On sufficient “local” conditions for existence results to generalized <i>p</i>(.)-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, give sufficient “local” conditions, which mean that growths between main operator and nonlinear term are locally assumed for -sublinear, -superlinear, sandwich-type cases. Compared constant exponent problems (e.g., p -Laplacian , q \left(p,q) -Laplacian), characterizes variable problems. We show by applying variants mountain pass theorem -sublinear -superlinear cases constructing values defined minimax argument in genus theory case. Moreover, also obtain nontrivial nonnegative solution case changing role parameters. Our work is generalization several existing works literature.