Existence And Symmetry Of Solutions For Some Classes Of The Fractional Laplacian Equation (System)

The thesis consists of six parts. In the Preface, we introduce the background and some physical application of the fractional Laplacian, we also give the definition of the fractional operator and the Sobolev space. In Chapter 1, we study the existence,concentration and multiplicity of the positive solutions for the fractional Choquard equation in the critical case with the variational method and Lusternik-Schnirelmann category theory. In Chapter 2, We consider a pseudo-differential system involving d-ifferent fractional orders. Through an iteration method, we obtain the key ingredients of the method of moving planes-the maximum principles. Then we derive symme-try on non-negative solutions without any decay assumption at infinity with the direct method of moving planes. In Chapter 3, we investigate a fractional Choquard equa-tion. Firstly we prove it is equivalent to a fractional system, then derive the symmetry and nonexistence of positive solutions by the same way as in the chapter 2. In Chapter 4, we consider the fractional henon system, using the direct method of moving planes,we derive the radial symmetry of positive solution, then derive the nonexistence of positive solutions in the critical and supercritical cases by the Pohozaev identity in the integral form. In Chapter 5, we first get the pohozaev identity for the fractional system in the star-shaped domain, then deduce the nonexistence of positive solution in the subcritical case.