A Priori Estimates And Nonexistence Of Solutions Of Nonlinear Differential Inequalities

This thesis deals with a priori estimates and nonexistence of solutions of nonlinear differential inequalities with gradient nonlinearities.The thesis is composed of four chapters:In Chapter 1, we summarize the background of the related problems and state the main results of the thesis.In Chapter 2, we get a priori estimates of solutions of nonlinear differential inequalities with a gradient nonlinearity in the exterior or interior domain by using test function method, and then obtain the Harnack-type inequalities for two-side nonlinear differential inequalities with a gradient nonlinearity. Here we shall use the ideas of Serrin and Zou ([51]) to give a priori estimates for u and its gradient, and then obtain the Harnack-type inequalities for two-side nonlinear differential inequalities.In Chapter 3, by using method developed by Bidaut-Veron, Mitidieri and Pohozaev in [64] and [8], we will show the new nonexistence theorems for coercive nonlinear differ-ential inequality div (g(u)A(x,u,▽u)▽u)≥|x|-αuq1|▽u|s1-uq2|Vu|s2 and anticoercive nonlinear differential inequality - div (g(u)A(x, u,▽u)▽u)≥|x|-αuq1|▽u|s1-uq2|▽u|s2 with two gradient terms included in the right-hand side, which have not covered by other works.In Chapter 4, we prove some nonexistence results of solutions of two class singular coercive nonlinear differential inequality div(A(x, u,▽u)Vu)≥a(x)uq|▽u|s and antico-ercive nonlinear differential inequality - div(A(x,u, Vu)Vu)≥a(x)uq|▽u|s with gradient nonlinearities by using the test function method developed by Mitidieri and Pohozaev. In particular, we don't use the comparison principles so that no conditions at infinity on the solution are required.