Abstract Using monotonicity methods and some variational argument we consider nonlinear problems which involve monotone potential mappings satisfying condition (S) and their strongly continuous perturbations. We investigate when functional whose minimum is obtained by a direct method of the calculus of variations satisfies the Palais-Smale condition, relate minimizing sequence and Galerkin approximaitons when both exist, then provide structure conditions on the derivative of the action functional under which bounded Palais-Smale sequences are convergent. Finally, we make some comment concerning the convergence of Palais-Smale sequence obtained in the mountain pass theorem due to Rabier.

中文翻译：

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具有单调算子的变分非线性方程

摘要 使用单调性方法和一些变分论证，我们考虑涉及满足条件（S）的单调势映射及其强连续扰动的非线性问题。我们研究了通过直接变分法获得最小值的泛函何时满足 Palais-Smale 条件，将最小化序列和 Galerkin 逼近当两者都存在时关联起来，然后提供作用函数的导数的结构条件，在此条件下有界 Palais- Smale 序列是收敛的。最后，我们对由Rabier在山口定理中得到的Palais-Smale序列的收敛性做一些评论。