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

TL;DR

This paper establishes local conditions for the existence of multiple solutions to generalized variable exponent p-Laplace equations with critical growth, extending previous constant exponent results using variational methods.

Contribution

It introduces local growth conditions for variable exponent problems and applies variational techniques to prove multiple solutions, generalizing prior constant exponent studies.

Findings

Multiple solutions are proven to exist under local growth conditions.

A nontrivial nonnegative solution is obtained for the sandwich-type case.

The work extends classical results to variable exponent settings.

Abstract

In this paper, we study the existence of multiple solutions to a generalized $p(\cdot)$-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 $p(\cdot)$-sublinear, $p(\cdot)$-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 $p(\cdot)$-sublinear and $p(\cdot)$-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.