Bounded solutions of a k -Hessian equation involving a weighted nonlinear source

Abstract

We consider the problemwhere B denotes the unit ball in R n , n > 2k (k ∈ N), λ > 0, q > k and σ ≥ 0. We study the existence, uniqueness and multiplicity of negative bounded radially symmetric solutions of (1). The methodology to obtain our results is based on a dynamical system approach. For this, we introduce a new transformation which reduces problem (1) to an autonomous two dimensional generalized Lotka-Volterra system.

Multiplicity of bounded solutions to the $k$-Hessian equation with a Matukuma-type source

“…This notion of maximal solution was recently introduced in [23] to prove existence results, see also [24]. Now we state our first main result concerning the existence and non-existence of solutions to problem (P λ ).…”

“…the Tso and Joseph-Lundgren type exponents, respectively. The generalized Joseph-Lundgren exponent, q JL (k, σ), was recently obtained in [24] in connection with the multiplicity of radial bounded solutions of a k-Hessian equation involving a weight of the form |x| σ . We point out that, for k = 1 and σ = 0, q JL (1, 0) coincides with the classical Joseph-Lundgren exponent [16].…”

“…[5,6,30,37]. In the framework of the k-Hessian operator, this transformation has been recently introduced in [24]. Further, for…”

“…It is not difficult to see that the (bounded) orbit (x(t), y(t)) of (LV S q,ρ− ) starts at P 3 (n − 2 + µ, 0). See [24].…”

“…We handle the existence and multiplicity of radially symmetric bounded solutions combining dynamical-systems tools with the approach of the intersection number between a regular solution and a singular solution. To this end, we first use a new transformation recently introduced in [24], which reduces the radial version of (1) (denoted (P λ )) to a two-dimensional non-autonomous Lotka-Volterra system (denoted (M S q,µ )). This non-autonomous system can be considered as an asymptotically autonomous system in the sense of Thieme [25].…”