Analysis of an Emden-Fowler-Hénon type equation

Abstract

This work investigates a Hénon-type equation with the cap-shaped weight $V_{\alpha }(|x|)=(4|x|(1-|x|){)}^{\alpha }$, which concentrates on the sphere of radius $1/2$ as $\alpha \to \infty$. Particular attention is devoted to ground state solutions and the phenomenon of symmetry breaking in the large-$\alpha$ regime. A complete description of the asymptotic behavior of ground state radial solutions is obtained as $\alpha \to \infty$. The analytical results are further supported by numerical simulations of the radial ground states.