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

Abstract

The primary objective of this article is to analyze the existence of infinitely many radial pp-kk-convex solutions to the boundary blow-up pp-kk-Hessian problem σk(λ(Di(∣Du∣p−2Dju)))=H(∣x∣)f(u)inΩ,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\}, σk(λ){\sigma }_{k}\left(\lambda ) is the kk-Hessian operator, and Ω\Omega is a ball in RN(N≥2){{\mathbb{R}}}^{N}\hspace{0.33em}\left(N\ge 2). Our methods are mainly based on the sub- and super-solutions method.

Keywords

• p-k-hessian equation
• boundary blow up
• sub-supersolution method
• p-k-convex solution
• 34b18
• 34b15
• 34a34