Positive solutions for φ-Laplace equations with discontinuous state-dependent forcing terms

Abstract

This paper concerns the existence, localization and multiplicity of positive solutions for a φ-Laplacian problem with a perturbed term that may have discontinuities in the state variable. First, the initial discontinuous differential equation is replaced by a differential inclusion with an upper semicontinuous term. Next, the existence and localization of a positive solution of the inclusion is obtained via a compression-expansion fixed point theorem for a composition of two multivalued maps, and finally, a suitable control of discontinuities allows to prove that any solution of the inclusion is a solution in the sense of Carathéodory of the initial discontinuous equation. No monotonicity assumptions on the nonlinearity are required.

Authors

Department of Mathematics Babes-Bolyai University, Cluj-Napoca, Romania

Jorge Rodríguez-López
Instituto de Matemáticas, Universidade de Santiago de Compostela, Facultade de Matemáticas, Campus Vida, Santiago, Spain

Keywords

discontinuous differential equation; φ-Laplacian problem; positive solution; fixed point; multivalued map; infinitely many solutions

Paper coordinates

R. Precup, J. Rodríguez-López, Positive solutions for φ-Laplace equations with discontinuous state-dependent forcing terms, Nonlinear Analysis: Modelling and Control 24 (2019), No. 3, 447-461, https://doi.org/10.15388/na.2019.3.8

PDF

Journal

Nonlinear Analysis: Modelling and Control

Publisher Name

Vilnius University Press

1392-5113

Online ISSN

2335-8963

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