Existence And Concentration Of Solutions For Some Classes Of Nonlinear Elliptic Equations With Fractional Order

The fractional Laplacian arises in several areas such as physics, biology, chem-istry and finance. In particularly, from a probabilistic point of view, the fractional Laplacian is the infinitesimal generators of Levy stable process. So, in recent years, the study of nonlinear equations involving a fractional Laplacian has attracted much attention from many mathematicians.In this thesis, we study the existence, multiplicity of some fractional elliptic equa-tions with critical exponent, and the concentration of the solutions of some fractional nonlinear Schrodinger equations, by using the critical point theory and Lyapunov-Schmidt reduction methods of nonlinear analysis. Some new results are obtained. The organization of this thesis is as follows.In Chapter 1, we apply Nehari manifold method and Ljusternik-Schnirelmann theory to consider the nonlinear fractional Schrodinger equation with critical expo-nent. We obtain the existence of ground state solutions and catΛδ (Λ) nontrivial solu-tions under some different case.In Chapter 2, we investigate the concentration phenomenon of solutions for the nonlinear fractional Schrodinger equation ε2s(-Δ)su＋V(x)u= K(x)｜u｜p-1u, x ∈ RN, where V(x) and K(x) are positive smooth functions. Let Γ(x)= [V(x)]p+1/P-1-N/2s[K(x)]-2/p-1. Under certain assumptions on V(x) and K(x), we show existence and multiplicity of solutions which concentrating near some critical points of Γ(x) by a perturbative variational method. We complement and improve the main results of [36,44], in the sense that we are considering the multiplicity results.In Chapter 3, we study the existence of positive multi-peak solutions to the semi-linear equation ε2s(-Δ)su＋u= Q(x)up-1, u> 0. u∈ Hs(RN), where Q(x) is a bounded positive continuous function. For any positive integer k, we prove the existence of a positive solution with k-peaks and concentrating near a given local minimum point of Q. For s= 1 this corresponds to the result of [65].In Chapter 4, we use harmonic extension technique and critical point theory to study the nonhomogeneous fractional Laplacian problem with critical exponent. We prove that this problem has at least two positive solutions. We also establish a nonexis-tence result for a positive solution in a class of linear positive-type domains are more general than star-shaped ones. This nonexistence result improve the corresponding result of Tan [85].