mathematics and its Applications, 162. Cambridge University Press, Cambridge, 2016.

[28]

A. C. Ponce, Elliptic PDE's, Measures and Capacities, Tracts in Mathematics 23, European Mathematical Society (EMS), Zurich, 2016.

[29]

P. H. Rabinowitz, Some global results for nonlinear eigenvalue problems, J. Funct. Analysis, 7 (1971), 487-513.  doi: 10.1016/0022-1236(71)90030-9.

[30]

R. Servadei and E. Valdinoci, The Brezis-Nirenberg result for the fractional Laplacian, Trans. Amer. Math. Soc., 367 (2015), 67-102. 

[31]

E. M. Stein and G. Weiss, Fractional integrals on n-dimensional Euclidean space, J. Math. Mech., 7 (1958), 503-514. 

[32]

S. Terracini, On positive entire solutions to a class of equations with a singular coefficient and critical exponent, Adv. Differential Equations, 1 (1996), 241-264. 

[33]

D. Yafaev, Sharp constants in the Hardy-Rellich inequalities, J. Functional Analysis, 168 (1999), 121-144.  doi: 10.1006/jfan.1999.3462.

show all references

To Rafa de la Llave in his 60th birthday, with our best wishes.

References:
[1]

B. Abdellaoui and R. Bentifour, Caffarelli-Kohn-Nirenberg Type Inequalities of Fractional Order and Applications, J. Funct. Anal., 272 (2017), 3998-4029.  doi: 10.1016/j.jfa.2017.02.007.

[2]

B. AbdellaouiA. Dieb and E. Valdinoci, A nonlocal concave-convex problem with nonlocal mixed boundary data, Comm. on Pure Appl. Analysis, 17 (2018), 1103-1120.  doi: 10.3934/cpaa.2018053.

[3]

B. AbdellaouiE. Colorado and I. Peral, Some remarks on elliptic equations with singular potential and mixed boundary conditions, Advanced Nonlinear Studies, 4 (2004), 503-533. 

[4]

B. AbdellaouiE. Colorado and I. Peral, Effect on the boundary conditions in the behaviour of the optimal constant of some Caffarelli-Kohn-Nirenberg inequalities. Application to some doubly critical nonlinear elliptic problems, Advances in Differential Equations, 11 (2006), 667-720. 

[5]

B. AbdellaouiM. MedinaI. Peral and A. Primo, A note on the effect of the Hardy potential in some Calderon-Zygmund properties for the fractional Laplacian, J. Differential Equations, 260 (2016), 8160-8206.  doi: 10.1016/j.jde.2016.02.016.

[6]

B. AbdellaouiM. MedinaI. Peral and A. Primo, Optimal results for the fractional heat equation involving the Hardy potential, Nonlinear Anal., 140 (2016), 166-207.  doi: 10.1016/j.na.2016.03.013.

[7]

B. AbdellaouiI. Peral and A. Primo, A remark on the fractional Hardy inequality with a remainder term, C. R. Acad. Sci. Paris, Ser. I, 352 (2014), 299-303.  doi: 10.1016/j.crma.2014.02.003.

[8]

B. Barrios and M. Medina, Strong maximum principles for fractional elliptic and parabolic problems with mixed boundary conditions. arXiv: 1607.01505.

[9]

W. Beckner, Pitt's inequality and the uncertainty principle, Proceedings of the American Mathematical Society, 123 (1995), 1897-1905. 

[10]

H. Brézis and E. Lieb, A relation between pointwise convergence of functions and convergence of functionals, Proc. Amer. Math. Soc., 88 (1983), 486-490. 

[11]

H. Brézis and L. Nirenberg, Positive solutions of nonlinear elliptic equations involving critical Sobolev exponents, Comm. Pure Appl. Math., 36 (1983), 437-477.  doi: 10.1002/cpa.3160360405.

[12]

C. Bucur and E. Valdinoci, Nonlocal Diffusion and Applications, Lecture Notes of the Unione Matematica Italiana, 20. Springer; Unione Matematica Italiana, Bologna, 2016.

[13]

X. Cabré and J. Tan, Positive solutions of nonlinear problems involving the square root of the Laplacian, Adv. Math., 224 (2010), 2052-2093.  doi: 10.1016/j.aim.2010.01.025.

[14]

L. Caffarelli and L. Silvestre, An extension problem related to the fractional Laplacian, Comm. Partial Differential Equations., 32 (2007), 1245-1260.  doi: 10.1080/03605300600987306.

[15]

E. Di NezzaG. Palatucci and E. Valdinoci, Hitchhiker's guide to the fractional Sobolev spaces, Bull. Sci. Math., 136 (2012), 521-573.  doi: 10.1016/j.bulsci.2011.12.004.

[16]

S. Dipierro, L. Montoro, I. Peral and D. Sciunzi, Qualitative properties of positive solutions to nonlocal critical problems involving the Hardy-Leray potential, Calc. Var. Partial Differential Equations, 55 (2016), Art. 99, 29 pp.

[17]

S. DipierroX. Ros-Oton and E. Valdinoci, Nonlocal problems with Neumann boundary conditions, Rev. Mat. Iberoam, 33 (2017), 377-416.  doi: 10.4171/RMI/942.

[18]

I. Ekeland, On the variational principle, J. Math. Anal. Appl., 47 (1974), 324-353.  doi: 10.1016/0022-247X(74)90025-0.

[19]

F. Ferrari and I. Verbitsky, Radial fractional Laplace operators and Hessian inequalities, J. Differential Equations, 253 (2012), 244-272.  doi: 10.1016/j.jde.2012.03.024.

[20]

R. L. Frank and R. Seiringer, Non-linear ground state representations and sharp Hardy inequalities, Journal of Functional Analysis, 255 (2008), 3407-3430.  doi: 10.1016/j.jfa.2008.05.015.

[21]

R. FrankE. H. Lieb and R. Seiringer, Hardy-Lieb-Thirring inequalities for fractional Schrödinger operators, J. Amer. Math. So, 21 (2008), 925-950. 

[22]

G. Grubb, Local and nonlocal boundary conditions for $mu$-transmission and fractional order elliptic pseudodifferential operators, Anal. PDE, 7 (2014), 1649-1682.  doi: 10.2140/apde.2014.7.1649.

[23]

I. W. Herbst, Spectral theory of the operator $(p^2+m^2)^{1/2}-Ze^2/r$, Commun. math. Phys., 53 (1977), 285-294. 

[24]

N. S. Landkof, Foundations of Modern Potential Theory, Translated from the Russian by A. P. Doohovskoy, Die Grundlehren der mathematischen Wissenschaften, Band 180, Springer-Verlag, New York-Heidelberg, 1972.

[25]

T. Leonori, M. Medina, I. Peral, A. Primo and F. Soria, Principal eigenvalue of mixed problem for the fractional Laplacian: Moving the boundary conditions, J. Differential Equations, 265 (2018), no. 2, 593-619. doi: 10.1016/j.jde.2018.03.001.

[26]

T. LeonoriI. PeralA. Primo and F. Soria, Basic estimates for solutions of a class of nonlocal elliptic and parabolic equations, Discrete Contin. Dyn. Syst., 35 (2015), 6031-6068.  doi: 10.3934/dcds.2015.35.6031.

[27]

G. Molica Bisci, V. Radulescu and R. Servadei, Variational Methods for Nonlocal Fractional Problems, Encyclopedia of Mathematics and its Applications, 162. Cambridge University Press, Cambridge, 2016.

[28]

A. C. Ponce, Elliptic PDE's, Measures and Capacities, Tracts in Mathematics 23, European Mathematical Society (EMS), Zurich, 2016.

[29]

P. H. Rabinowitz, Some global results for nonlinear eigenvalue problems, J. Funct. Analysis, 7 (1971), 487-513.  doi: 10.1016/0022-1236(71)90030-9.

[30]

R. Servadei and E. Valdinoci, The Brezis-Nirenberg result for the fractional Laplacian, Trans. Amer. Math. Soc., 367 (2015), 67-102. 

[31]

E. M. Stein and G. Weiss, Fractional integrals on n-dimensional Euclidean space, J. Math. Mech., 7 (1958), 503-514. 

[32]

S. Terracini, On positive entire solutions to a class of equations with a singular coefficient and critical exponent, Adv. Differential Equations, 1 (1996), 241-264. 

[33]

D. Yafaev, Sharp constants in the Hardy-Rellich inequalities, J. Functional Analysis, 168 (1999), 121-144.  doi: 10.1006/jfan.1999.3462.

Figure 1.  Example 1
Figure 2.  Example 2
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