Topological Methods in Nonlinear Analysis
- Topol. Methods Nonlinear Anal.
- Volume 46, Number 1 (2015), 191-221.
Partially symmetric solutions of the generalized Hénon equation in symmetric domains
Full-text: Access denied (no subscription detected)
We're sorry, but we are unable to provide you with the full text of this article because we are not able to identify you as a subscriber. If you have a personal subscription to this journal, then please login. If you are already logged in, then you may need to update your profile to register your subscription. Read more about accessing full-text
Abstract
We study the generalized Hénon equation in a symmetric domain $\Omega$. Let $H$ and $G$ be closed subgroups of the orthogonal group such that $H \varsubsetneq G$ and $\Omega$ is $G$ invariant. Then we show the existence of a positive solution which is $H$ invariant but $G$ non-invariant under suitable assumptions of $H,G$ and the coefficient function of the equation.
Article information
Source
Topol. Methods Nonlinear Anal., Volume 46, Number 1 (2015), 191-221.
Dates
First available in Project Euclid: 30 March 2016
Permanent link to this document
https://projecteuclid.org/euclid.tmna/1459343891
Digital Object Identifier
doi:10.12775/TMNA.2015.043
Mathematical Reviews number (MathSciNet)
MR3443684
Zentralblatt MATH identifier
06712684
Citation
Kajikiya, Ryuji. Partially symmetric solutions of the generalized Hénon equation in symmetric domains. Topol. Methods Nonlinear Anal. 46 (2015), no. 1, 191--221. doi:10.12775/TMNA.2015.043. https://projecteuclid.org/euclid.tmna/1459343891
References
- N. Ackermann, M. Clapp and F. Pacella, Self-focusing multibump standing waves in expanding waveguides, Milan J. Math. 79 (2011), 221–232. Mathematical Reviews (MathSciNet): MR2831447
Digital Object Identifier: doi:10.1007/s00032-011-0147-6 - D. Arcoya, Positive solutions for semilinear Dirichlet problems in an annulus, J. Differential Equations 94 (1991), 217–227. Mathematical Reviews (MathSciNet): MR1137613
Digital Object Identifier: doi:10.1016/0022-0396(91)90090-V - M.A. Armstrong, Groups and symmetry, Springer, New York, 1988. Mathematical Reviews (MathSciNet): MR965514
- M. Badiale and E. Serra, Multiplicity results for the supercritical Hénon equation, Adv. Nonlinear Stud. 4 (2004), 453–467.
- T. Bartsch, M. Clapp, M. Grossi and F. Pacella, Asymptotically radial solutions in expanding annular domains, Math. Ann. 352 (2012), 485–515. Mathematical Reviews (MathSciNet): MR2874964
Digital Object Identifier: doi:10.1007/s00208-011-0646-3 - T. Bartch, M. Schneider and T. Weth, Multiple solutions of a critical polyharmonic equation, J. Reine Angew. Math. 571 (2004), 131–143. Mathematical Reviews (MathSciNet): MR2070146
- V. Barutello, S. Secchi and E. Serra, A note on the radial solutions for the supercritical Hénon equation, J. Math. Anal. Appl. 341 (2008), 720–728. Mathematical Reviews (MathSciNet): MR2394119
Digital Object Identifier: doi:10.1016/j.jmaa.2007.10.052 - A. Borel, Le plan projectif des octaves et les sphères comme espaces homogènes, C.R. Acad. Sci. Paris 230 (1950), 1378–1380. Mathematical Reviews (MathSciNet): MR34768
- J. Byeon, Existence of many nonequivalent nonradial positive solutions of semilinear elliptic equations on three-dimensional annuli, J. Differential Equations 136 (1997), 136–165.
- J. Byeon, S. Cho and J. Park, On the location of a peak point of a least energy solution for Hénon equation, Discrete Contin. Dyn. Syst. 30 (2011), 1055–1081. Mathematical Reviews (MathSciNet): MR2812953
Digital Object Identifier: doi:10.3934/dcds.2011.30.1055 - J. Byeon and K. Tanaka, Multi-bump positive solutions for a nonlinear elliptic problem in expanding tubular domains, Calc. Var. PartialDifferential Equations 50 (2014). 365–397. Mathematical Reviews (MathSciNet): MR3194687
Digital Object Identifier: doi:10.1007/s00526-013-0639-z - J. Byeon and Z.-Q. Wang, On the Hénon equation: asymptotic profile of ground states, I, Ann. Inst. H. Poincaré Anal. Non Linéaire 23 (2006), 803–828. Mathematical Reviews (MathSciNet): MR2271694
Digital Object Identifier: doi:10.1016/j.anihpc.2006.04.001 - ––––, On the Hénon equation: Asymptotic profile of ground states, II. J. Differential Equations 216 (2005), 78–108. Mathematical Reviews (MathSciNet): MR2158918
Digital Object Identifier: doi:10.1016/j.jde.2005.02.018 - M. Calanchi, S. Secchi and E. Terraneo, Multiple solutions for a Hénon-like equation on the annulus, J. Differential Equations 245 (2008), 1507–1525. Mathematical Reviews (MathSciNet): MR2436451
Digital Object Identifier: doi:10.1016/j.jde.2008.06.018 - D. Cao and S. Peng, The asymptotic behaviour of the ground state solutions for Hénon equation, J. Math. Anal. Appl. 278 (2003), 1–17. Mathematical Reviews (MathSciNet): MR1963460
Digital Object Identifier: doi:10.1016/S0022-247X(02)00292-5 - D. Castorina and F. Pacella, Symmetry of positive solutions of an almost-critical problem in an annulus, Calc. Var. Partial Differential Equations 23 (2005), 125–138. Mathematical Reviews (MathSciNet): MR2138079
Digital Object Identifier: doi:10.1007/s00526-004-0292-7 - F. Catrina and Z.-Q. Wang, Nonlinear elliptic equations on expanding symmetric domains, J. Differential Equations 156 (1999), 153–181.
- J.-L. Chern and C.-S. Lin, The symmetry of least-energy solutions for semilinear elliptic equations, J. Differential Equations 187 (2003), 240–268. Mathematical Reviews (MathSciNet): MR1949440
Digital Object Identifier: doi:10.1016/S0022-0396(02)00080-3 - C.V. Coffman, A nonlinear boundary value problem with many positive solutions, J. Differential Equations 54 (1984), 429–437. Mathematical Reviews (MathSciNet): MR760381
Digital Object Identifier: doi:10.1016/0022-0396(84)90153-0 - H.S.M. Coxeter, Regular Polytopes, third edition, Dover, New York, 1973. Mathematical Reviews (MathSciNet): MR370327
- E.N. Dancer, On the number of positive solutions of some weakly nonlinear equations on annular regions, Math. Z. 206 (1991), 551–562.
- ––––, Global breaking of symmetry of positive solutions on two-dimensional annuli, Differential Integral Equations 5 (1992), 903–913.
- ––––, Some singularly perturbed problems on annuli and a counterexample to a problem of Gidas, Ni and Nirenberg, Bull. London Math. Soc. 29 (1997), 322–326. Mathematical Reviews (MathSciNet): MR1435567
Digital Object Identifier: doi:10.1112/S0024609396002391 - E.N. Dancer and S. Yan, Multibump solutions for an elliptic problem in expanding domains, Comm. Partial Differential Equations 27 (2002), 23–55.
- P. Esposito, A. Pistoia and J. Wei, Concentrating solutions for the Hénon equation in $\mathbb{R}^2$, J. Anal. Math. 100 (2006), 249–280.
- V.V. Gorbatsevich, A.L. Onishchik and E.B. Vinberg, Foundations of Lie Theory and Lie Transformation Groups, Springer, Berlin, 1997. Mathematical Reviews (MathSciNet): MR1631937
- N. Hirano, Existence of positive solutions for the Hénon equation involving critical Sobolev terms, J. Differential Equations 247 (2009), 1311–1333. Mathematical Reviews (MathSciNet): MR2541412
Digital Object Identifier: doi:10.1016/j.jde.2009.06.008 - R. Kajikiya, Orthogonal group invariant solutions of the Emden–Fowler equation, Nonlinear Anal. 44 (2001), 845–896. Mathematical Reviews (MathSciNet): MR1827891
- ––––, Least energy solutions of the generalized Hénon equation in reflectionally symmetric or point symmetric domains, J. Differential Equations 253 (2012), 1621–1646. Mathematical Reviews (MathSciNet): MR2927393
Digital Object Identifier: doi:10.1016/j.jde.2012.05.004 - ––––, Least energy solutions of the Emden–Fowler equation in hollow thin symmetric domains, J. Math. Anal. Appl. 406 (2013), 277–286. Mathematical Reviews (MathSciNet): MR3062420
Digital Object Identifier: doi:10.1016/j.jmaa.2013.04.068 - ––––, Least energy solutions without group invariance for the generalized Hénon equation in symmetric domains, Submitted for publication.
- ––––, Multiple positive solutions of the Emden–Fowler equation in hollow thin symmetric domains, Calc. Var. Partial Differential Equations 52 (2015), 681–704. Mathematical Reviews (MathSciNet): MR3311910
Digital Object Identifier: doi:10.1007/s00526-014-0729-6 - A. Kristály and W. Marzantowicz, Multiplicity of symmetrically distinct sequences of solutions for a quasilinear problem in ${\mathbb R}^N$, NoDEA Nonlinear Differential Equations Appl. 15 (2008), 209–226. Mathematical Reviews (MathSciNet): MR2408352
Digital Object Identifier: doi:10.1007/s00030-007-7015-7 - Y.Y. Li, Existence of many positive solutions of semilinear elliptic equations in annulus, J. Differential Equations 83 (1990), 348–367. Mathematical Reviews (MathSciNet): MR1033192
Digital Object Identifier: doi:10.1016/0022-0396(90)90062-T - S.-S. Lin, Positive radial solutions and non-radial bifurcation for semilinear elliptic equations in annular domains, J. Differential Equations 86 (1990), 367–391. Mathematical Reviews (MathSciNet): MR1064016
Digital Object Identifier: doi:10.1016/0022-0396(90)90035-N - ––––, Existence of positive nonradial solutions for nonlinear elliptic equations in annular domains, Trans. Amer. Math. Soc. 332 (1992), 775–791. Mathematical Reviews (MathSciNet): MR1055571
Digital Object Identifier: doi:10.1090/S0002-9947-1992-1055571-1 - ––––, Existence of many positive nonradial solutions for nonlinear elliptic equations on an annulus, J. Differential Equations 103 (1993), 338–349.
- ––––, Asymptotic behavior of positive solutions to semilinear elliptic equations on expanding annuli, J. Differential Equations 120 (1995), 255–288.
- N. Mizoguchi, Generation of infinitely many solutions of semilinear elliptic equation in two-dimensional annulus, Comm. Partial Differential Equations 21 (1996), 221–227. Mathematical Reviews (MathSciNet): MR1373771
Digital Object Identifier: doi:10.1080/03605309608821181 - N. Mizoguchi and T. Suzuki, Semilinear elliptic equations on annuli in three and higher dimensions, Houston J. Math. 22 (1996), 199–215. Mathematical Reviews (MathSciNet): MR1434392
- D. Montgomery and H. Samelson, Transformation groups of spheres, Ann. of Math. 44 (1943), 454–470.
- D. Montgomery and L. Zippin, Topological Transformation Groups, Robert E. Krieger Publishing, New York, 1974. Mathematical Reviews (MathSciNet): MR379739
- M. Musso, F. Pacard and J. Wei, Finite-energy sign-changing solutions with dihedral symmetry for the stationary nonlinear Schrödinger equation, J. Eur. Math. Soc. 14 (2012), 1923–1953.
- A.L. Onishchik, Topology of Transitive Transformation Groups, Johann Ambrosius Barth, Leipzig, 1994. Mathematical Reviews (MathSciNet): MR1266842
- F. Pacard, Radial and nonradial solutions of $-\Delta u=\lambda f(u)$, on an annulus of $\mathbb{R}^n$, $n\geq 3$, J. Differential Equations 101 (1993), 103–138.
- A. Pistoia and E. Serra, Multi-peak solutions for the Hénon equation with slightly subcritical growth, Math. Z. 256 (2007), 75–97. Mathematical Reviews (MathSciNet): MR2282260
Digital Object Identifier: doi:10.1007/s00209-006-0060-9 - L.S. Pontrjagin, Topological Groups, third edition. translated from the Russian by A. Brown, Gordon and Breach Science Publishers, New York, 1986. Mathematical Reviews (MathSciNet): MR201557
- E. Serra, Non radial positive solutions for the Hénon equation with critical growth, Calc. Var. Partial Differential Equations 23 (2005), 301–326. Mathematical Reviews (MathSciNet): MR2142066
Digital Object Identifier: doi:10.1007/s00526-004-0302-9 - D. Smets, M. Willem and J. Su, Non-radial ground states for the Hénon equation, Commun. Contemp. Math. 4 (2002), 467–480. Mathematical Reviews (MathSciNet): MR1918755
Digital Object Identifier: doi:10.1142/S0219199702000725 - T. Suzuki, Positive solutions for semilinear elliptic equations on expanding annuli: mountain pass approach, Funkc. Ekvac. 39 (1996), 143–164. Mathematical Reviews (MathSciNet): MR1401657
- Z.-Q. Wang and M. Willem, Existence of many positive solutions of semilinear elliptic equations on an annulus, Proc. Amer. Math. Soc. 127 (1999), 1711–1714. Mathematical Reviews (MathSciNet): MR1476398
Digital Object Identifier: doi:10.1090/S0002-9939-99-04708-5
- You have access to this content.
- You have partial access to this content.
- You do not have access to this content.
More like this
- Existence for critical Hénon-type equations
Long, Wei and Yang, Jianfu, Differential and Integral Equations, 2012 - Hénon type equations
with one-sided exponential growth
do Ó, João Marcos, Barboza, Eudes Mendes, and Ribeiro, Bruno, Topological Methods in Nonlinear Analysis, 2017 - Linear perturbations for the critical Hénon problem
Gladiali, Francesca and Grossi, Massimo, Differential and Integral Equations, 2015
- Existence for critical Hénon-type equations
Long, Wei and Yang, Jianfu, Differential and Integral Equations, 2012 - Hénon type equations
with one-sided exponential growth
do Ó, João Marcos, Barboza, Eudes Mendes, and Ribeiro, Bruno, Topological Methods in Nonlinear Analysis, 2017 - Linear perturbations for the critical Hénon problem
Gladiali, Francesca and Grossi, Massimo, Differential and Integral Equations, 2015 - Bifurcation and symmetry breaking for the Henon equation
Amadori, Anna Lisa and Gladiali, Francesca, Advances in Differential Equations, 2014 - High regularity of solutions of compressible Navier-Stokes equations
Cho, Yonggeun, Advances in Differential Equations, 2007 - The existence of solutions for Hénon equation in
hyperbolic space
He, Haiyang, Proceedings of the Japan Academy, Series A, Mathematical Sciences, 2013 - Prescribing Gauss-Kronecker curvature on group invariant convex hypersurfaces
Mikula, Richard, Differential and Integral Equations, 2006 - Multiple positive solutions for the $N$-Laplace equation with nonlinear Neumann
boundary conditions
Giacomoni, J., Prashanth, S., and Sreenadh, K., Differential and Integral Equations, 2010 - Multiple positive symmetric solutions of a singularly perturbed elliptic equation
Clapp, Mónica and Izquierdo, Gustavo, Topological Methods in Nonlinear Analysis, 2001 - Singular boundary value problems for nonlinear elliptic equations in nonsmooth
domains
Fabbri, Jean and Veron, Laurent, Advances in Differential Equations, 1996