2022/01/28 by Ky Ho, Ho, Ky, Inbo Sim +1
Computer Science · Mathematics · #35B33 #35J20 #35J25 #35J62 #46E35 #Advanced Mathematical Modeling in Engineering #Analysis of PDEs (math.AP) #Boundary value problem #Combinatorics #Critical exponent #Exponent #FOS: Mathematics #Generalization #Geometry #Laplace operator #Laplace transform #Mathematical analysis #Mathematical physics #Mathematics #Minimax #Mountain pass theorem #Nonlinear Differential Equations Analysis #Nonlinear Partial Differential Equations #Nonlinear system #Operator (biology) #Physics #Pure mathematics #Quantum mechanics #Sublinear function #Type (biology) #math.AP #msc:35B33 #msc:35J20 #msc:35J25 #msc:35J62 #msc:46E35 #p-Laplacian
paper · pdf · doi:10.48550/arxiv.2201.12148
published in arXiv (Cornell University) (Cornell University) · 30 pages
arxiv created 2022/01/28 · openalex publication_date 2022/01/28 · arxiv updated 2022/01/31 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
In this paper, we study the existence of multiple solutions to a generalized p(⋅)-Laplace equation with two parameters involving critical growth. More precisely, we give sufficient "local" conditions, which mean that growths between the main operator and nonlinear term are locally assumed for the cases p(⋅)-sublinear, p(⋅)-superlinear, and sandwich-type. Compared to constant exponent problems (for examples, p-Laplacian and (p,q)-Laplacian), this characterizes the study of variable exponent problems. We show this by applying variants of the Mountain Pass Theorem for p(⋅)-sublinear and p(⋅)-superlinear cases and constructing critical values defined by a minimax argument in the genus theory for sandwich-type case. Moreover, we also obtain a nontrivial nonnegative solution for sandwich-type case changing a role of parameters. Our work is a generalization of several existing works in the literature.