Ricerc@Sapienza

We study the asymptotic behavior of positive solutions of fully nonlinear elliptic equations in a ball, as the exponent of the power nonlinearity approaches a critical value. We show that solutions concentrate and blow up at the center of the ball, while a suitable associated energy remains invariant.

Abstract. In this paper we study existence, nonexistence and classification of radial positive
solutions of some weighted fully nonlinear equations involving Pucci extremal operators. Our
results are entirely based on the analysis of the dynamics induced by an autonomous quadratic
system which is obtained after a suitable transformation. This method allows to treat both regular
and singular solutions in a unified way, without using energy arguments. In particular we recover

We study radial sign-changing solutions of a class of fully nonlinear elliptic Dirichlet problems in a ball, driven by the extremal Pucci's operators and with a power nonlinear term. We first determine a new critical exponent related to the existence or nonexistence of such solutions. Then we analyze the asymptotic behavior of the radial nodal solutions as the exponents approach the critical values, showing that new concentration phenomena occur. Finally we define a suitable weighted energy for these solutions and compute its limit value.

We give necessary and sufficient conditions for the existence of positive radial solutions for a class of fully nonlinear uniformly elliptic equations posed in the complement of a ball in RN, and equipped with homogeneous Dirichlet boundary conditions.

In this paper we consider sign-changing radial solutions u ε
to the problem

We consider the semilinear Lane–Emden problem [Equation not available: see fulltext.]where B is the unit ball of RN, N? 2 , centered at the origin and 1 < p< pS, with pS= + ? if N= 2 and pS=N+2N-2 if N? 3. Our main result is to prove that in dimension N= 2 the Morse index of the least energy sign-changing radial solution up of (Ep) is exactly 12 if p is sufficiently large. As an intermediate step we compute explicitly the first eigenvalue of a limit weighted problem in RN in any dimension N? 2. © 2016, Springer-Verlag Berlin Heidelberg.