Singular Trudinger–Moser inequality and fractional [formula omitted]-Laplace equations in [formula omitted

Abstract

In this paper, we establish a singular Trudinger–Moser inequality in RN. Applying that result, we study the existence of solution to N∕s-fractional Laplacian equation LN∕ssu(x)+V(x)|u|Ns−2u=f(x,u)|x|γ,where γ∈[0,N), 0<s<1,LN∕ss is a nonlocal fractional operator with singular kernel K:RN∖{0}→R+,f is a continuous function on RN×R which does not satisfy the Ambrosetti–Rabinowitz condition. By using Mountain Pass Theorem, we obtain the existence of solution to above equation. Furthermore, when f satisfies the Ambrosetti–Rabinowitz condition, we obtain the existence to solution of that equation with negative energy. In our best knowledge, it is the first time the existence to solution of above equation is studied.