Opuscula Math. 31, no. 3 (2011), 411-424
http://dx.doi.org/10.7494/OpMath.2011.31.3.411

Opuscula Mathematica

# Existence and asymptotic behavior of solutions for Hénon type equations

Wei Long
Jianfu Yang

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 $$n\geq 3$$.

Keywords: Hénon equation, cylindrical symmetry, non-cylindrical symmetry, asymptotic behavior.

Mathematics Subject Classification: 35J50, 35J55, 35J60.

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• Wei Long
• Jiangxi Normal University, Department of Mathematics, Nanchang, Jiangxi 330022, People’s Republic of China
• Jianfu Yang
• Jiangxi Normal University, Department of Mathematics, Nanchang, Jiangxi 330022, People’s Republic of China