Solutions for a class of singular quasilinear equations involving critical growth in R2

Using a variational approach, we study the existence of solutions for the following class of quasilinear Schrodinger equations: -Delta u + V(x)u - Delta(vertical bar u vertical bar(2 beta)) vertical bar u vertical bar(2 beta-2)u = g(u)/vertical bar x vertical bar(a) in R-2, where beta > 1/2, a is an element of[0,2), V(x) is a positive potential bounded away from zero and can be "large" at infinity, the nonlinearity g(s) is allowed to satisfy the exponential critical growth with respect to the Trudinger-Moser inequality. Precisely, g(s) behaves like exp (alpha(0) vertical bar s vertical bar|(4 beta)) as vertical bar s vertical bar -> infinity for some alpha(0) > 0. This model of equation has been proposed in the theory of superfluid films in plasma physics. As for as we know, this the first work involving this class of operators and singular non-linearities with exponential critical growth. Moreover, we are able to deal with exponents beta > 1/2.