×

Concentrating solutions for the Hénon equation in \(\mathbb R^{2}\). (English) Zbl 1173.35504

Summary: We consider the boundary value problem \(\Delta u+|x|^{2\alpha} u^p =0\), \(\alpha >0\), in the unit ball \(B\) with homogeneous Dirichlet boundary condition and\(p\) a large exponent. We find a condition which ensures the existence of a positive solution \(u_p\) concentrating outside the origin at \(k\) symmetric points as \(p\) goes to \(+\infty \). The same techniques lead also to a more general result on general domains. In particular, we find that concentration at the origin is always possible, provided \(\alpha \notin\mathbb{N}\).

MSC:

35J70 Degenerate elliptic equations
35B25 Singular perturbations in context of PDEs
35B65 Smoothness and regularity of solutions to PDEs
35H20 Subelliptic equations
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Adimurthi and M. Grossi,Asymptotic estimates for a two-dimensional problem with polynomial nonlinearity, Proc. Amer. Math. Soc.132 (2004), 1013–1019. · Zbl 1083.35035
[2] S. Baraket and F. Pacard,Construction of singular limits for a semilinear elliptic equation in dimension 2, Calc. Var. Partial Differential Equations6 (1998), 1–38. · Zbl 0890.35047
[3] J. Byeon and Z.-Q. Wang,On the Hénon equation: asymptotic profile of ground states. preprint (2002).
[4] J. Byeon and Z.-Q. Wang,On the Hénon equation: asymptotic profile of ground states, II., J. Differential Equations216 (2005), 78–108. · Zbl 1114.35070
[5] D. Cao and S. Peng,The asymptotic behavior of the ground state solutions for Hénon equation, J. Math. Anal. Appl.,278 (2003), 1–17. · Zbl 1086.35036
[6] D. Chae and O. Imanuvilov,The existence of non-topological multivortex solutions in the relativistic self-dual Chern-Simons theory, Comm. Math. Phys.215 (2000), 119–142. · Zbl 1002.58015
[7] W. Chen and C. Li,Classification of solutions of some nonlinear elliptic equations, Duke Math. J.63 (1991), 615–623. · Zbl 0768.35025
[8] G. Chen, W.-M. Ni and J. Zhou,Algorithms and visualization for solutions of nonlinear elliptic equations, Internat. J. Bifur. Chaos Appl. Sci. Engrg.10 (2000), 1565–1612. · Zbl 1090.65549
[9] M. del Pino, M. Kowalczyk and M. Musso,Singular limits in Liouville-type equations, Calc. Var. Partial Differential Equations24 (2005), 47–81. · Zbl 1088.35067
[10] K. El Mehdi and M. Grossi,Asymptotic estimates and qualitative properties of an elliptic problem in dimension two, Adv. Nonlinear Stud.4 (2004), 15–36. · Zbl 1065.35112
[11] P. Esposito,Blow up solutions for a Liouville equation with singular data, SIAM J. Math. Anal.36 (2005), 1310–1345. · Zbl 1162.35350
[12] P. Esposito,Blow up solutions for a Liouville equation with singular data, inRecent Advances in Elliptic and Parabolic Problems, eds. Chiun-Chuan Chen, Michel Chipot and Chang-Shou Lin, World Sci. Publ., Hackensack, NJ, 2005, pp. 61–79. · Zbl 1147.35313
[13] P. Esposito, M. Grossi and A. Pistoia,On the existence of blowing-up solutions for a mean field equation, Ann. Inst. H. Poincaré Anal. Non Linéaire22 (2005), 227–257. · Zbl 1129.35376
[14] P. Esposito, M. Musso and A. Pistoia,Concentrating solutions for a planar elliptic problem involving nonlinearities with large exponent, J. Differential Equation227 (2006), 29–68. · Zbl 1254.35083
[15] P. Esposito, M. Musso and A. Pistoia,On the existence and profile of nodal solutions for a two-dimensional elliptic problem with large exponent in nonlinearity, Proc. London Math. Soc., to appear. · Zbl 1387.35219
[16] M. Hénon,Numerical experiments on the stability of spherical stellar systems, Astronomy and Astrophysics24 (1973), 229–238.
[17] L. Ma and J. Wei,Convergence for a Liouville equation, Comment. Math. Helv.76 (2001), 506–514. · Zbl 0987.35056
[18] W.-M. Ni,A nonlinear Dirichlet problem on the unit ball and its applications, Indiana Univ. Math. J.6 (1982), 801–807. · Zbl 0515.35033
[19] S. Peng,Multiple boundary concentrating solutions to Dirichlet problem of Hénon equation, Acta Math. Appl. Sin. Engle. Ser.22 (2006), 137–162. · Zbl 1153.35325
[20] A. Pistoia and E. Serra,Multi-peak solutions for the Hénon equation with slightly subcritical growth, Math. Z., to appear. · Zbl 1134.35047
[21] J. Prajapat and G. Tarantello,On a class of elliptic problem in \(\mathbb{R}\) N :symmetry and uniqueness results, Proc. Roy. Soc. Edinburgh Sect. A131 (2001), 967–985. · Zbl 1009.35018
[22] X. Ren and J. Wei,On a two-dimensional elliptic problem with large exponent in nonlinearity, Trans. Amer. Math. Soc.343 (1994), 749–763. · Zbl 0804.35042
[23] X. Ren and J. Wei,Single point condensation and least energy solutions, Proc. Amer. Math. Soc.124 (1996), 111–120. · Zbl 0845.35012
[24] D. Smets, J. Su and M. Willem,Non-radial ground states for the Hénon equation, Commun. Contemp. Math.4 (2002), 467–480. · Zbl 1160.35415
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.