Liouville Type Theorem For Elliptic Equations And The Existence Of Problems With Fractional Laplacian

In this thesis, we investigate Liouvile type theorem for boundedness solutions with finite Morse index of mixed boundary value problems and the existence of solutions of the critical elliptic problem with a Hardy term and fractional Laplacian.In chapter 1, we introduce the background of problems and main results of the thesis.In chapter 2, we establish Liouvile type theorem for boundedness solutions with finite Morse index of the following mixed boundary value problems: and where R+N={x ?R^N:x_N> 0}, ?₁={x ?R^N:x_N= 0,X1< 0}, and ?₀={x ? R^N:x_N= 0,x₁> 0}. The exponents p,q satisfy 1< p?N-2/N+2,1< q?N-2/N and ?p,q???N-2/N+2,N-2/N?.In chapter 3, We consider the existence of solutions of the critical elliptic prob-lem with a Hardy term and fractional Laplacian where ????R^N is a smooth bounded domain and 0 ? ?,? is a positive parameter, N? 2s and s ? ?0,1?,2_s* is the critical exponent. ?-??^s stands for the spectral fractional Laplacian. Assuming that ? is non-contractible, we show that there exists a solution.