Existence and boundary behaviour of radial solutions for weighted elliptic systems with gradient terms

2022/06/04 by Singh, Gurpreet, Devine, Daniel
#Analysis of PDEs (math.AP) #FOS: Mathematics

paper · doi:10.48550/arxiv.2206.01868

Abstract

We are concerned with the existence and boundary behaviour of positive radial solutions for the system \ \beginaligned Δuamp;=|x|^av^p amp;amp; in Ω,
Δvamp;=|x|^bv^qf(|∇ u|) amp;amp; in Ω, \endaligned . where Ω⊂ \bR^N is either a ball centered at the origin or the whole space \bR^N, a, b, p, q> 0, and f ∈ C¹[0, ∞) is an increasing function such that f(t)> 0 for all t> 0. Firstly, we study the existence of positive radial solutions in case when the system is posed in a ball corresponding to their behaviour at the boundary. Next, we take f(t) = t^s, s> 1, Ω= \bR^N and by the use of dynamical system techniques we are able to describe the behaviour at infinity for such positive radial solutions.

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