Fibered nonlinearities for $p(x)$-Laplace equations

TL;DR

This paper investigates a class of $p(x)$-Laplace equations in higher dimensions, establishing geometric inequalities and symmetry properties for solutions, advancing understanding of nonlinear PDEs with variable exponents.

Contribution

It introduces new geometric inequalities and symmetry results for solutions to $p(x)$-Laplace equations, extending previous work to variable exponent settings.

Findings

Proved a geometric inequality for solutions.

Established symmetry properties of solutions.

Extended analysis to variable exponent PDEs.

Abstract

In $\R^m\times\R^{n-m}$, endowed with coordinates $X=(x,y)$, we consider the PDE $$ -{\rm div} \big(\alpha(\x) |\nabla u(\X)|^{p(x)-2}\nabla u(\X)\big)=f(x,u(\X)).$$ We prove a geometric inequality and a symmetry result.