Liouville-type results in exterior domains for radial solutions of fully nonlinear equations

Abstract

In this paper 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 R N , and equipped with homogeneous Dirichlet boundary conditions. 2010 Mathematics Subject Classification. 35J60; 35B33; 34B53.

New concentration phenomena for a class of radial fully nonlinear equations

“…Problem (2.7) has a unique solution u α = u(α, p, r), defined and positive on a maximal interval [1, ρ α ), for some 1 < ρ α ≤ +∞. In [10] it has been proved that there exists [9,11]). Concerning the asymptotic properties with respect to the parameter α, we recall that ρ α → 1 as α → +∞, while ρ α → +∞ as α → 0 (see […”

“…In [11] it has been proved that (2.9) has positive radial solutions if and only if p > p * − (see [11,Theorem 1.1]). More precisely, we have the following (see [11,Sect. 6 and Theorem 6.2]): Theorem 2.2.…”

“…which exists by the results of [11] because p * − < p * * + . Then, setting Σ * * + := E * * (W − ), (1.13) where E * * (W − ) is the (finite) energy in R N \ B of W − (see (10.9)), we have the following.…”

“…If k is odd the previous argument works only for p ≤ p * − (see [11,Theorem 1.2]). For this reason we proceed in a different way.…”

“…The above result will be proved in Section 3. Let us observe that while it is easy to obtain i), using Theorem 3.1 of [11], the proof of ii) is quite involved and requires several steps.…”