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

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.