Nonlinear elliptic equations in dimension two with potentials which can vanish at infinity

We will focus on the existence of nontrivial solutions to the following nonlinear elliptic equation -Δu + V (x)u = f (u), x ∈ R², where V is a nonnegative function which can vanish at infinity or be unbounded from above, and f have exponential growth range. The proof involves a truncation argument combined with Mountain Pass Theorem and a Trudinger-Moser type inequality.