# American Institute of Mathematical Sciences

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:
 [1] Adimurthi and M. Grossi, Asymptotic estimates for a two-dimensional problem with polynomial nonlinearity, Proc. Amer. Math. Soc., 132 (2003), 1013-1019.  Google Scholar [2] C. S. Lin, Uniqueness of least energy solutions to a semilinear elliptic equation in $R^2$, Manuscripta Math., 84 (1994), 13-19.  Google Scholar [3] K. El Mehdi and M. Grossi, Asymptotic estimates and qualitative properties of an elliptic problem in dimension two, Advances in Nonlinear Stud., 4 (2004), 15-36.  Google Scholar [4] X. Ren and J. Wei, On a two-dimensional elliptic problem with large exponent in nonlinearity, Trans. Amer. Math. Soc., 343 (1994), 749-763.  Google Scholar [5] X. Ren and J. Wei, Single-point condensation and least-energy solutions, Proc. Amer. Math. Soc., 124 (1996), 111-120.  Google Scholar [6] F. Takahashi, Morse indices and the number of maximum points of some solutions to a two-dimensional elliptic problem, Archiv der Math., 93 (2009), 191-197.  Google Scholar [7] F. Takahashi, Blow up points and the Morse indices of solutions to the Liouville equation in two-dimension, Advances in Nonlinear Studies, 12 (2012), 115-122. Google Scholar [8] F. Takahashi, Blow up points and the Morse indices of solutions to the Liouville equation: inhomogeneous case,, submitted., ().   Google Scholar [9] C. Zhao, Some results on two-dimensional Hénon equation with large exponent in nonlinearity,, preprint., ().   Google Scholar

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##### References:
 [1] Adimurthi and M. Grossi, Asymptotic estimates for a two-dimensional problem with polynomial nonlinearity, Proc. Amer. Math. Soc., 132 (2003), 1013-1019.  Google Scholar [2] C. S. Lin, Uniqueness of least energy solutions to a semilinear elliptic equation in $R^2$, Manuscripta Math., 84 (1994), 13-19.  Google Scholar [3] K. El Mehdi and M. Grossi, Asymptotic estimates and qualitative properties of an elliptic problem in dimension two, Advances in Nonlinear Stud., 4 (2004), 15-36.  Google Scholar [4] X. Ren and J. Wei, On a two-dimensional elliptic problem with large exponent in nonlinearity, Trans. Amer. Math. Soc., 343 (1994), 749-763.  Google Scholar [5] X. Ren and J. Wei, Single-point condensation and least-energy solutions, Proc. Amer. Math. Soc., 124 (1996), 111-120.  Google Scholar [6] F. Takahashi, Morse indices and the number of maximum points of some solutions to a two-dimensional elliptic problem, Archiv der Math., 93 (2009), 191-197.  Google Scholar [7] F. Takahashi, Blow up points and the Morse indices of solutions to the Liouville equation in two-dimension, Advances in Nonlinear Studies, 12 (2012), 115-122. Google Scholar [8] F. Takahashi, Blow up points and the Morse indices of solutions to the Liouville equation: inhomogeneous case,, submitted., ().   Google Scholar [9] C. Zhao, Some results on two-dimensional Hénon equation with large exponent in nonlinearity,, preprint., ().   Google Scholar
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