Qualitative Properties for Solutions of Nonlinear Equations and Systems involving Higher Order and Fractional Order Laplacians

Abstract

In this thesis, We prove general symmetry and uniqueness results for positive solutions of nonlinear equations and systems involving higher order and fractional order Laplacians.;First, we study weighted higher order Laplacians in n-dimensional Euclidean space Rn. We establish the equivalence between the partial differential equations (PDEs) and integral equations (IEs). In the proof of equivalence, we employ a new idea in order to weaken some strong degeneracy assumptions. By using the method of moving planes in integral forms and the Pohozaev-type identity in integral forms, we prove nonexistence of positive solutions for the IEs. Combining with the equivalence result, we derive the uniqueness of solutions for the PDEs.;Next, we consider the same PDEs as above with Navier boundary conditions in a half space Rn+ = {lcub} x = (x1, x2,··· , xn) Eepsilon Rn/ xn > 0{rcub}. Utilizing the super poly-harmonic properties, we obtain the equivalence between PDEs and the corresponding IEs under very mild growth conditions. Combining the Kelvin transform with the method of moving planes, we deduce rotational symmetry and monotonicity for the positive solutions of the integral equations. Furthermore, applying the Pohozaev-type identity in integral forms, we arrive at the nonexistence of positive solutions for the integral equations.;We also analyze the Schrodinger system with two different boundary conditions in the half space Rn+, and obtain the symmetry and nonexistence for the positive solutions.;Finally, we consider nonlinear equations involving fractional Laplacian, a nonlocal pseudo-differential operator. Using the Liouville Theorem and the maximum principle for the fractional Laplacian, we show equivalence between pseudo-differential equations and integral equations. Furthermore, a combination of the equivalence with the method of moving planes leads to much more general results on the qualitative properties of solutions for pseudo-differential equations.