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Volume 12
Issue 23
10.3390/math12233668
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Article

Existence, Nonexistence and Multiplicity of Positive Solutions for Generalized Laplacian Problems with a Parameter

by
Jeongmi Jeong
1 and
Chan-Gyun Kim
2,*
1
Department of mathematics, Pusan National University, Busan 46241, Republic of Korea
2
Department of Mathematics Education, Chinju National University of Education, Jinju 52673, Republic of Korea
*
Author to whom correspondence should be addressed.
Mathematics 2024, 12(23), 3668; https://doi.org/10.3390/math12233668
Submission received: 27 October 2024 / Revised: 17 November 2024 / Accepted: 19 November 2024 / Published: 22 November 2024
(This article belongs to the Section Difference and Differential Equations)
Versions Notes

Abstract

We investigate the homogeneous Dirichlet boundary value problem for generalized Laplacian equations with a singular, potentially non-integrable weight. By examining asymptotic behaviors of the nonlinear term near 0 and ∞, we establish the existence, nonexistence, and multiplicity of positive solutions for all positive values of the parameter λ. Our proofs rely on the fixed point theorem concerning cone expansion and compression of norm type and the Leray–Schauder’s fixed point theorem.
Keywords: generalized Laplacian problems; multiplicity of positive solutions; singular weight function generalized Laplacian problems; multiplicity of positive solutions; singular weight function

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MDPI and ACS Style

Jeong, J.; Kim, C.-G. Existence, Nonexistence and Multiplicity of Positive Solutions for Generalized Laplacian Problems with a Parameter. Mathematics 2024, 12, 3668. https://doi.org/10.3390/math12233668

AMA Style

Jeong J, Kim C-G. Existence, Nonexistence and Multiplicity of Positive Solutions for Generalized Laplacian Problems with a Parameter. Mathematics. 2024; 12(23):3668. https://doi.org/10.3390/math12233668

Chicago/Turabian Style

Jeong, Jeongmi, and Chan-Gyun Kim. 2024. "Existence, Nonexistence and Multiplicity of Positive Solutions for Generalized Laplacian Problems with a Parameter" Mathematics 12, no. 23: 3668. https://doi.org/10.3390/math12233668

APA Style

Jeong, J., & Kim, C. -G. (2024). Existence, Nonexistence and Multiplicity of Positive Solutions for Generalized Laplacian Problems with a Parameter. Mathematics, 12(23), 3668. https://doi.org/10.3390/math12233668

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