On a $p(\cdot)$-biharmonic problem with no-flux boundary condition

TL;DR

This paper introduces and analyzes a novel $p( abla)$-biharmonic PDE with no-flux boundary conditions, demonstrating existence and multiplicity of solutions using variational methods without common restrictive conditions.

Contribution

It is the first to study a $p( abla)$-biharmonic problem with no-flux boundary conditions, expanding the understanding of fourth order PDEs with variable exponent.

Findings

Existence of weak solutions established

Multiple solutions proven using mountain pass theorem

No need for Ambrosetti-Rabinowitz or symmetry conditions

Abstract

The study of fourth order partial differential equations has flourished in the last years, however, a $p(\cdot)$-biharmonic problem with no-flux boundary condition has never been considered before, not even for constant $p$. This is an important step further, since surfaces that are impermeable to some contaminants are appearing quite often in nature, hence the significance of such boundary condition. By relying on several variational arguments, we obtain the existence and the multiplicity of weak solutions to our problem. We point out that, although we use a mountain pass type theorem in order to establish the multiplicity result, we do not impose an Ambrosetti-Rabinowitz type condition, nor a symmetry condition, on our nonlinearity $f$.