Admissible solutions to Hessian equations with exponential growth

Abstract

The aim of this paper is to prove the existence of radially symmetric $k$-admissible solutions for the following Dirichlet problem associated with the $k$-th Hessian operator:

where $B$ is the unit ball of ${\mathbb{R}}^N,$ $N=2k$ $(k\in \mathbb{N})$, and $f:\overline{B}\times \mathbb{R}\to \mathbb{R}$ behaves like $\exp (u^{(N+2)/N})$ when $u\to \infty$ and satisfies the Ambrosetti–Rabinowitz condition. Our results constitute the exponential counterpart of the existence theorems of Tso (1990) for power-type nonlinearities under the condition $N>2k$.