A sharp Moser-Trudinger type inequality involving L norm in Rn with degenerate potential

In this paper, we establish a Moser-Trudinger type inequality with a special degenerate potential V. Specifically, we demonstrate that for any potentials V(x) satisfying the conditions (V1) and (V2), the following inequalityMT(V,α,p)≔supu∈E,‖u‖≤1⁡∫RnΦ(αn(1+α‖u‖pn)1n−1unn−1)dx<∞ holds if α<λ1(V); where E is the usual weighted Sobolev space associated to the potential V, the norm ‖u‖n:=∫Rn|∇u|n+V(x)|u|ndx, and λ1(V) is the first Dirichlet eigenvalue of the corresponding operator −Δ+V. The potential conditions are as follows. 0≤m=infx∈Rn⁡V(x)n and α≥λ1(V), there holds MT(V,α,p)=∞. Meanwhile, if α>λ1(V), then MT(V,α,n)=∞. Furthermore, via a subtle blow-up analysis, we also prove the existence of an extremal function for the inequality above when α is sufficiently small.