Three weak solutions for double phase elliptic problem with indefinite weight

Khaled Kefi; Mohammed M. Al-Shomrani; Khaled Kefi; Mohammed M. Al-Shomrani

doi:10.3934/mmc.2026007

Mathematical Modelling and Control

2026, Volume 6, Issue 1: 88-96. doi: 10.3934/mmc.2026007

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Research article

Three weak solutions for double phase elliptic problem with indefinite weight

Khaled Kefi ^{1
,
,},
Mohammed M. Al-Shomrani ²

1.
Center for Scientific Research and Entrepreneurship, Northern Border University, Arar 73213, Saudi Arabia
2.
Department of Mathematics, Faculty of Science, King Abdulaziz University, P.O. Box 80203, Jeddah, 21589, Saudi Arabia

Received: 18 March 2025 Revised: 16 September 2025 Accepted: 23 October 2025 Published: 04 March 2026

This paper investigates the multiplicity of weak solutions for a double-phase elliptic problem with indefinite interaction, where the nonlinearity involves a potential term $ k(x) $ that may change sign and be singular within the domain. The problem is set up as a Dirichlet boundary value problem, where the differential operator has two different phases, involving two exponents $ p $ and $ q $ that meet a certain condition. Employing critical point theory, we demonstrate that at least one solution exists, and under suitable conditions, there are at least three solutions. These results come from applying abstract variational approaches, particularly the critical point theorems by Bonanno and Marano. The manuscript also presents a detailed variational framework and sets up the necessary preliminaries to support the main results.
- double phase,
- variational methods,
- critical point theorem,
- $ p $-Laplacian
Citation: Khaled Kefi, Mohammed M. Al-Shomrani. Three weak solutions for double phase elliptic problem with indefinite weight[J]. Mathematical Modelling and Control, 2026, 6(1): 88-96. doi: 10.3934/mmc.2026007

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Abstract

This paper investigates the multiplicity of weak solutions for a double-phase elliptic problem with indefinite interaction, where the nonlinearity involves a potential term $ k(x) $ that may change sign and be singular within the domain. The problem is set up as a Dirichlet boundary value problem, where the differential operator has two different phases, involving two exponents $ p $ and $ q $ that meet a certain condition. Employing critical point theory, we demonstrate that at least one solution exists, and under suitable conditions, there are at least three solutions. These results come from applying abstract variational approaches, particularly the critical point theorems by Bonanno and Marano. The manuscript also presents a detailed variational framework and sets up the necessary preliminaries to support the main results.

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