An elliptic problem of the Prandtl-Batchelor type with a singularity

TL;DR

This paper proves the existence of multiple solutions for a Prandtl-Batchelor type elliptic problem involving a power nonlinearity and a singular term, using a novel approach due to the nondifferentiability of the energy functional.

Contribution

It introduces a new method employing a sequence of $C^1$ functionals and a cutoff function to handle nondifferentiability in elliptic problems.

Findings

Existence of at least two solutions established.

Novel approach overcomes nondifferentiability of the energy functional.

Application of mountain pass theorem and elliptic regularity theory.

Abstract

We establish the existence of at least two solutions of the {\it Prandtl-Batchelor} like elliptic problem driven by a power nonlinearity and a singular term. The associated energy functional is nondifferentiable and hence the usual variational techniques do not work. We shall use a novel approach in tackling the associated energy functional by a sequence of $C^1$ functionals and a {\it cutoff function}. Our main tools are fundamental elliptic regularity theory and the mountain pass theorem.