A dynamical system approach to a class of radial weighted fully nonlinear equations

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 known results on regular solutions for the fully nonlinear non weighted problem by alternative proofs. We also slightly improve the classification of the solutions for the extremal operator M − .

On a class of fully nonlinear elliptic equation in dimension two

“…Analogously it is defined the ω-limit ω(τ ) at +∞. With the same proofs as in [12] we have Lemma 3.4. For every p > 1, any regular solution of (22) satisfying the initial conditions: u p (0) = γ > 0 , u p (0) = 0, corresponds to the unique trajectory Γ p of (25) whose α-limit is the stationary point N 0 .…”

“…Remark 2.5. The uniqueness of the radius y p where u (r) = 0 can be obtained also as in [12] analyzing the flow induced by an associated dynamical system. This method also works in dimension two, and we will use it to study the problem for M − λ,Λ .…”

“…The eigenvalues of D 2 u are the simple eigenvalue u (r) and u (r) r which has multiplicity (N − 1) (see [8]). Thus u satisfies a corresponding ODE from which it is easy to deduce that u is decreasing as long as it is positive, concave in an interval (0, τ 0 ) and changes concavity at least once (see [8] or [12]) Hence ( 20) can be written as:…”

“…Before proving Theorem 1.2 we recall the following definitions. As shown in [12] in terms of the dynamical system (25), a fast decaying, slow decaying or pseudo-slow decaying solution corresponds to an orbit Γ p such that ω(Γ p ) = A 0 , or M 0 or a periodic orbit around M 0 respectively.…”

“…The proof that p * − = Ñ−+2 Ñ−−2 is more delicate and can be carried out as in [12] (see Theorem 5.2 and Theorem 4.7 therein). The fact that p * − ≤ p− comes from Theorem 3.5, therefore using Theorem 3.3 we complete the estimate (7).…”