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

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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 - 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, H^s 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, H^s 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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