Multiple positive solutions for a double phase system with singular nonlinearity

TL;DR

This work addresses multiplicity in a double phase system with singular nonlinearities under Dirichlet conditions, formulating the problem variationally in Musielak–Orlicz spaces. By dissecting the fibering map on the Nehari manifold and leveraging a careful decomposition into $\mathcal{N}_{\lambda}^{+}$ and $\mathcal{N}_{\lambda}^{-}$, the authors establish two distinct positive weak solutions for small $\lambda$, one with negative energy and one with positive energy, and observe a saddle-point structure. The analysis first handles the case $m=2$ and then extends to general $m$ using a multilinear bound on the mixed term, thereby broadening the scope of double phase systems with singular terms. These results advance the variational treatment of double phase problems in Musielak–Orlicz spaces and provide a framework for future extensions to more complex systems and nonlinearities.

Abstract

In this paper, we study a class of double phase systems which contain the singular and mixed nonlinear terms. Unlike the single equation, the mixed nonlinear terms make the problem more complicate. The geometry of the fibering mapping has multiple possibilities. To overcome the difficulties posed by the mixed nonlinear terms, we need to repeatedly construct concave functions, discuss different cases, and use the properties of concave functions and basic inequalities such as Holder inequality, Poincares inequality and Youngs inequality. By the use of the Nehari manifold, the existence and multiplicity of positive solutions which have nonnegative energy are obtained. It is worth mentioning that we note the existence of saddle point solution(a station point that is not a local minimum), see Remark 3.1.

Theorems & Definitions (36)

• Lemma 2.1
• Lemma 2.2
• Lemma 2.3
• Lemma 2.4
• Lemma 2.5
• Lemma 3.1
• proof
• Lemma 3.2
• proof
• Lemma 3.3