JournalsrmiVol. 37, No. 2pp. 749–773

Admissible solutions to Hessian equations with exponential growth

  • José Francisco de Oliveira

    Universidade Federal do Piauí, Teresina, Brazil
  • Pedro Ubilla

    Universidad de Santiago de Chile, Chile
Admissible solutions to Hessian equations with exponential growth cover

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Abstract

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

{Sk[u]=f(x,u)u<0}\mboxinB,  u=0\mboxonB,\left\{\begin{aligned} &\left.\begin{aligned} &S_k[u]=f(x,-u)\\ &u<0 \\ \end{aligned}\right\}&\mbox{in }& B,\\ &\;u=0&\mbox{on }& \partial B, \end{aligned}\right.

where BB is the unit ball of RN,\mathbb{R}^{N}, N=2kN=2k (kN)(k\in \mathbb{N}), and f ⁣:B×RRf\colon \overline{B}\times\mathbb{R}\rightarrow\mathbb{R} behaves like exp(u(N+2)/N)\exp (u^{(N+2)/N}) when uu\rightarrow\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>2kN > 2k.

Cite this article

José Francisco de Oliveira, Pedro Ubilla, Admissible solutions to Hessian equations with exponential growth. Rev. Mat. Iberoam. 37 (2020), no. 2, pp. 749–773

DOI 10.4171/RMI/1215