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

We consider a class of weighted Emden-Fowler equations [EQUATION PRESENTED] posed on the unit ball B = B(0,1)⊂RN, N ≤ 1. We prove that symmetry breaking occurs for the groundstate solutions as the parameter α ← &inf;. The above problem reads as a possibly large perturbation of the classical Hénon equation. We consider a radial function Vα having a spherical shell of zeroes at |x| = R &insin; (0,1). For N ≤ 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 H10(B) ⊂ Lp+1(B). In the case N = 2 we highlight a similar phenomenon when R = R(α) 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 (Pα) as α ← &inf;.