Liouville Type Theorems For Higher-order Nonlinear Elliptic Systems

This thesis mainly deals with Liouville type theorems (i.e., non-existence of positive so-lutions) for some nonlinear high-order elliptic systems. The substantial difficulty is that the General Maximum Principle of second order elliptic equation does not hold any more for the high order cases. The main idea is to establish the symmetry and Liouville type theorem for the corresponding integral systems, and then prove the equivalence between the higher order elliptic equations and involved integral systems. The first problem is a higher-order elliptic sys-tem with Navier boundary value in half space. By using the Moving Plane Method in integral forms, we establish a Liouville type theorem for the related integral system in the half-space, and furthermore, the results are valid for the higher-order differential system as well via showing the equivalence between the higher order elliptic equations and the integral systems with funda-mental solution’s estimates. The second problem is to obtain non-existence of non-trivial pos-itive solutions to higher-order nonlinear elliptic systems with general power function coupling. Lastly consider non-existence of positive solutions for the doubly weighted integral system in half space under super-critical case.The thesis is composed of five chapters:Chapter1is to describe the background of the related issues and to briefly summarize the main results of the present thesis.Chapter2studies the Liouville type theorem for a2m-order elliptic equations coupled with the Navier boundary conditions in the half-space. This is realized via establishing the non-existence of positive solutions for the equivalent integral system by the moving plane method in integral forms. When proving the equivalence between IEs and PDEs, we remove an additional restriction, similar a recent result for scalar problem by Fang and Chen.Chapter3is devoted to studying the Liouville type theorem for higher-order nonlinear elliptic systems with power function coupling term. Combining the method of moving planes in integral forms with Hardy-Littlewood-Sobolev inequality and estimates of fundamental solution involved, we proved that the solutions of the related integral systems are symmetric, and thus depend on xn only. So we get the non-existence of positive solutions for the integral system. Combining with the equivalence between the differential and the integral systems, we get the Liouville type theorem for the higher order differential system. Chapter4deals with the nonexistence of positive solutions to a doubly weighted integral system in half space. We show that the solutions u(x) and v(x) are monotonously increasing in xn by using the moving plane method in integral forms. Combining the global integral condition, we obtain the non-existence of positive solutions under the super-critical case.Chapter5summarize the main conclusions with innovative points of thesis, and proposes the works to be done in the future.