Nehari manifold approach for a singular multi-phase variable exponent problem

TL;DR

The paper addresses a singular multi-phase Dirichlet problem with variable exponents in a Musielak–Orlicz Sobolev setting, combining a Hardy-type potential and a blow-up term as $u\to0$. The authors employ a Nehari manifold approach applied to the energy functional $\mathcal{J}_{\lambda}$ to obtain two positive weak solutions, one with negative energy and one with positive energy, for sufficiently small $\lambda$. The work extends variational methods to variable-exponent, multi-phase operators and provides a priori control for singular terms, along with a concrete application to groundwater flow in heterogeneous porous media. The results contribute to the theory of nonlinear elliptic problems with singular and spatially heterogeneous growth, offering potential implications for modeling in heterogeneous materials and porous media.

Abstract

This paper is concerned with a singular multi-phase problem with variable singularities. The main tool used is the Nehari manifold approach. Existence of at least two positive solutions with positive-negative energy levels are obtained.

Theorems & Definitions (21)

• Proposition 2.1
• Proposition 2.2
• Proposition 2.3
• Proposition 2.4
• Proposition 2.5
• Proposition 2.6
• Proposition 2.7
• Remark 1
• Lemma 2.8
• Lemma 2.9