Monotonicity of the Morse index of radial solutions of the Henon equation in dimension two

We consider the equation -Delta u = vertical bar x vertical bar(alpha)vertical bar u vertical bar(p-1)u, x epsilon B, u = 0on partial derivative B, where B subset of R-2 is the unit ball centered at the origin, alpha >= 0, p > 1, and we prove some results on the Morse index of radial solutions. The contribution of this paper is twofold. Firstly, fixed the number of nodal sets n >= 1 of the solution ua,n, we prove that the Morse index m(u(alpha,n)) is monotone non-decreasing with respect to a. Secondly, we provide a lower bound for the Morse indices m(u(alpha,n)), which shows that m(u(alpha,n)) -> + infinity as alpha -> + infinity. (C) 2019 Elsevier Ltd. All rights reserved.