Topological Methods in Nonlinear Analysis

Partially symmetric solutions of the generalized Hénon equation in symmetric domains

Ryuji Kajikiya

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


Export citation

References

  • N. Ackermann, M. Clapp and F. Pacella, Self-focusing multibump standing waves in expanding waveguides, Milan J. Math. 79 (2011), 221–232.
  • D. Arcoya, Positive solutions for semilinear Dirichlet problems in an annulus, J. Differential Equations 94 (1991), 217–227.
  • M.A. Armstrong, Groups and symmetry, Springer, New York, 1988.
  • 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.
  • T. Bartch, M. Schneider and T. Weth, Multiple solutions of a critical polyharmonic equation, J. Reine Angew. Math. 571 (2004), 131–143.
  • 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.
  • A. Borel, Le plan projectif des octaves et les sphères comme espaces homogènes, C.R. Acad. Sci. Paris 230 (1950), 1378–1380.
  • 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.
  • 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.
  • 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.
  • ––––, On the Hénon equation: Asymptotic profile of ground states, II. J. Differential Equations 216 (2005), 78–108.
  • M. Calanchi, S. Secchi and E. Terraneo, Multiple solutions for a Hénon-like equation on the annulus, J. Differential Equations 245 (2008), 1507–1525.
  • D. Cao and S. Peng, The asymptotic behaviour of the ground state solutions for Hénon equation, J. Math. Anal. Appl. 278 (2003), 1–17.
  • 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.
  • 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.
  • C.V. Coffman, A nonlinear boundary value problem with many positive solutions, J. Differential Equations 54 (1984), 429–437.
  • H.S.M. Coxeter, Regular Polytopes, third edition, Dover, New York, 1973.
  • 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.
  • 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.
  • N. Hirano, Existence of positive solutions for the Hénon equation involving critical Sobolev terms, J. Differential Equations 247 (2009), 1311–1333.
  • R. Kajikiya, Orthogonal group invariant solutions of the Emden–Fowler equation, Nonlinear Anal. 44 (2001), 845–896.
  • ––––, Least energy solutions of the generalized Hénon equation in reflectionally symmetric or point symmetric domains, J. Differential Equations 253 (2012), 1621–1646.
  • ––––, Least energy solutions of the Emden–Fowler equation in hollow thin symmetric domains, J. Math. Anal. Appl. 406 (2013), 277–286.
  • ––––, 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.
  • 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.
  • Y.Y. Li, Existence of many positive solutions of semilinear elliptic equations in annulus, J. Differential Equations 83 (1990), 348–367.
  • S.-S. Lin, Positive radial solutions and non-radial bifurcation for semilinear elliptic equations in annular domains, J. Differential Equations 86 (1990), 367–391.
  • ––––, Existence of positive nonradial solutions for nonlinear elliptic equations in annular domains, Trans. Amer. Math. Soc. 332 (1992), 775–791.
  • ––––, 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.
  • N. Mizoguchi and T. Suzuki, Semilinear elliptic equations on annuli in three and higher dimensions, Houston J. Math. 22 (1996), 199–215.
  • 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.
  • 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.
  • 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.
  • L.S. Pontrjagin, Topological Groups, third edition. translated from the Russian by A. Brown, Gordon and Breach Science Publishers, New York, 1986.
  • E. Serra, Non radial positive solutions for the Hénon equation with critical growth, Calc. Var. Partial Differential Equations 23 (2005), 301–326.
  • D. Smets, M. Willem and J. Su, Non-radial ground states for the Hénon equation, Commun. Contemp. Math. 4 (2002), 467–480.
  • T. Suzuki, Positive solutions for semilinear elliptic equations on expanding annuli: mountain pass approach, Funkc. Ekvac. 39 (1996), 143–164.
  • 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.