A k-Hessian equation with a power nonlinearity source and self-similarity

2025/03/05 by Sánchez, Justino
#34A34 #35A01 #35B07 #35C06 #35J60 #Analysis of PDEs (math.AP) #FOS: Mathematics

paper · doi:10.48550/arxiv.2503.03661

Abstract

We study existence and uniqueness of spherically symmetric solutions of S_k(D²v)+beta xi⋅∇ v+αv+\absv^q-1v=0 in Rⁿ, where α,βare real parameters, n>2, q>k≥ 1 and S_k(D²v) stands for the k-Hessian operator of v. Our results are based mainly on the analysis of an associated dynamical system and energy methods. We derive some properties of the solutions of the above equation for different ranges of the parameters αand β. In particular, we describe with precision its asymptotic behavior at infinity. Further, according to the position of q with respect to the first critical exponent ((n+2)k)/(n) and the Tso critical exponent ((n+2)k)/(n-2k) we study the existence of three classes of solutions: crossing, slow decay or fast decay solutions. In particular, if k>1 all the fast decay solutions have a compact support in Rⁿ. The results also apply to construct self-similar solutions of type I to a related nonlinear evolution equation. These are self-similar functions of the form u(t,x)=t^-αv(xt^-β) with suitable αand β.

Related

• Existence of solutions for a k-Hessian equation and its connection with self-similar solutions
• Existence, multiplicity and classification results for solutions to k-Hessian equations with general weights
• Asymptotic behavior of solutions of a k-Hessian evolution equation
• Radially bounded solutions of a k-Hessian equation involving a weighted nonlinear source
• Bounded solutions of a k-Hessian equation in a ball