American Institute of Mathematical Sciences

Advanced
January  2019, 39(1): 503-519. doi: 10.3934/dcds.2019021

Non-local sublinear problems: Existence, comparison, and radial symmetry

Antonio Greco , and  Vincenzino Mascia

Department of Mathematics and Informatics, via Ospedale 72, 09124 Cagliari, Italy

* Corresponding author

Received  April 2018 Revised  July 2018 Published  October 2018

We establish a symmetry result for a non-autonomous overdetermined problem associated to a sublinear fractional equation. To this purpose we prove, in particular, that the solution of the corresponding Dirichlet problem is monotonically increasing with respect to the domain. We also obtain a strong minimum principle and a boundary-point lemma for linear fractional equations that may have an independent interest.

Keywords: Overdetermined problem, radial symmetry, fractional Laplacian, comparison principle, boundary-point lemma.
Mathematics Subject Classification: Primary: 35N25; Secondary: 35B51, 35S15.
Citation: Antonio Greco, Vincenzino Mascia. Non-local sublinear problems: Existence, comparison, and radial symmetry. Discrete & Continuous Dynamical Systems - A, 2019, 39 (1) : 503-519. doi: 10.3934/dcds.2019021
References:
[1]

H. Brézis and L. Oswald, Remarks on sublinear elliptic equations, Nonlinear Anal., 10 (1986), 55-64.  doi: 10.1016/0362-546X(86)90011-8.  Google Scholar

[2]

X. Cabré and Y. Sire, Nonlinear equations for fractional Laplacians I: Regularity, maximum principles, and Hamiltonian estimates, Ann. Inst. H. Poincaré Anal. Non Linéaire, 31 (2014), 23-53.  doi: 10.1016/j.anihpc.2013.02.001.  Google Scholar

[3]

L. Cesari, Optimization - Theory and Applications, Springer-Verlag, 1983. doi: 10.1007/978-1-4613-8165-5.  Google Scholar

[4]

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.  Google Scholar

[5]

J. I. Díaz and J. E. Saá, Existence et unicité de solutions positives pour certaines équations elliptiques quasilinéaires, C. R. Acad. Sci. Paris Sér. I Math., 305 (1987), 521-524.   Google Scholar

[6]

A. Erdélyi (Editor), Higher Transcendental Functions, McGraw-Hill, 1953. Google Scholar

[7]

M. M. Fall and S. Jarohs, Overdetermined problems with fractional Laplacian, ESAIM Control Optim. Calc. Var., 21 (2015), 924-938.  doi: 10.1051/cocv/2014048.  Google Scholar

[8]

A. Fiscella, R. Servadei and E. Valdinoci, Density properties for fractional Sobolev spaces, Annales Academiae Scientiarum Fennicae Mathematica, 40 (2015), 235-253.  doi: 10.5186/aasfm.2015.4009.  Google Scholar

[9]

R. K. Getoor, First passage times for symmetric stable processes in space, Trans. Amer. Math. Soc., 101 (1961), 75-90.  doi: 10.1090/S0002-9947-1961-0137148-5.  Google Scholar

[10]

A. Greco, A characterization of the ellipsoid through the torsion problem, J. Appl. Math. Phys. (ZAMP), 59 (2008), 753-765.  doi: 10.1007/s00033-007-7040-8.  Google Scholar

[11]

A. Greco, Boundary point lemmas and overdetermined problems, J. Math. Anal. Appl., 278 (2003), 214-224.  doi: 10.1016/S0022-247X(02)00656-X.  Google Scholar

[12]

A. Greco, Comparison principle and constrained radial symmetry for the subdiffusive p-Laplacian, Publ. Mat., 58 (2014), 485-498.  doi: 10.5565/PUBLMAT_58214_24.  Google Scholar

[13]

A. Greco, Constrained radial symmetry for the infinity-Laplacian, Nonlinear Analysis: Real World Applications, 37 (2017), 239-248.  doi: 10.1016/j.nonrwa.2017.02.016.  Google Scholar

[14]

A. Greco, Symmetry around the origin for some overdetermined problems, Adv. Math. Sci. Appl., 13 (2003), 387-399.   Google Scholar

[15]

A. Greco and R. Servadei, Hopf's lemma and constrained radial symmetry for the fractional Laplacian, Math. Res. Lett., 23 (2016), 863-885.  doi: 10.4310/MRL.2016.v23.n3.a14.  Google Scholar

[16]

A. Iannizzotto, S. Mosconi and M. Squassina, Hs versus C0-weighted minimizers, Nonlinear Differ. Equ. Appl., 22 (2015), 477-497.  doi: 10.1007/s00030-014-0292-z.  Google Scholar

[17]

S. Jarohs and T. Weth, On the strong maximum principle for nonlocal operators, preprint, arXiv: 1702.08767. Google Scholar

[18]

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.  Google Scholar

[19]

V. Mascia, Un Problema Sublineare Non Locale (Italian), Thesis. University of Cagliari, 2017. Google Scholar

[20]

G. Molica Bisci and V. D. Rădulescu, Multiplicity results for elliptic fractional equations with subcritical term, Nonlinear Differ. Equ. Appl., 22 (2015), 721-739.  doi: 10.1007/s00030-014-0302-1.  Google Scholar

[21]

X. Ros-Oton and J. Serra, The Dirichlet problem for the fractional Laplacian: Regularity up to the boundary, J. Math. Pures Appl., 101 (2014), 275-302.  doi: 10.1016/j.matpur.2013.06.003.  Google Scholar

[22]

R. Servadei and E. Valdinoci, The Brezis-Nirenberg result for the fractional Laplacian, Trans. Amer. Math. Soc., 367 (2015), 67-102.  doi: 10.1090/S0002-9947-2014-05884-4.  Google Scholar

[23]

E. M. Stein, The characterization of functions arising as potentials, Bull. Amer. Math. Soc., 67 (1961), 102-104.  doi: 10.1090/S0002-9904-1961-10517-X.  Google Scholar

[24]

R. L. Wheeden, On hypersingular integrals and Lebesgue spaces of differentiable functions, Trans. Amer. Math. Soc., 134 (1968), 421-435.  doi: 10.1090/S0002-9947-1968-0232249-1.  Google Scholar

show all references

References:
[1]

H. Brézis and L. Oswald, Remarks on sublinear elliptic equations, Nonlinear Anal., 10 (1986), 55-64.  doi: 10.1016/0362-546X(86)90011-8.  Google Scholar

[2]

X. Cabré and Y. Sire, Nonlinear equations for fractional Laplacians I: Regularity, maximum principles, and Hamiltonian estimates, Ann. Inst. H. Poincaré Anal. Non Linéaire, 31 (2014), 23-53.  doi: 10.1016/j.anihpc.2013.02.001.  Google Scholar

[3]

L. Cesari, Optimization - Theory and Applications, Springer-Verlag, 1983. doi: 10.1007/978-1-4613-8165-5.  Google Scholar

[4]

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.  Google Scholar

[5]

J. I. Díaz and J. E. Saá, Existence et unicité de solutions positives pour certaines équations elliptiques quasilinéaires, C. R. Acad. Sci. Paris Sér. I Math., 305 (1987), 521-524.   Google Scholar

[6]

A. Erdélyi (Editor), Higher Transcendental Functions, McGraw-Hill, 1953. Google Scholar

[7]

M. M. Fall and S. Jarohs, Overdetermined problems with fractional Laplacian, ESAIM Control Optim. Calc. Var., 21 (2015), 924-938.  doi: 10.1051/cocv/2014048.  Google Scholar

[8]

A. Fiscella, R. Servadei and E. Valdinoci, Density properties for fractional Sobolev spaces, Annales Academiae Scientiarum Fennicae Mathematica, 40 (2015), 235-253.  doi: 10.5186/aasfm.2015.4009.  Google Scholar

[9]

R. K. Getoor, First passage times for symmetric stable processes in space, Trans. Amer. Math. Soc., 101 (1961), 75-90.  doi: 10.1090/S0002-9947-1961-0137148-5.  Google Scholar

[10]

A. Greco, A characterization of the ellipsoid through the torsion problem, J. Appl. Math. Phys. (ZAMP), 59 (2008), 753-765.  doi: 10.1007/s00033-007-7040-8.  Google Scholar

[11]

A. Greco, Boundary point lemmas and overdetermined problems, J. Math. Anal. Appl., 278 (2003), 214-224.  doi: 10.1016/S0022-247X(02)00656-X.  Google Scholar

[12]

A. Greco, Comparison principle and constrained radial symmetry for the subdiffusive p-Laplacian, Publ. Mat., 58 (2014), 485-498.  doi: 10.5565/PUBLMAT_58214_24.  Google Scholar

[13]

A. Greco, Constrained radial symmetry for the infinity-Laplacian, Nonlinear Analysis: Real World Applications, 37 (2017), 239-248.  doi: 10.1016/j.nonrwa.2017.02.016.  Google Scholar

[14]

A. Greco, Symmetry around the origin for some overdetermined problems, Adv. Math. Sci. Appl., 13 (2003), 387-399.   Google Scholar

[15]

A. Greco and R. Servadei, Hopf's lemma and constrained radial symmetry for the fractional Laplacian, Math. Res. Lett., 23 (2016), 863-885.  doi: 10.4310/MRL.2016.v23.n3.a14.  Google Scholar

[16]

A. Iannizzotto, S. Mosconi and M. Squassina, Hs versus C0-weighted minimizers, Nonlinear Differ. Equ. Appl., 22 (2015), 477-497.  doi: 10.1007/s00030-014-0292-z.  Google Scholar

[17]

S. Jarohs and T. Weth, On the strong maximum principle for nonlocal operators, preprint, arXiv: 1702.08767. Google Scholar

[18]

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.  Google Scholar

[19]

V. Mascia, Un Problema Sublineare Non Locale (Italian), Thesis. University of Cagliari, 2017. Google Scholar

[20]

G. Molica Bisci and V. D. Rădulescu, Multiplicity results for elliptic fractional equations with subcritical term, Nonlinear Differ. Equ. Appl., 22 (2015), 721-739.  doi: 10.1007/s00030-014-0302-1.  Google Scholar

[21]

X. Ros-Oton and J. Serra, The Dirichlet problem for the fractional Laplacian: Regularity up to the boundary, J. Math. Pures Appl., 101 (2014), 275-302.  doi: 10.1016/j.matpur.2013.06.003.  Google Scholar

[22]

R. Servadei and E. Valdinoci, The Brezis-Nirenberg result for the fractional Laplacian, Trans. Amer. Math. Soc., 367 (2015), 67-102.  doi: 10.1090/S0002-9947-2014-05884-4.  Google Scholar

[23]

E. M. Stein, The characterization of functions arising as potentials, Bull. Amer. Math. Soc., 67 (1961), 102-104.  doi: 10.1090/S0002-9904-1961-10517-X.  Google Scholar

[24]

R. L. Wheeden, On hypersingular integrals and Lebesgue spaces of differentiable functions, Trans. Amer. Math. Soc., 134 (1968), 421-435.  doi: 10.1090/S0002-9947-1968-0232249-1.  Google Scholar

[1]

Lingyu Jin, Yan Li. A Hopf's lemma and the boundary regularity for the fractional p-Laplacian. Discrete & Continuous Dynamical Systems - A, 2019, 39 (3) : 1477-1495. doi: 10.3934/dcds.2019063
[2]

Francesca Colasuonno, Fausto Ferrari. The Soap Bubble Theorem and a $ p $-Laplacian overdetermined problem. Communications on Pure & Applied Analysis, 2020, 19 (2) : 983-1000. doi: 10.3934/cpaa.2020045
[3]

Orlando Lopes. Uniqueness and radial symmetry of minimizers for a nonlocal variational problem. Communications on Pure & Applied Analysis, 2019, 18 (5) : 2265-2282. doi: 10.3934/cpaa.2019102
[4]

Erik Ekström, Johan Tysk. A boundary point lemma for Black-Scholes type operators. Communications on Pure & Applied Analysis, 2006, 5 (3) : 505-514. doi: 10.3934/cpaa.2006.5.505
[5]

Lizhi Zhang. Symmetry of solutions to semilinear equations involving the fractional laplacian. Communications on Pure & Applied Analysis, 2015, 14 (6) : 2393-2409. doi: 10.3934/cpaa.2015.14.2393
[6]

Tingzhi Cheng. Monotonicity and symmetry of solutions to fractional Laplacian equation. Discrete & Continuous Dynamical Systems - A, 2017, 37 (7) : 3587-3599. doi: 10.3934/dcds.2017154
[7]

Zhigang Wu, Hao Xu. Symmetry properties in systems of fractional Laplacian equations. Discrete & Continuous Dynamical Systems - A, 2019, 39 (3) : 1559-1571. doi: 10.3934/dcds.2019068
[8]

Patricio Felmer, César Torres. Radial symmetry of ground states for a regional fractional Nonlinear Schrödinger Equation. Communications on Pure & Applied Analysis, 2014, 13 (6) : 2395-2406. doi: 10.3934/cpaa.2014.13.2395
[9]

Nicolas Forcadel, Mamdouh Zaydan. A comparison principle for Hamilton-Jacobi equation with moving in time boundary. Evolution Equations & Control Theory, 2019, 8 (3) : 543-565. doi: 10.3934/eect.2019026
[10]

Tomás Sanz-Perela. Regularity of radial stable solutions to semilinear elliptic equations for the fractional Laplacian. Communications on Pure & Applied Analysis, 2018, 17 (6) : 2547-2575. doi: 10.3934/cpaa.2018121
[11]

Yu Tian, John R. Graef, Lingju Kong, Min Wang. Existence of solutions to a multi-point boundary value problem for a second order differential system via the dual least action principle. Conference Publications, 2013, 2013 (special) : 759-769. doi: 10.3934/proc.2013.2013.759
[12]

Mikko Kemppainen, Peter Sjögren, José Luis Torrea. Wave extension problem for the fractional Laplacian. Discrete & Continuous Dynamical Systems - A, 2015, 35 (10) : 4905-4929. doi: 10.3934/dcds.2015.35.4905
[13]

Ran Zhuo, Wenxiong Chen, Xuewei Cui, Zixia Yuan. Symmetry and non-existence of solutions for a nonlinear system involving the fractional Laplacian. Discrete & Continuous Dynamical Systems - A, 2016, 36 (2) : 1125-1141. doi: 10.3934/dcds.2016.36.1125
[14]

CÉSAR E. TORRES LEDESMA. Existence and symmetry result for fractional p-Laplacian in $\mathbb{R}^{n}$. Communications on Pure & Applied Analysis, 2017, 16 (1) : 99-114. doi: 10.3934/cpaa.2017004
[15]

Ran Zhuo, Yan Li. Nonexistence and symmetry of solutions for Schrödinger systems involving fractional Laplacian. Discrete & Continuous Dynamical Systems - A, 2019, 39 (3) : 1595-1611. doi: 10.3934/dcds.2019071
[16]

Leyun Wu, Pengcheng Niu. Symmetry and nonexistence of positive solutions to fractional p-Laplacian equations. Discrete & Continuous Dynamical Systems - A, 2019, 39 (3) : 1573-1583. doi: 10.3934/dcds.2019069
[17]

Rossella Bartolo, Anna Maria Candela, Addolorata Salvatore. Infinitely many radial solutions of a non--homogeneous $p$--Laplacian problem. Conference Publications, 2013, 2013 (special) : 51-59. doi: 10.3934/proc.2013.2013.51
[18]

Patricio Cerda, Leonelo Iturriaga, Sebastián Lorca, Pedro Ubilla. Positive radial solutions of a nonlinear boundary value problem. Communications on Pure & Applied Analysis, 2018, 17 (5) : 1765-1783. doi: 10.3934/cpaa.2018084
[19]

E. N. Dancer, Zhitao Zhang. Critical point, anti-maximum principle and semipositone p-laplacian problems. Conference Publications, 2005, 2005 (Special) : 209-215. doi: 10.3934/proc.2005.2005.209
[20]

Virginia Agostiniani, Rolando Magnanini. Symmetries in an overdetermined problem for the Green's function. Discrete & Continuous Dynamical Systems - S, 2011, 4 (4) : 791-800. doi: 10.3934/dcdss.2011.4.791

2018 Impact Factor: 1.143

Tools

Metrics

  • PDF downloads (117)
  • HTML views (87)
  • Cited by (0)

Other articles
by authors

[Back to Top]

Copyright © 2019 American Institute of Mathematical Sciences