Existence of solutions for a singular double phase in Sobolev-Orlicz spaces with variable exponents in a complete manifold

TL;DR

This paper proves the existence of multiple positive solutions for a class of singular double phase problems with variable exponents on complete manifolds, extending previous results from constant exponent spaces.

Contribution

It introduces a novel approach using Nehari manifold and fibering maps to handle singular double phase problems in Sobolev-Orlicz spaces with variable exponents on manifolds.

Findings

Existence of at least two positive solutions for small parameter values.

Extension of previous constant exponent results to variable exponent Sobolev-Orlicz spaces.

Application of Nehari manifold method to singular double phase problems.

Abstract

The purpose of this paper is to study a class of double phase problems, with a singular term and a superlinear parametric term on the right-hand side. Using the method of Nehari manifold combined with the fibering maps, we prove that for all small values of the parameter {\lambda} > 0, there exist at least two non-trivial positive solutions. Our results extend the previous works Papageorgiou, Repov\u{s}, and Vetro [24] and Liu, Dai, Papageorgiou, and Winkert [21], from the case of Musielak-Orlicz Sobolev space, when exponents p and q are constant, to the case of Sobolev-Orlicz spaces with variable exponents in a complete manifold.