# Equivalent conditions for existence of three solutions for a problem with discontinuous and strongly-singular terms

## TL;DR

This paper establishes conditions for the existence of three solutions to a Kirchhoff problem involving strongly-singular and discontinuous nonlinear terms within Orlicz-Sobolev spaces, using convex analysis and Clarke's generalized gradient.

## Contribution

It introduces an optimal condition that enables the use of variational methods despite the challenges posed by singular and discontinuous nonlinearities.

## Key findings

- Derived an optimal existence condition for solutions.
- Applied convex analysis and Clarke's generalized gradient.
- Addressed difficulties due to non-differentiability of the energy functional.

## Abstract

In this paper, we are concerned with a Kirchhoff problem in the presence of a strongly-singular term perturbed by a discontinuous nonlinearity of the Heaviside type in the setting of Orlicz-Sobolev space. The presence of both strongly-singular and non-continuous terms bring up difficulties in associating a differentiable functional to the problem with finite energy in the whole space $W_0^{1,\Phi}(\Omega)$. To overcome this obstacle, we established an optimal condition for the existence of $W_0^{1,\Phi}(\Omega)$-solutions to a strongly-singular problem, which allows us to constrain the energy functional to a subset of $W_0^{1,\Phi}(\Omega)$ to apply techniques of convex analysis and generalized gradient in Clarke sense.