{"title":"The existence of infinitely many boundary blow-up solutions to the p-k-Hessian equation","authors":"M. Feng, Xuemei Zhang","doi":"10.1515/ans-2022-0074","DOIUrl":null,"url":null,"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.","PeriodicalId":7191,"journal":{"name":"Advanced Nonlinear Studies","volume":" ","pages":""},"PeriodicalIF":2.1000,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"1","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Advanced Nonlinear Studies","FirstCategoryId":"100","ListUrlMain":"https://doi.org/10.1515/ans-2022-0074","RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q1","JCRName":"MATHEMATICS","Score":null,"Total":0}
引用次数: 1
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.
期刊介绍:
Advanced Nonlinear Studies is aimed at publishing papers on nonlinear problems, particulalry those involving Differential Equations, Dynamical Systems, and related areas. It will also publish novel and interesting applications of these areas to problems in engineering and the sciences. Papers submitted to this journal must contain original, timely, and significant results. Articles will generally, but not always, be published in the order when the final copies were received.