Positive solutions for phi-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.

Keywords

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