    May  2013, 12(3): 1237-1241. doi: 10.3934/cpaa.2013.12.1237

## On the number of maximum points of least energy solution to a two-dimensional Hénon equation with large exponent

 1 Department of Mathematics, Osaka City University, 3-3-138 Sugimoto, Sumiyoshi-ku, Osaka, 558-8585

Received  December 2011 Revised  May 2012 Published  September 2012

In this note, we prove that least energy solutions of the two-dimensional Hénon equation \begin{eqnarray*} -\Delta u = |x|^{2\alpha} u^p \quad x \in \Omega, \quad u 0 \quad x \in \Omega, \quad u = 0 \quad x \in \partial \Omega, \end{eqnarray*} where $\Omega$ is a smooth bounded domain in $R^2$ with $0 \in \Omega$, $\alpha \ge 0$ is a constant and $p >1$, have only one global maximum point when $\alpha > e-1$ and the nonlinear exponent $p$ is sufficiently large. This answers positively to a recent conjecture by C. Zhao (preprint, 2011).
Citation: Futoshi Takahashi. On the number of maximum points of least energy solution to a two-dimensional Hénon equation with large exponent. Communications on Pure & Applied Analysis, 2013, 12 (3) : 1237-1241. doi: 10.3934/cpaa.2013.12.1237
##### References:
  Adimurthi and M. Grossi, Asymptotic estimates for a two-dimensional problem with polynomial nonlinearity,, Proc. Amer. Math. Soc., 132 (2003), 1013. Google Scholar  C. S. Lin, Uniqueness of least energy solutions to a semilinear elliptic equation in $R^2$,, Manuscripta Math., 84 (1994), 13. Google Scholar  K. El Mehdi and M. Grossi, Asymptotic estimates and qualitative properties of an elliptic problem in dimension two,, Advances in Nonlinear Stud., 4 (2004), 15. Google Scholar  X. Ren and J. Wei, On a two-dimensional elliptic problem with large exponent in nonlinearity,, Trans. Amer. Math. Soc., 343 (1994), 749. Google Scholar  X. Ren and J. Wei, Single-point condensation and least-energy solutions,, Proc. Amer. Math. Soc., 124 (1996), 111. Google Scholar  F. Takahashi, Morse indices and the number of maximum points of some solutions to a two-dimensional elliptic problem,, Archiv der Math., 93 (2009), 191. Google Scholar  F. Takahashi, Blow up points and the Morse indices of solutions to the Liouville equation in two-dimension,, Advances in Nonlinear Studies, 12 (2012), 115. Google Scholar  F. Takahashi, Blow up points and the Morse indices of solutions to the Liouville equation: inhomogeneous case,, submitted., (). Google Scholar  C. Zhao, Some results on two-dimensional Hénon equation with large exponent in nonlinearity,, preprint., (). Google Scholar

show all references

##### References:
  Adimurthi and M. Grossi, Asymptotic estimates for a two-dimensional problem with polynomial nonlinearity,, Proc. Amer. Math. Soc., 132 (2003), 1013. Google Scholar  C. S. Lin, Uniqueness of least energy solutions to a semilinear elliptic equation in $R^2$,, Manuscripta Math., 84 (1994), 13. Google Scholar  K. El Mehdi and M. Grossi, Asymptotic estimates and qualitative properties of an elliptic problem in dimension two,, Advances in Nonlinear Stud., 4 (2004), 15. Google Scholar  X. Ren and J. Wei, On a two-dimensional elliptic problem with large exponent in nonlinearity,, Trans. Amer. Math. Soc., 343 (1994), 749. Google Scholar  X. Ren and J. Wei, Single-point condensation and least-energy solutions,, Proc. Amer. Math. Soc., 124 (1996), 111. Google Scholar  F. Takahashi, Morse indices and the number of maximum points of some solutions to a two-dimensional elliptic problem,, Archiv der Math., 93 (2009), 191. Google Scholar  F. Takahashi, Blow up points and the Morse indices of solutions to the Liouville equation in two-dimension,, Advances in Nonlinear Studies, 12 (2012), 115. Google Scholar  F. Takahashi, Blow up points and the Morse indices of solutions to the Liouville equation: inhomogeneous case,, submitted., (). Google Scholar  C. Zhao, Some results on two-dimensional Hénon equation with large exponent in nonlinearity,, preprint., (). Google Scholar
  Jaeyoung Byeon, Sungwon Cho, Junsang Park. On the location of a peak point of a least energy solution for Hénon equation. Discrete & Continuous Dynamical Systems - A, 2011, 30 (4) : 1055-1081. doi: 10.3934/dcds.2011.30.1055  Chunyi Zhao. Some results on two-dimensional Hénon equation with large exponent in nonlinearity. Communications on Pure & Applied Analysis, 2013, 12 (2) : 803-813. doi: 10.3934/cpaa.2013.12.803  Jaeyoung Byeon, Sangdon Jin. The Hénon equation with a critical exponent under the Neumann boundary condition. Discrete & Continuous Dynamical Systems - A, 2018, 38 (9) : 4353-4390. doi: 10.3934/dcds.2018190  Linfeng Mei, Zongming Guo. Morse indices and symmetry breaking for the Gelfand equation in expanding annuli. Discrete & Continuous Dynamical Systems - B, 2017, 22 (4) : 1509-1523. doi: 10.3934/dcdsb.2017072  Shun Kodama. A concentration phenomenon of the least energy solution to non-autonomous elliptic problems with a totally degenerate potential. Communications on Pure & Applied Analysis, 2017, 16 (2) : 671-698. doi: 10.3934/cpaa.2017033  Björn Sandstede, Arnd Scheel. Relative Morse indices, Fredholm indices, and group velocities. Discrete & Continuous Dynamical Systems - A, 2008, 20 (1) : 139-158. doi: 10.3934/dcds.2008.20.139  Pedro Duarte, Silvius Klein, Manuel Santos. A random cocycle with non Hölder Lyapunov exponent. Discrete & Continuous Dynamical Systems - A, 2019, 39 (8) : 4841-4861. doi: 10.3934/dcds.2019197  Shoichi Hasegawa. A critical exponent of Joseph-Lundgren type for an Hénon equation on the hyperbolic space. Communications on Pure & Applied Analysis, 2017, 16 (4) : 1189-1198. doi: 10.3934/cpaa.2017058  Futoshi Takahashi. Morse indices and the number of blow up points of blowing-up solutions for a Liouville equation with singular data. Conference Publications, 2013, 2013 (special) : 729-736. doi: 10.3934/proc.2013.2013.729  Henri Berestycki, Juncheng Wei. On least energy solutions to a semilinear elliptic equation in a strip. Discrete & Continuous Dynamical Systems - A, 2010, 28 (3) : 1083-1099. doi: 10.3934/dcds.2010.28.1083  Zhaohi Huo, Yueling Jia, Qiaoxin Li. Global well-posedness for the 3D Zakharov-Kuznetsov equation in energy space $H^1$. Discrete & Continuous Dynamical Systems - S, 2016, 9 (6) : 1797-1851. doi: 10.3934/dcdss.2016075  Konstantin Mischaikow, Marian Mrozek, Frank Weilandt. Discretization strategies for computing Conley indices and Morse decompositions of flows. Journal of Computational Dynamics, 2016, 3 (1) : 1-16. doi: 10.3934/jcd.2016001  Luis Hernández-Corbato, Francisco R. Ruiz del Portal. Fixed point indices of planar continuous maps. Discrete & Continuous Dynamical Systems - A, 2015, 35 (7) : 2979-2995. doi: 10.3934/dcds.2015.35.2979  Hirotada Honda. Global-in-time solution and stability of Kuramoto-Sakaguchi equation under non-local Coupling. Networks & Heterogeneous Media, 2017, 12 (1) : 25-57. doi: 10.3934/nhm.2017002  Miaomiao Niu, Zhongwei Tang. Least energy solutions for nonlinear Schrödinger equation involving the fractional Laplacian and critical growth. Discrete & Continuous Dynamical Systems - A, 2017, 37 (7) : 3963-3987. doi: 10.3934/dcds.2017168  Sihem Guerarra. Positive and negative definite submatrices in an Hermitian least rank solution of the matrix equation AXA*=B. Numerical Algebra, Control & Optimization, 2019, 9 (1) : 15-22. doi: 10.3934/naco.2019002  Yavdat Il'yasov. On critical exponent for an elliptic equation with non-Lipschitz nonlinearity. Conference Publications, 2011, 2011 (Special) : 698-706. doi: 10.3934/proc.2011.2011.698  Grzegorz Graff, Piotr Nowak-Przygodzki. Fixed point indices of iterations of $C^1$ maps in $R^3$. Discrete & Continuous Dynamical Systems - A, 2006, 16 (4) : 843-856. doi: 10.3934/dcds.2006.16.843  Gianluca Mola. Recovering a large number of diffusion constants in a parabolic equation from energy measurements. Inverse Problems & Imaging, 2018, 12 (3) : 527-543. doi: 10.3934/ipi.2018023  Guangyu Xu, Jun Zhou. Global existence and blow-up of solutions to a singular Non-Newton polytropic filtration equation with critical and supercritical initial energy. Communications on Pure & Applied Analysis, 2018, 17 (5) : 1805-1820. doi: 10.3934/cpaa.2018086

2018 Impact Factor: 0.925