On a class of elliptic free boundary problems with multiple solutions

We prove that a certain class of elliptic free boundary problems, which includes the Prandtl–Batchelor problem from fluid dynamics as special case, has two distinct nontrivial solutions for large values parameter. The first solution is global minimizer energy. energy functional nondifferentiable, so standard variational arguments cannot be used directly to obtain second solution. our limit mountain pass points sequence $$C^1$$ -functionals approximating use careful estimates corresponding levels show this neither trivial nor minimizer.