Analysis of an Emden-Fowler-H\'enon type equation

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.