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Least energy solutions of the generalized Hénon equation in reflectionally symmetric or point symmetric domains. (English) Zbl 1248.35062
The paper under review deals with the generalized Hénon equation in reflectionally symmetric or point symmetric domains. The main result establishes that a least energy solution is neither reflectionally symmetric nor even. Moreover, the existence of a positive solution with prescribed exact symmetry is argued. In the first part of this paper a function is constructed that has a lower energy than the symmetric least energy, provided that a symmetric least energy solution satisfies a certain inequality. A central auxiliary result is an a priori \(L^\infty\) estimate of a symmetric least energy solution. The proofs of the main results are done in the final part of this paper.

MSC:
35J20 Variational methods for second-order elliptic equations
35J25 Boundary value problems for second-order elliptic equations
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[1] Ambrosetti, A.; Rabinowitz, P.H., Dual variational methods in critical point theory and applications, J. funct. anal., 14, 349-381, (1973) · Zbl 0273.49063
[2] Badiale, M.; Serra, E., Multiplicity results for the supercritical Hénon equation, Adv. nonlinear stud., 4, 453-467, (2004) · Zbl 1113.35047
[3] Barutello, V.; Secchi, S.; Serra, E., A note on the radial solutions for the supercritical Hénon equation, J. math. anal. appl., 341, 720-728, (2008) · Zbl 1135.35031
[4] Byeon, J.; Wang, Z.-Q., On the Hénon equation: asymptotic profile of ground states, I, Ann. inst. H. Poincaré anal. non linéaire, 23, 803-828, (2006) · Zbl 1114.35071
[5] 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
[6] Chern, J.-L.; Lin, C.-S., The symmetry of least-energy solutions for semilinear elliptic equations, J. differential equations, 187, 240-268, (2003) · Zbl 1247.35038
[7] Cao, D.; Peng, S., The asymptotic behaviour of the ground state solutions for Hénon equation, J. math. anal. appl., 278, 1-17, (2003) · Zbl 1086.35036
[8] Calanchi, M.; Secchi, S.; Terraneo, E., Multiple solutions for a Hénon-like equation on the annulus, J. differential equations, 245, 1507-1525, (2008) · Zbl 1158.35347
[9] Esposito, P.; Pistoia, A.; Wei, J., Concentrating solutions for the Hénon equation in \(\mathbb{R}^2\), J. anal. math., 100, 249-280, (2006) · Zbl 1173.35504
[10] Gilbarg, D.; Trudinger, N.S., Elliptic partial differential equations of second order, (1983), Springer Berlin · Zbl 0691.35001
[11] Hénon, M., Numerical experiments on the stability of spherical stellar systems, Astronom. astrophys., 24, 229-237, (1973)
[12] Hirano, N., Existence of positive solutions for the Hénon equation involving critical Sobolev terms, J. differential equations, 247, 1311-1333, (2009) · Zbl 1176.35083
[13] Kajikiya, R., Non-even least energy solutions of the Emden-Fowler equation, Proc. amer. math. soc., 140, 1353-1362, (2012) · Zbl 1248.34023
[14] Kajikiya, R., Non-radial least energy solutions of the generalized Hénon equation, J. differential equations, 252, 1987-2003, (2012) · Zbl 1233.35080
[15] R. Kajikiya, Non-even positive solutions of the Emden-Fowler equations with sign-changing weights, Proc. Roy. Soc. Edinburgh Sect. A, in press. · Zbl 1302.34042
[16] Moore, R.A.; Nehari, Z., Nonoscillation theorems for a class of nonlinear differential equations, Trans. amer. math. soc., 93, 30-52, (1959) · Zbl 0089.06902
[17] Palais, R.S., The principle of symmetric criticality, Comm. math. phys., 69, 19-30, (1979) · Zbl 0417.58007
[18] Pistoia, A.; Serra, E., Multi-peak solutions for the Hénon equation with slightly subcritical growth, Math. Z., 256, 75-97, (2007) · Zbl 1134.35047
[19] 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
[20] Smets, D.; Willem, M.; Su, J., Non-radial ground states for the Hénon equation, Commun. contemp. math., 4, 467-480, (2002) · Zbl 1160.35415
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