Positive Solution To A Critical Elliptic Equation With Logarithmic Term

In this paper,we consider the existence of positive solution for a critical problem with logarithmic term:where Ω?RN is a bounded smooth domain,λ,μ∈R,N≥3,2*=2N/N-2 is the critical Sobolev exponent for the embedding H01(Ω)(?)/L2*(Ω).The uncertainty of the sign of slog s2 in(0,+∞)makes this problem much more interesting.We will show that if N≥4,λ∈R and μ>0,then the above problem has a Mountain pass solution,which is positive and also a ground state solution,and if N=3 and λ,μ satisfy some assumptions,then it has a positive solution.Combining with[4],our results can reflect the influence of the logarithmic term on the existence of the positive solution to the critical problem.