Existence and asymptotic behavior of solutions for Hénon type equations

Abstract. 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 \(\Omega=B^k(0,1)\times B^{n-k}(0,1)\subset \mathbb{R}^n\) and \(x=(y,z) \in \mathbb{R}^k \times \mathbb{R}^{n-k}\). 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 \(n\geq 3\).