On The One-dimensional Symmetry Of Solutions Of Some Elliptic Equations

This paper mainly studies two problems. The first one is the one-dimensional symmetry of bounded entire solutions of elliptic equations: with F∈C2(R), p≥2, and n=2,3.The method is Liuville Property on Laplace operate offered by H. Berestycki, L. Caffarelli and L. Nirenberg, we extend it to the P-Laplace operator. Combining with the theory of calibrations and extremal field of the Calculus of Variations, and using the extended Liuville Property, we get the one-dimensional symmetry of the bounded solutions, with the condition of higher regularity of the solutions.On the other hand, we study the monotonicity and symmetry of the solutions of nonlinear function F(x, u(x):▽u(x),▽2u(x))=0 in the special cube without finite points, using method of moving plans and maximum principles.