Study On The Existence Of Weak Solutions For Two Fractional P-Laplacian Equations

In this paper we study on the existence of solutions of two fractional p-Laplacian equations,and established the existence theorem of weak solutions of equations under the condition of subcritical and critical.The fist case:devoted to the existence of week solutions for the following frac-tional p-Laplacian equation(-△)psu + V(x)|u|p-2u = f(x,u),x ∈ RN,(3)where p ≥ 2,N ≥ 2 and s ∈(0,1),V(x)∈ C(RN)is a function possibly changing sign,(-△)ps stands for the fractional p-Laplacian and nonlinearity f:RN × R → R is a Caratheodory function.By using mountain pass theorem,the existence theorem of the week solutions for the equation is established.The second case:devoted to the existence of week solutions for the following fractional p-Laplacian equation with critical exponent(-△)psu + V(x)|u|P-2u = |u|ps*-2u +f(x,u),x ∈ RN,(4)where p ≥ 2,N ≥ 2,N>Ps,ps*Np/(N-ps)and s ∈(0,1),(-△)ps stands for the fractional p-Laplacian and nonlinearity f:RN × R→R is a Caratheodory function.By using mountain pass theorem,the existence theorem of the week solutions of the equation is established.This thesis consists of three chapters.The first chapter is devoted to discuss the introduction including research background and prerequisite knowledge.The second chapter deals with the existence of the solutions for the equation under the first case.In the last chapter we research the the existence of the solutions for the equation about the second case.