Radial solutions for Hénon type fully nonlinear equations in annuli and exterior domains

Liliane Maia; Gabrielle Nornberg; Liliane Maia; Gabrielle Nornberg

doi:10.3934/mine.2022055

Mathematics in Engineering

2022, Volume 4, Issue 6: 1-18. doi: 10.3934/mine.2022055

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Radial solutions for Hénon type fully nonlinear equations in annuli and exterior domains

Liliane Maia ^{1
,
,},
Gabrielle Nornberg ²

1.
Departamento de Matemática, Universidade de Brasília, Campus Universitário Darcy Ribeiro, Asa Norte, 70910900 - Brasília, DF, Brazil
2.
Departamento de Matemática, Instituto de Ciências Matemáticas e de Computação, Universidade de São Paulo, Avenida Trabalhador São-carlense, 400, Centro, 13566590 - São Carlos, SP, Brazil

Received: 12 July 2021 Accepted: 23 November 2021 Published: 24 December 2021

In this note we study existence of positive radial solutions in annuli and exterior domains for a class of nonlinear equations driven by Pucci extremal operators subject to a Hénon type weight. Our approach is based on the shooting method applied to the corresponding ODE problem, energy arguments, and the associated flow of an autonomous quadratic dynamical system.
- fully nonlinear equations,
- radial positive solutions,
- dynamical system,
- shooting method
Citation: Liliane Maia, Gabrielle Nornberg. Radial solutions for Hénon type fully nonlinear equations in annuli and exterior domains[J]. Mathematics in Engineering, 2022, 4(6): 1-18. doi: 10.3934/mine.2022055

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Abstract

In this note we study existence of positive radial solutions in annuli and exterior domains for a class of nonlinear equations driven by Pucci extremal operators subject to a Hénon type weight. Our approach is based on the shooting method applied to the corresponding ODE problem, energy arguments, and the associated flow of an autonomous quadratic dynamical system.

References

[1]	C. Bandle, C. Coffman, M. Marcus, Nonlinear elliptic problems in annular domains, J. Differ. Equations, 69 (1987), 322–345. doi: 10.1016/0022-0396(87)90123-9
[2]	J. Busca, M. J. Esteban, A. Quaas, Nonlinear eigenvalues and bifurcation problems for Pucci's operators, Ann. Inst. H. Poincaré Anal. Non Linéaire, 22 (2005), 187–206. doi: 10.1016/j.anihpc.2004.05.004
[3]	F. Da Lio, B. Sirakov, Symmetry results for viscosity solutions of fully nonlinear uniformly elliptic equations, J. Eur. Math. Soc., 9 (2007), 317–330.
[4]	P. Felmer, A. Quaas, M. Tang, On the complex structure of positive solutions to Matukuma-type equations, Ann. Inst. H. Poincaré Anal. Non Linéaire, 26 (2009), 869–887. doi: 10.1016/j.anihpc.2008.03.006
[5]	G. Galise, A. Iacopetti, F. Leoni, Liouville-type results in exterior domains for radial solutions of fully nonlinear equations, J. Differ. Equations, 269 (2020), 5034–5061. doi: 10.1016/j.jde.2020.03.051
[6]	G. Galise, F. Leoni, F. Pacella, Existence results for fully nonlinear equations in radial domains, Commun. Part. Diff. Eq., 42 (2017), 757–779. doi: 10.1080/03605302.2017.1306076
[7]	J. K. Hale, H. Koçak, Dynamics and bifurcations, New York: Springer-Verlag, 1991.
[8]	S.-S. Lin, F.-M. Pai, Existence and multiplicity of positive radial solutions for semilinear elliptic equations in annular domains, SIAM J. Math. Anal., 22 (1991), 1500–1515. doi: 10.1137/0522097
[9]	L. Maia, G. Nornberg, F. Pacella, A dynamical system approach to a class of radial weighted fully nonlinear equations, Commun. Part. Diff. Eq., 46 (2021), 573–610. doi: 10.1080/03605302.2020.1849281
[10]	E. Moreira dos Santos, G. Nornberg, D. Schiera, H. Tavares, Principal spectral curves for Lane-Emden fully nonlinear type systems and applications, 2020, arXiv: 2012.07794.
[11]	W.-M. Ni, R. D. Nussbaum, Uniqueness and nonuniqueness for positive radial solutions of $\Delta u+f(u, r) = 0$, Commun. Pure Appl. Math., 38 (1985), 67–108. doi: 10.1002/cpa.3160380105
[12]	F. Pacella, D. Stolnicki, On a class of fully nonlinear elliptic equations in dimension two, J. Differ. Equations, 298 (2021), 463–479. doi: 10.1016/j.jde.2021.07.004
[13]	A. Quaas, B. Sirakov, Principal eigenvalues and the Dirichlet problem for fully nonlinear elliptic operators, Adv. Math., 218 (2008), 105–135. doi: 10.1016/j.aim.2007.12.002

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