Quantitative symmetry breaking of groundstates for a class of weighted Emden-Fowler equations

Abstract

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 $\alpha \rightarrow \infty.$