Some remarks on the method of moving planes. (English) Zbl 1040.35032

The article deals with a nonlinear elliptic equation in divergence form given in a bounded domain with the homogeneous Dirichlet boundary condition. The domain is symmetric with respect to a hyperplane, symmetry and monotonicity properties of coefficient functions of the equation are supposed. The author proposes a variational approach to the method of moving planes. The approach is based on comparison principles for nonlinear operators and does not rely on the Alexandrov-Bakelman-Pucci’s inequality. The method is applied to weak solutions of the nonlinear elliptic equation in a general domain (i.e., without supposing any smoothness of the boundary). The symmetry result for a weak solution of the problem is proved.


35J65 Nonlinear boundary value problems for linear elliptic equations
35J20 Variational methods for second-order elliptic equations