On sufficient “local” conditions for existence results to generalized p(.)-Laplace equations involving critical growth

Abstract

In this article, we study the existence of multiple solutions to a generalized p(⋅)p\left(\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 p(⋅)p\left(\cdot )-sublinear, p(⋅)p\left(\cdot )-superlinear, and sandwich-type cases. Compared to constant exponent problems (e.g., pp-Laplacian and (p,q)\left(p,q)-Laplacian), this characterizes the study of variable exponent problems. We show this by applying variants of the mountain pass theorem for p(⋅)p\left(\cdot )-sublinear and p(⋅)p\left(\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 the role of parameters. Our work is a generalization of several existing works in the literature.

Keywords

• leray-lions-type operators
• critical growth
• concentration-compactness principle
• variational methods
• 35b33
• 35j20
• 35j25
• 35j62
• 46e35