×

On the Hénon equation: asymptotic profile of ground states. II. (English) Zbl 1114.35070

Summary: This paper is concerned with analyzing the limiting behavior of the least energy solutions for the Hénon equation
\[ -\Delta u=| x|^\alpha u^{p-1},\quad u>0 \text{ in }\Omega, \quad u=0 \text{ on } \partial\Omega \] where \(\Omega\) is a bounded domain in \(\mathbb R^N\) with \(N\geq 3\), \(2<p<2^*\cong {2N\over N-2}\). We study the problems in bounded domains with smooth or non-smooth boundaries, and in particular we examine the effect of the boundary on the limiting profile of the solutions.
Part I, cf. Ann. Inst. Henri Poincaré, Anal. Non Linéaire 23, No. 6, 803–828 (2006; Zbl 1114.35071).

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

Citations:

Zbl 1114.35071
Full Text: DOI

References:

[1] 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
[2] J. Byeon, Z.-Q. Wang, On the Hénon equation: asymptotic profile of ground states, preprint, 2002; J. Byeon, Z.-Q. Wang, On the Hénon equation: asymptotic profile of ground states, preprint, 2002
[3] Caffarelli, L.; Kohn, R.; Nirenberg, L., First order interpolation inequalities with weights, Compositio Math., 53, 259-275 (1984) · Zbl 0563.46024
[4] Caffarelli, L. A.; Gidas, B.; Spruck, J., Asymptotic symmetry and local behavior of semilinear elliptic equations with critical Sobolev growth, Comm. Pure Appl. Math., 42, 271-297 (1989) · Zbl 0702.35085
[5] D. Cao, S. Peng, The asymptotic behavior of the ground state solutions for Hénon equation, J. Math. Anal. Appl., to appear.; D. Cao, S. Peng, The asymptotic behavior of the ground state solutions for Hénon equation, J. Math. Anal. Appl., to appear.
[6] D. Cao, S. Peng, S. Yan, Asymptotic behavior of ground state solutions for the Hénon equation, preprint.; D. Cao, S. Peng, S. Yan, Asymptotic behavior of ground state solutions for the Hénon equation, preprint.
[7] Catrina, F.; Wang, Z.-Q., On the Caffarelli-Kohn-Nirenberg Inequalities, C.R. Acad. Sci. Paris Sér. I Math., 330, 437-442 (2000) · Zbl 0954.35050
[8] Catrina, F.; Wang, Z.-Q., On the Caffarelli-Kohn-Nirenberg Inequalitiessharp constants, existence (and nonexistence) and symmetry of extremal functions, Comm. Pure Appl. Math., 54, 229-258 (2001) · Zbl 1072.35506
[9] F. Catrina, Z.-Q. Wang, Asymptotic uniqueness and exact symmetry of \(k\); F. Catrina, Z.-Q. Wang, Asymptotic uniqueness and exact symmetry of \(k\) · Zbl 1301.35014
[10] Chen, G.; Ni, W.-M.; Zhou, J., Algorithms and visualization for solutions of nonlinear elliptic equations, Internat. J. Bifurc. Chaos, 10, 1565-1612 (2000) · Zbl 1090.65549
[11] Gilbarg, D.; Trudinger, N., Elliptic Partial Differential Equations of Second Order (1983), Springer: Springer Berlin · Zbl 0562.35001
[12] Hénon, M., Numerical experiments on the stability of spherical stellar systems, Astronom. Astrophys., 24, 229-238 (1973)
[13] B. Kawohl, Rearrangements and Convexity of Level Sets in PDE, Lecture Notes in Mathematics, vol. 1150, Springer, Berlin, Heidelberg, NY, 1985.; B. Kawohl, Rearrangements and Convexity of Level Sets in PDE, Lecture Notes in Mathematics, vol. 1150, Springer, Berlin, Heidelberg, NY, 1985. · Zbl 0593.35002
[14] P.L. Lions, The concentration-compactness principle in the calculus of variations, The limit case, Rev. Mat. Ibero Amer. 1 (1985) 145-201, 45-120.; P.L. Lions, The concentration-compactness principle in the calculus of variations, The limit case, Rev. Mat. Ibero Amer. 1 (1985) 145-201, 45-120. · Zbl 0704.49005
[15] Smets, D.; Su, J.; Willem, M., Non-radial ground states for the Hénon equation, Comm. Contemp. Math., 4, 467-480 (2002) · Zbl 1160.35415
[16] 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
[17] Wang, Z.-Q., Existence and symmetry of multi-bump solutions for nonlinear Schrödinger equations, J. Differential Equations, 159, 102-137 (1999) · Zbl 1005.35083
[18] Willem, M., Minimax Theorem (1996), Birkhäuser: Birkhäuser Basel
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.