Existence and asymptotic behavior of solutions for Hénon type equations

This paper is concerned with ground state solutions for the Hénon type equation $-\Delta u(x)=|y|^{\alpha} u^{p-1}(x)$ in $\Omega$, where $\Omega=B^k(0,1)\times B^{n-k}(0,1)\subset \r^n$ and $x=(y,z) \in \r^k imes \r^{n-k}.$ We study the existence of cylindrically symmetric and non-cylindrically symmetric ground state solutions for the problem. We also investigate asymptotic behavior of the ground state solution when $p$ tends to the critical exponent $2^*=\frac {2n}{n-2}$ if $n\geq 3.$