# American Institute of Mathematical Sciences

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 , $D\subset \mathbb R^d\setminus \Omega$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 $
. 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. Abdellaoui, A. 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. Abdellaoui, E. 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. Abdellaoui, E. 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. Abdellaoui, M. Medina, I. 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. Abdellaoui, M. Medina, I. 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. Abdellaoui, I. 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 Nezza, G. 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. Dipierro, X. 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. Frank, E. 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. Leonori, I. Peral, A. 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. Abdellaoui, A. 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. Abdellaoui, E. 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. Abdellaoui, E. 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. Abdellaoui, M. Medina, I. 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. Abdellaoui, M. Medina, I. 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. Abdellaoui, I. 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 Nezza, G. 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. Dipierro, X. 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. Frank, E. 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. Leonori, I. Peral, A. 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.
Example 1
Example 2
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