Classification of radial solutions for elliptic systems driven by the k-Hessian operator

2020/02/26 by Ghergu, Marius
#35B40 #35J47 #70G60 #Analysis of PDEs (math.AP) #FOS: Mathematics

paper · doi:10.48550/arxiv.2002.12170

Abstract

We are concerned with non-constant positive radial solutions of the system \ \beginaligned S_k(D² u)amp;=|∇ u|^m v^pamp;amp; in Ω,
S_k(D² v)amp;=|∇ u|^q v^s amp;amp; in Ω, \endaligned . where S_k(D²u) is the k-Hessian operator of u∈ C²(Ω) (1≤ k≤ N) and Ω⊂ℝ^N (N≥ 2) is either a ball or the whole space. The exponents satisfy q>0, m,s≥ 0, p≥ s≥ 0 and (k-m)(k-s)≠ pq. In the case where Ω is a ball, we classify all the positive radial solutions according to their behavior at the boundary. Further, we consider the case Ω=ℝ^N and find that the above system admits non-constant positive radial solutions if and only if 0≤ m

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