Byeon, Jaeyoung; Wang, Zhi-Qiang On the Hénon equation: asymptotic profile of ground states. I. (English) Zbl 1114.35071 Ann. Inst. Henri Poincaré, Anal. Non Linéaire 23, No. 6, 803-828 (2006). Summary: This paper is concerned with the qualitative property of the ground state solutions for the Hénon equation. By studying a limiting equation on the upper half space \(\mathbb R_+^N\), we investigate the asymptotic energy and the asymptotic profile of the ground states for the Hénon equation. The limiting problem is related to a weighted Sobolev type inequality which we establish in this paper. Cited in 1 ReviewCited in 75 Documents MSC: 35J60 Nonlinear elliptic equations 35J20 Variational methods for second-order elliptic equations 35J25 Boundary value problems for second-order elliptic equations 47J30 Variational methods involving nonlinear operators Keywords:symmetry breaking; minimal energy solutions; asymptotic behaviour × Cite Format Result Cite Review PDF Full Text: DOI Numdam EuDML References: [1] Brezis, H., Symmetry in nonlinear PDE’s, (Differential Equations: La Pietra 1996 (Florence). Differential Equations: La Pietra 1996 (Florence), Proc. Sympos. Pure Math., vol. 65 (1999), Amer. Math. Soc.: Amer. Math. Soc. Providence), 1-12 · Zbl 0927.35038 [2] Byeon, J., Existence of large positive solutions of some nonlinear elliptic equations on singularly perturbed domains, Comm. Partial Differential Equations, 22, 1731-1769 (1997) · Zbl 0883.35040 [3] Byeon, J.; Wang, Z.-Q., On the Hénon equation: asymptotic profile of ground states, II, J. Differential Equations, 216, 78-108 (2005) · Zbl 1114.35070 [4] Cao, D.; Peng, S., The asymptotic behavior of the ground state solutions for Hénon equation, J. Math. Anal. Appl., 278, 1-17 (2003) · Zbl 1086.35036 [5] D. Cao, S. Peng, S. Yan, Asymptotic behavior of ground state solutions for the Hénon equation; D. Cao, S. Peng, S. Yan, Asymptotic behavior of ground state solutions for the Hénon equation [6] Chen, G.; Zhou, J.; Ni, W. M., Algorithms and visualization for solutions of nonlinear elliptic equations, Int. J. Bifurcation and Chaos, 10, 1731-1769 (1997) [7] Gidas, B.; Ni, W. N.; Nirenberg, L., Symmetry and related properties via the maximum principle, Comm. Math. Phys., 68, 209-243 (1979) · Zbl 0425.35020 [8] Gilbarg, D.; Trudinger, N., Elliptic Partial Differential Equations of Second Order (1983), Springer-Verlag · Zbl 0562.35001 [9] Hénon, M., Numerical experiments on the stability of spherical stellar systems, Astronom. Astrophys., 24, 229-238 (1973) [10] Kawohl, B., Rearrangements and Convexity of Level Sets in PDE, Lecture Notes in Math., vol. 1150 (1985), Springer-Verlag: Springer-Verlag Berlin · Zbl 0593.35002 [11] Ni, W. M., A nonlinear Dirichlet problem on the unit ball and its applications, Indiana Univ. Math. J., 31, 801-807 (1982) · Zbl 0515.35033 [12] Protter, H.; Weinberger, H. F., Maximum Principles in Differential Equations (1984), Springer-Verlag: Springer-Verlag New York · Zbl 0549.35002 [13] Serra, E., Non radial positive solutions for the Hénon equation with critical growth, Calc. Var. Partial Differential Equations, 23, 301-326 (2005) · Zbl 1207.35147 [14] Smets, D.; Su, J.; Willem, M., Non-radial ground states for the Hénon equation, Comm. Contemp. Math., 4, 467-480 (2002) · Zbl 1160.35415 [15] Smets, D.; Willem, M., Partial symmetry and asymptotic behavior for some elliptic variational problems, Calc. Var. Partial Differential Equations, 18, 57-75 (2003) · Zbl 1274.35026 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.