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., p p -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.