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 \(C^1\)-functionals approximating the energy. We use careful estimates of the corresponding energy levels to show that this limit is neither trivial nor a minimizer.

我们证明，一类特定的椭圆自由边界问题，包括特殊情况下的流体动力学中的Prandtl–Batchelor问题，对于参数的大值具有两个截然不同的非平凡解。第一个解决方案是全局最小化能量。能量泛函是不可微的，因此标准变分自变量不能直接用于获得第二个非平凡解。我们获得第二个解，作为\（C ^ 1 \） -函数逼近能量序列的山峰通过点的极限。我们对相应的能级进行了仔细的估计，以表明该极限既不重要，也不是最小化。