Existence And Asymptotical Behavior Of Solutions For Elliptic Equations With Singularity In R~3

In this paper,we mainly investigate the existence and asymptotical behavior of solutions to elliptic equation with singularity in R3.The thesis consists of the following five parts.1.We consider the following Choquard equation with weak singularity:where Iα is the Riesz potential of order α∈(0,3)and 1+α/3≤p<3+α,0<γ<1.Under certain assumptions on V and f,we study the existence and uniqueness of positive solution uλ。for equation(Pλ)with λ>0 in Chapter 1 and show that uλconverges to the unique solution of equation(P0)as λ→0+.In Chapter 2,we obtain that equation(Pλ)with λ<0 has at least two solutions:a positive ground state solution uλ(1)and a positive solution uλ(2)Moreover,as λ→0-,these solutions have the following convergence:uλ(2)is divergent and uλ(1)converges to the unique solution of equation(P0).2.We consider the following critical Choquard equation with weak singularity:where Iα is the Riesz potential of order α∈(0,3)and 1+α/3≤p<3,0<γ<1,λ>0.In Chapter 3,under certain assumptions on V and f,we obtain that equation(CPλ)has at least two solutions:a positive ground state solution uλ and a positive solution vλ.Moreover,as λ→0+,these solutions have the following convergence:uλand vλ tend to a positive ground state solution and a positive solution of equation(CP0),respectively.3.We consider the following critical elliptic equation with weak singularity:where 0<γ<1,λ>0 and Q(x)>0.In Chapter 4,under certain assumptions on f and assume that the maximum of Q(x)is achieved at k different points a1,a2,…,ak,we obtain that equation(KPλ)has,besides a ground state solution uλ which tends to 0 in D1,2(R3)as λ→0+,at least k positive distinct solutions uλ,i(i=1,2,…,k)satisfying in the sense of measure as λ→0+,where δai is the Dirac measure assigned to ai and S is the best Sobolev constant for the embedding of D1,2(R3)in L6(R3).4.We consider the following fractional Schrodinger-Poisson system with singular-ity:where λ>0 and 0<s≤t<1 with 4s+2t>3.(-Δ)s is the fractional Laplacian operator.Under certain assumptions on V and f,the existence and uniqueness of a positive solution uλ for weak singular case(i.e.0<γ<1)and strong singular case(i.e.γ>1)are obtained in Chapter 5 and Chapter 6,respectively.Moreover,asλ→0+,the unique solution uλ obtained in both two cases converges to the unique solution of the corresponding equation(SP0).5.We consider the following fractional Kirchhoff type equation with singularity:where(-Δ)s is the fractional Laplacian with 0<s<1,b>0 is a constant.Under certain assumptions on V and f,the existence and uniqueness of a positive solution ub for weak singular case(i.e.0<γ<1)and strong singular case(i.e.γ>1)are obtained in Chapter 7 and Chapter 8,respectively.Moreover,as b→0+,the unique solution ub obtained in both two cases converges to the unique solution of the corresponding equation(K0).