Classification of radial solutions to Hénon type equation on the hyperbolic space
Keywords
Semilinear elliptic equation, decay rate, sign-changing solutionsAbstract
We devote this paper to classifying radial solutions of a weighted semilinear elliptic equation on the hyperbolic space. More precisely, for a weighted Lane-Emden equation on the hyperbolic space, we shall study the sign and asymptotic behavior of the radial solutions. We shall also show the existence of fast-decay sign-changing solutions to the Lane-Emden equation on the hyperbolic space.References
C. Bandle and Y. Kabeya, On the positive, “radial” solutions of a semilinear elliptic equation in HN , Adv. Nonlinear Anal. 1 (2012), no. 1, 1–25.
E. Berchio, A. Ferrero and G. Grillo, Stability and qualitative properties of radial solutions of the Lane–Emden–Fowler equation on Riemannian models, J. Math. Pures Appl. (9) 102(2014), no. 1, 1–35.
M. Bhakta and K. Sandeep, Poincaré–Sobolev equations in the hyperbolic space, Calc. Var. Partial Differential Equations 44 (2012), no. 1–2, 247–269.
M. Bonforte, F. Gazzola, G. Grillo and J. L. Vázquez, Classification of radial solutions to the Emden–Fowler equation on the hyperbolic space, Calc. Var. Partial Differential Equations 46 (2013), no. 1–2, 375–401.
W.-Y. Ding, and W.-M. Ni, On the elliptic equation ∆u+Ku(n+2)/(n−2) = 0 and related topics, Duke Math. J. 52 (1985), no. 2, 485–506.
S. Hasegawa, A critical exponent for Hénon type equation on the hyperbolic space, Nonlinear Anal. 129 (2015), 343–370.
S. Hasegawa, A critical exponent of Joseph–Lundgren type for an Hénon equation on the hyperbolic space, Commun. Pure Appl. Anal. 16 (2017), no. 4, 1189–1198.
S. Hasegawa, Remarks on two critical exponents for Hénon type equation on the hyperbolic space, RIMS Kôkyurôku 2032 (2017), 109–124.
H. He, The existence of solutions for Hénon equation in hyperbolic space, Proc. Japan Acad. Ser. A Math. Sci. 89 (2013), no. 2, 24–28.
Y. Kabeya, A unified approach to Matukuma type equations on the hyperbolic space or on a sphere, Discrete Contin. Dyn. Syst. Dynamical Systems, Differential Equations and Applications. 9th AIMS Conference. Suppl., 2013, 385–391.
N. Kawano, On bounded entire solutions of semilinear elliptic equations, Hiroshima Math. J. 14 (1984), no. 1, 125–158.
N. Kawano, J. Satsuma and S. Yotsutani, Existence of positive entire solutions of an Emden-type elliptic equation, Funkcial. Ekvac. 31 (1988), no. 1, 121–145.
T. Kusano and M. Naito, Oscillation theory of entire solutions of second order superlinear elliptic equations, Funkcial. Ekvac. 30 (1987), no. 2–3, 269–282.
G. Mancini and K. Sandeep, On a semilinear elliptic equation in Hn , Ann. Sc. Norm. Super. Pisa Cl. Sci. (5) 7 (2008), no. 4, 635–671.
M. Naito, A note on bounded positive entire solutions of semilinear elliptic equations, Hiroshima Math. J. 14 (1984), no. 1, 211–214.
W.-M. Ni, On the elliptic equation ∆u + K(x)u(n+2)/(n−2) = 0, its generalizations, and applications in geometry, Indiana Univ. Math. J. 31 (1982), no. 4, 493–529.
W.-M. Ni and S. Yotsutani, Semilinear elliptic equations of Matukuma-type and related topics, Japan J. Appl. Math. 5 (1988), no. 1, 1–32.
F. Punzo, On well-posedness of semilinear parabolic and elliptic problems in the hyperbolic space, J. Differential Equations 251 (2011), no. 7, 1972–1989.
S. Stapelkamp, The Brézis–Nirenberg problem on Hn . Existence and uniqueness of solutions, Elliptic and Parabolic Problems (Rolduc/Gaeta, 2001), World Sci. Publ., River Edge, NJ, (2002), 283–290.
E. Yanagida, Structure of radial solutions to ∆u + K(|x|)|u|p−1 u = 0 in Rn , SIAM J. Math. Anal. 27 (1996), no. 4, 997–1014.
E. Yanagida and S. Yotsutani, Classification of the structure of positive radial solutions to ∆u + K(|x|)up = 0 in Rn , Arch. Rational Mech. Anal. 124 (1993), no. 3, 239–259.
E. Yanagida and S. Yotsutani, Existence of nodal fast-decay solutions to ∆u + K(|x|)|u|p−1 u = 0 in Rn , Nonlinear Anal. 22 (1994), no. 8, 1005–1015.
E. Yanagida and S. Yotsutani, Recent topics on nonlinear partial differential equations: structure of radial solutions for semilinear elliptic equations, Amer. Math. Soc. Transl. Ser. 2 211 (2003), 121–137.
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