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Symmetry via the moving plane method for a class of quasilinear elliptic problems involving the Hardy potential

Part of: Partial differential equations

Published online by Cambridge University Press:  09 December 2022

Giusy Chirillo ,
Luigi Montoro ,
Luigi Muglia  and
Berardino Sciunzi
Giusy Chirillo
Affiliation:
Dipartimento di Matematica e Informatica, Università della Calabria, Via P. Bucci, 87036 Arcavacata di Rende (CS), Italy chirillo@mat.unical.it, montoro@mat.unical.it, muglia@mat.unical.it, sciunzi@mat.unical.it
Luigi Montoro
Affiliation:
Dipartimento di Matematica e Informatica, Università della Calabria, Via P. Bucci, 87036 Arcavacata di Rende (CS), Italy chirillo@mat.unical.it, montoro@mat.unical.it, muglia@mat.unical.it, sciunzi@mat.unical.it
Luigi Muglia
Affiliation:
Dipartimento di Matematica e Informatica, Università della Calabria, Via P. Bucci, 87036 Arcavacata di Rende (CS), Italy chirillo@mat.unical.it, montoro@mat.unical.it, muglia@mat.unical.it, sciunzi@mat.unical.it
Berardino Sciunzi
Affiliation:
Dipartimento di Matematica e Informatica, Università della Calabria, Via P. Bucci, 87036 Arcavacata di Rende (CS), Italy chirillo@mat.unical.it, montoro@mat.unical.it, muglia@mat.unical.it, sciunzi@mat.unical.it
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Abstract

We consider positive solutions to a class of quasilinear elliptic problems involving the Hardy potential under zero Dirichlet boundary condition. Via moving plane method, proving a weak comparison principle, we prove symmetry and monotonicity properties for the solutions defined on strictly convex symmetric domains.

Keywords

Quasilinear elliptic equationsHardy potentialSupercritical problemsSimmetry and Monotonicity

MSC classification

Primary: 35B06: Symmetries, invariants, etc.
Secondary: 35B09: Positive solutions 35B51: Comparison principles

Information

Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 153 , Issue 6 , December 2023 , pp. 1858 - 1882
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Copyright
Copyright © The Author(s), 2022. Published by Cambridge University Press on behalf of The Royal Society of Edinburgh

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References

Alexandrov, A. D.. A characteristic property of the spheres. Ann. Mat. Pura Appl. 58 (1962), 303–354.CrossRefGoogle Scholar
Berestycki, H. and Nirenberg, L.. On the method of moving planes and the sliding method. Bull. Brazil. Math. Soc. 22 (1991), 1–37.CrossRefGoogle Scholar
Canino, A., Montoro, L. and Sciunzi, B.. The moving plane method for singular semilinear elliptic problems. Nonlinear Anal. TMA 156 (2017), 61–69.CrossRefGoogle Scholar
Cianchi, A. and Maz'ya, V.. Global Lipschitz regularity for a class of quasilinear elliptic equations. Commun. Partial Differ. Equ. 36 (2011), 100–133.CrossRefGoogle Scholar
Cianchi, A. and Maz'ya, V.. Global boundedness of the gradient for a class of nonlinear elliptic systems. Arch. Ration. Mech. Anal. 212 (2014), 129–177.CrossRefGoogle Scholar
Cianchi, A. and Maz'ya, V.. Second-order two-sided estimates in nonlinear elliptic problems. Arch. Ration. Mech. Anal. 229 (2018), 569–599.CrossRefGoogle Scholar
Chirillo, G., Montoro, L., Muglia, L. and Sciunzi, B.. Existence and regularity for a general class of quasilinear elliptic problems involving the Hardy potential. https://arxiv.org/pdf/2205.05734.pdf.Google Scholar
Colombo, M. and Mingione, G.. Bounded minimisers of double phase variational integrals. Arch. Ration. Mech. Anal. 215 (2015), 443–496.CrossRefGoogle Scholar
Colombo, M. and Mingione, G.. Bounded minimisers of double phase variational integrals. Arch. Ration. Mech. Anal. 218 (2015), 219–273.CrossRefGoogle Scholar
Colombo, M. and Mingione, G.. Calderón-Zygmund estimates and non-uniformly elliptic operators. J. Funct. Anal. 270 (2016), 1416–1478.CrossRefGoogle Scholar
Colombo, M. and Mingione, G.. Regularity for double phase variational problems. Arch. Ration. Mech. Anal. 215 (2015), 443–496.CrossRefGoogle Scholar
Cupini, G., Marcellini, P. and Mascolo, E.. Existence for elliptic equations under $p, q$-growth. Adv. Differ. Equ. 19 (2014), 693–724.Google Scholar
Damascelli, L.. Comparison theorems for some quasilinear degenerate elliptic operators and applications to symmetry and monotonicity results. Ann. Inst. Henri. Poincare Anal. Nonlinear 15 (1998), 493–516.CrossRefGoogle Scholar
Damascelli, L. and Pacella, F.. Monotonicity and symmetry of solutions of $p$-Laplace equations, $1< p<2$, via the moving plane method. Ann. Sc. Norm. Super. Pisa Cl. Sci. 26 (1998), 689–707.Google Scholar
Damascelli, L. and Sciunzi, B.. Regularity, monotonicity and symmetry of positive solutions of $m$-Laplace equations. J. Differ. Equ. 206 (2004), 483–515.CrossRefGoogle Scholar
DiBenedetto, E.. $C^1+\alpha$ local regularity of weak solutions of degenerate elliptic equations. Nonlinear Anal. TMA 7 (1983), 827–850.CrossRefGoogle Scholar
Esposito, F., Montoro, L. and Sciunzi, B.. Monotonicity and symmetry of singular solutions to quasilinear problems. J. Math. Pure Appl. 126 (2019), 214–231.CrossRefGoogle Scholar
Esposito, F., Sciunzi, B. and Trombetta, A.. Regularity and symmetry results for nonlinear degenerate elliptic equations. J. Differ. Equ. 336 (2022), 315–333.CrossRefGoogle Scholar
Esposito, L., Leonetti, F. and Mingione, G.. Sharp regularity for functionals with $(p, q)$ growth. J. Differ. Equ. 204 (2004), 5–55.CrossRefGoogle Scholar
Farina, A., Montoro, L. and Sciunzi, B.. Monotonicity of solutions of quasilinear degenerate elliptic equation in half-spaces. Math. Ann. 357 (2013), 855–893.CrossRefGoogle Scholar
Gidas, B., Ni, W. M. and Nirenberg, L.. Symmetry and related properties via the maximum principle. Commun. Math. Phys. 68 (1979), 209–243.CrossRefGoogle Scholar
Gilbarg, D. and Trudinger, N. S.. Elliptic partial differential equations of second order, 2nd edn (Berlin: Springer, 1983).Google Scholar
Lieberman, G. M.. Boundary regularity for solutions of degenerate elliptic equations. Nonlinear Anal. TMA 12 (1988), 1203–1219.CrossRefGoogle Scholar
Marcellini, P.. Regularity and existence of solutions of elliptic equations with $p, q$-growth conditions. J. Differ. Equ. 90 (1991), 1–30.CrossRefGoogle Scholar
Merchàn, S., Montoro, L., Peral, I. and Sciunzi, B.. Existence and qualitative properties of solutions to a quasilinear elliptic equation involving the Hardy-Leray potential. Ann. Inst. Henri Poincare C 31 (2014), 1–22.Google Scholar
Merchàn, S., Montoro, L. and Sciunzi, B.. On the Harnack inequality for quasilinear elliptic equations with a first-order term. Proc. R. Soc. Edinb. Sect. A: Math. 148 (2018), 1075–1095.CrossRefGoogle Scholar
Pucci, P. and Serrin, J.. The maximum principle. Progress in Nonlinear Differential Equations and their Applications, vol. 73 (Basel: Birkhäuser Verlag, 2007).CrossRefGoogle Scholar
Serrin, J.. A symmetry problem in potential theory. Arch. Ration. Mech. Anal. 43 (1971), 304–318.CrossRefGoogle Scholar
Tolksdorf, P.. Regularity for a more general class of quasilinear elliptic equations. J. Differ. Equ. 51 (1984), 126–150.CrossRefGoogle Scholar
Vazquez, J. L.. A strong maximum principle for some quasilinear elliptic equations. Appl. Math. Optim. 12 (1984), 191–202.CrossRefGoogle Scholar
Zhikov, V. V.. Weighted Sobolev spaces. Sbornik Math. 189 (1998), 1139–1170.CrossRefGoogle Scholar