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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, 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

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