Asymptotic behavior of the ground state Solutions for Hénon equation with Robin boundary condition

Haiyang He

doi:10.3934/cpaa.2013.12.2393

Article Contents

2013, Volume 12, Issue 6: 2393-2408. Doi: 10.3934/cpaa.2013.12.2393

This issue Previous Article Multiple solutions for singular N-Laplace equations with a sign changing nonlinearity Next Article Oblique derivative problems for elliptic and parabolic equations

Asymptotic behavior of the ground state Solutions for Hénon equation with Robin boundary condition

Haiyang He^1,

1.
College of Mathematics and Computer Science, Key Laboratory of High Performance Computing, and Stochastic Information Processing(Ministry of Education of China), Hunan Normal University, Changsha, Hunan 410081

Received: March 31, 2012

Revised: November 30, 2012

Published: October 2013

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Abstract

In this paper, we consider the problem \begin{eqnarray} -\Delta u=|x|^\alpha u^{p-1}, x \in \Omega,\\ u>0, x \in \Omega,\\ \frac{\partial u}{\partial \nu }+\beta u=0, x\in \partial \Omega, \end{eqnarray} where $\Omega$ is the unit ball in $R^N$ centered at the origin with $N\geq 3$, $\alpha>0, \beta>\frac{N-2}{2}, p\geq 2$ and $\nu $ is the unit outward vector normal to $\partial \Omega$. We investigate the asymptotic behavior of the ground state solutions $u_p$ of (1) as $p\to \frac{2N}{N-2}$. We show that the ground state solutions $u_p$ has a unique maximum point $x_p\in \bar\Omega$. In addition, the ground state solutions is non-radial provided that $p\to \frac{2N}{N-2}$.

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Mathematics Subject Classification: Primary: 35J20, 35J61.

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