Asymptotic behavior of the ground state Solutions for Hénon equation with Robin boundary condition
Haiyang He
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 $p\to \frac{2N}{N-2}$.
keywords: blow-up Robin boundary condition asymptotic behavior. Hénon equation
Positive solutions for critical elliptic systems in non-contractible domains
Haiyang He Jianfu Yang
In this paper, we consider the existence of solutions for a critical elliptic system. We show that the problem possesses at least a high energy positive solution in non-contractible domains.
keywords: Critical Sobolev exponents Palais-Smale condition high energy solution.

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