Multiple Solutions For A Class Of Elliptic Equations With Bi-nonlocal Problems

In the present paper,we investigate the existence of multiple positive solutions for bi-nonlocal problems with singularity and critical growth,by the variational method,the critical point theory and the analytical technique.Firstly,the following bi-nonlocal Kirchhoff type equations with singular terms is considered:(?)where Ω(?)R3 is a smooth bounded domain with boundary (?)Ω,a,b>0,2<2(r+1)<p(β+1)<6,β>0 and 0<γ<1,λ>0 is a real parameter.We will perturbation technique is used to solve the problem that the functional with singular term cannot be differentiable at zero,by applying the Ekeland variational principal and the mountain pass lemma,the perturbation functional corresponding to this problem has a local minimum and critical point of the mountain path type,the existence of two positive solutions for the equation are obtained by estimating the sequence with a consistent lower bound and taking the limit of perturbation.Secondly,the solvability of the following bi-nonlocal Schr(?)dinger-Poisson system with critical growth is studied:(?)where Ω(?)R3 is a smooth bounded domain,λ>0,1<r<2,0<s<(r-1)/3 and F(x,u)=∫0uf(x,ξ)dξ,f(x,u)satisfy some suitable assumptions,the existence of multiple positive solutions of the system is obtained by the concentration compactness principle,the truncation technique,the Ekeland variational principal,and the mountain pass lemma.