Existence and asymptotic behavior of solutions for Henon equations in hyperbolic spaces

He, Haiyang ; Wang, Wei

Electronic journal of differential equations, Tome 2013 (2013)

Résumé

In this article, we consider the existence and asymptotic behavior of solutions for the Henon equation

$\displaylines{ -\Delta_{\mathbb{B}^N}u=(d(x))^{\alpha}|u|^{p-2}u, \quad x\in \Omega\cr u=0 \quad x\in \partial \Omega, }$

where $\Delta_{\mathbb{B}^N}$ denotes the Laplace Beltrami operator on the disc model of the Hyperbolic space $\mathbb{B}^N, d(x)=d_{\mathbb{B}^N}(0,x), \Omega \subset \mathbb{B}^N$ is geodesic ball with radius $1, \alpha>0, N\geq 3$. We study the existence of hyperbolic symmetric solutions when $2$. We also investigate asymptotic behavior of the ground state solution when p tends to the critical exponent $2^* =\frac{2N}{N-2}$ with $N\geq 3$.

Détail

Classification : 35J20, 35J60
Keywords: henon equation, hyperbolic space, asymptotic behavior, blow up