We investigate nodal radial solutions to semilinear problems of typewhere Ω is a bounded radially symmetric domain of R N (N ≥ 2) and f is a real function. We characterize both the Morse index and the degeneracy in terms of a singular one dimensional eigenvalue problem, and describe the symmetries of the eigenfunctions. Next we use this characterization to give a lower bound for the Morse index; in such a way we give an alternative proof of an already known estimate for the autonomous problem and we furnish a new estimate for Hénon type problems with f (|x|, u) = |x| α f (u). Concerning the real Hénon problem, f (|x|, u) = |x| α |u| p−1 u, we prove radial nondegeneracy and show that the radial Morse index is equal to the number of nodal zones.

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In this paper we consider the Hénon problem in the unit disc with Dirichlet boundary conditions. We study the asymptotic profile of least energy and nodal least energy radial solutions and then deduce the exact computation of their Morse index for large values of the exponent p. As a consequence of this computation a multiplicity result for positive and nodal solutions is obtained.

Abstract. We prove a comparison principle for unbounded semicontinuous viscosity sub-and supersolutions of nonlinear degenerate parabolic integro-partial differential equations coming from applications in mathematical finance in which geometric Lévy processes act as the underlying stochastic processes for the assets dynamics. As a consequence of the "geometric form" of these processes, the comparison principle holds without assigning spatial boundary data. We present applications of our result to (i) backward stochastic differential equations and (ii) pricing of European and American derivatives via backward stochastic differential equations. Regarding (i), we extend previous results on backward stochastic differential equations in a Lévy setting and the connection to semilinear integro-partial differential equations.

We compute the Morse index of nodal radial solutions to the Hénon problem −∆u = |x| α |u| p−1 u in B, u = 0 on ∂B, 1