Existence and Uniqueness of Weak solution of $ p( x ) $- laplacian in Sobolev spaces with variable exponents in complete manifolds

TL;DR

This paper proves the existence and uniqueness of solutions to variable exponent p(x)-Laplacian equations on Riemannian manifolds using variational methods, specifically the mountain pass theorem.

Contribution

It establishes the existence and uniqueness of weak solutions for p(x)-Laplacian equations in variable exponent Sobolev spaces on complete manifolds, extending previous results to a geometric setting.

Findings

Existence of non-trivial solutions is proven.

Uniqueness of solutions is established.

Application of mountain pass theorem in a geometric context.

Abstract

The paper deals with the existence and uniqueness of a non-trivial solution to non-homogeneous $ p ( x ) -$laplacian equations, managed by non polynomial growth operator in the framework of variable exponent Sobolev spaces on Riemannian manifolds. The mountain pass Theorem is used.