On a singular class of elliptic systems involving critical growth in R2R2

In this paper, the Ekeland variational principle, the mountain-pass theorem and a suitable Trudinger–Moser inequality with a singular weight are employed to establish sufficient conditions for the existence of solutions for the following class of systems −Δui=fi(x,u1,…,um)|x|a+hi(x),x∈Ω,i=1,…,m, where ui∈H01(Ω),a∈[0,2),Ω is a bounded domain in R2R2 containing the origin with smooth boundary, fifi behaves like eβs2eβs2 when |s|→+∞|s|→+∞ for some β>0β>0 and hi∈((H01(Ω))∗,‖⋅‖∗).