The existence of infinitely many boundary blow-up solutions to the p-k-Hessian equation

Abstract

Abstract The primary objective of this article is to analyze the existence of infinitely many radial p p - k k -convex solutions to the boundary blow-up p p - k k -Hessian problem σ k ( λ ( D i ( ∣ D u ∣ p − 2 D j u ) ) ) = H ( ∣ x ∣ ) f ( u ) in Ω , u = + ∞ on ∂ Ω . {\sigma }_{k}\left(\lambda \left({D}_{i}\left({| Du| }^{p-2}{D}_{j}u)))=H\left(| x| )f\left(u)\hspace{0.33em}\hspace{0.1em}\text{in}\hspace{0.1em}\hspace{0.33em}\Omega ,\hspace{0.33em}u=+\infty \hspace{0.33em}\hspace{0.1em}\text{on}\hspace{0.1em}\hspace{0.33em}\partial \Omega . Here, k ∈ { 1 , 2 , … , N } k\in \left\{1,2,\ldots ,N\right\} , σ k ( λ ) {\sigma }_{k}\left(\lambda ) is the k k -Hessian operator, and Ω \Omega is a ball in R N ( N ≥ 2 ) {{\mathbb{R}}}^{N}\hspace{0.33em}\left(N\ge 2) . Our methods are mainly based on the sub- and super-solutions method.