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

TL;DR

The paper analyzes radial solutions to the fully nonlinear $k$-Hessian equation with a power-type source, establishing existence, uniqueness, and detailed asymptotics of self-similar type I profiles. By a strategic change of variables and an energy-based dynamical-systems approach, it derives the finite limit $L=\lim_{r\to\infty} r^{\kappa}v(r)$ and classifies solutions into crossing, slow decay, and fast decay regimes, with critical exponents $q_c(k)=\frac{(n+2)k}{n}$ and $q^*(k)=\frac{(n+2)k}{n-2k}$. It proves the existence of nonnegative solutions in various regimes, demonstrates that fast decay implies compact support when $k>1$, and provides precise asymptotic expansions for slow decay solutions, along with explicit profiles in a semilinear or Monge–Ampère-related setting. The results contribute to the theory of self-similar solutions of fully nonlinear PDEs and have implications for related nonlinear evolution equations with $k$-Hessian operators.

Abstract

We study existence and uniqueness of spherically symmetric solutions of S_k(D^2v)+beta xi\cdot\nabla v+αv+\abs{v}^{q-1}v=0 in R^n, where α,βare real parameters, n>2,\, q>k\geq 1 and S_k(D^2v) 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 \frac{(n+2)k}{n} and the Tso critical exponent \frac{(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 R^n. 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^{-α}v(xt^{-β}) with suitable αand β.

Figures (1)

• Figure 1: The figure shows the exponent $q$ as a function of the parameter $\kappa$. Also shows the localization of the regions $E$ and $\sim E$ on the quadrant $(\kappa,q)$, respectively, with or without effect of the nonlinear source on the first two terms in the asymptotic representation of the profiles.

Theorems & Definitions (35)

• Theorem 1.1
• Remark 1.1
• Remark 1.2
• Remark 1.3
• Remark 1.4
• Proposition 2.1
• proof
• Proposition 2.2
• proof
• Proposition 2.3