On a Generalized Prandtl–Batchelor Free Boundary Problem With a Singularity on a Stratified Lie Group

We investigate a class of elliptic free boundary problems, including a generalized Prandtl–Batchelor type problem with a singularity on a stratified Lie group. The associated energy functional is nondifferentiable, which precludes the direct application of standard variational techniques. To address this challenge, we approximate the energy functional using a sequence of <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" overflow="scroll"> <mml:msup> <mml:mi>C</mml:mi> <mml:mn>1</mml:mn> </mml:msup> </mml:math> -functionals and establish the existence of at least two distinct nontrivial solutions. Our analysis combines elliptic regularity theory with the mountain pass theorem in a non-Euclidean setting.