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

We consider a class of weighted Emden-Fowler equations {-Delta u = V-alpha(x)u(p) in B, u > 0 in B, (P-alpha) u = 0 on partial derivative B, posed on the unit ball B = B(0, 1) subset of R-N, N >= 1. We prove that symmetry breaking occurs for the groundstate solutions as the parameter alpha -> infinity. The above problem reads as a possibly large perturbation of the classical Henon equation. We consider a radial function V-alpha having a spherical shell of zeroes at vertical bar x vertical bar = R is an element of (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 H-0(1)(B) subset of Lp+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 (P-alpha) as alpha -> infinity.