We consider a class of weighted Emden-Fowler equations \begin{ equation} \tag{ $\mathcal P_{ \alpha}$} \label{ eqab} \left\{ \begin{ array}{ ll} -\Delta u=V_{ \alpha} (x) \, u^p & \text{ in} \,\,B,\\ u>0 & \text{ in} \,\,B,\\ u=0 & \text{ on}\,\,\partial B, \end{ array}\right. \end{ equation} posed on the unit ball $B=B(0,1)\subset \mathbb R^N$, $N \geq1$. We prove that symmetry breaking occurs for the groundstate solutions as the parameter $\alpha \rightarrow \infty.$ The above problem reads as a possibly large perturbation of the classical H\'enon equation. We consider a radial function $V_\alpha$ having a spherical shell of zeroes at $|x|=R \in (0,1].$ For $N \geq 3$, a quantitative condition on $R$ for this phenomenon to occur is given by means of universal constants, such as the best constant for the subcritical Sobolev's embedding $H^1_0(B)\subset L^{ p+1}(B).$ In the case $N=2$ we highlight a similar phenomenon when $R=R(\alpha)$ is a function with a suitable decay. Moreover, combining energy estimates and Liouville type theorems we study some qualitative and quantitative properties of the groundstate solutions to (\ref{ eqab}) as $\alpha \rightarrow \infty.$
Categories: math.AP