Existence Results for double phase problem in Sobolev-Orlicz spaces with variable exponents in Complete Manifold

TL;DR

This paper investigates the existence of solutions to double-phase problems within Sobolev-Orlicz spaces with variable exponents on complete Riemannian manifolds, using variational methods and embedding theorems.

Contribution

It introduces new existence results for double-phase problems in Sobolev-Orlicz spaces on manifolds, employing Nehari manifold techniques and establishing key inequalities.

Findings

Existence of non-negative solutions established.

Development of embedding and inequality results in this setting.

Application of variational methods to complex geometric contexts.

Abstract

In this paper, we study the existence of non-negative non-trivial solutions for a class of double-phase problems where the source term is a Caratheodory function that satisfies the Ambrosetti-Rabinowitz type condition in the framework of Sobolev-Orlicz spaces with variable exponents in complete compact Riemannian n-manifolds. Our approach is based on the Nehari manifold and some variational techniques. Furthermore, the H\"older inequality, continuous and compact embedding results are proved.