On sufficient "local" conditions for existence results to generalized p(⋅)-Laplace equations involving critical growth

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) #FOS: Mathematics #Nonlinear Differential Equations Analysis #Nonlinear Partial Differential Equations

paper · pdf · doi:10.48550/arxiv.2201.12148

openalex publication_date 2022/01/28 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

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.

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