On a class of elliptic free boundary problems with multiple solutions
Abstract
We prove that a certain class of elliptic free boundary problems, which
includes the Prandtl-Batchelor problem from fluid dynamics as a special case,
has two distinct nontrivial solutions for large values of a parameter. The
first solution is a global minimizer of the energy. The energy functional is
nondifferentiable, so standard variational arguments cannot be used directly
to obtain a second nontrivial solution. We obtain our second solution as the
limit of mountain pass points of a sequence of C1-functionals approximating
the energy. We use careful estimates of the corresponding energy levels to
show that this limit is neither trivial nor a minimizer.