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

Abstract

We study existence and uniqueness of spherically symmetric solutions of S_k(D^2v)+beta xi\cdot\nabla v+\alpha v+\abs{v}^{q-1}v=0 in R^n, where \alpha,\beta are 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 \alpha and \beta. 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^{-\alpha}v(xt^{-\beta}) with suitable \alpha and \beta.