A critical Trudinger-Moser inequality involving a degenerate potential and nonlinear Schrodinger equations

The classical critical Trudinger-Moser inequality in Double-struck capital R-2 under the constraint R-2(vertical bar del u vertical bar(2) + vertical bar u vertical bar(2))dx <= 1 was established through the technique of blow-up analysis or the rearrangement-free argument: for any tau > 0, it holds that sup(integral R2(tau vertical bar u vertical bar 2+vertical bar del u vertical bar 2)dx <= 1u is an element of H1(R2))integral(R2)(e(4 pi vertical bar u vertical bar 2)-1)dx <= C(tau) < +infinity, and 4 pi is sharp. However, if we consider the less restrictive constraint integral(R2)(vertical bar del u vertical bar(2) + V(x)u(2))dx <= 1, where V(x) is nonnegative and vanishes on an open set in Double-struck capital R-2, it is unknown whether the sharp constant of the Trudinger-Moser inequality is still 4 pi. The loss of a positive lower bound of the potential V(x) makes this problem become fairly nontrivial. The main purpose of this paper is two-fold. We will first establish the Trudinger-Moser inequality sup(u is an element of H1(R2),integral R2(vertical bar del u vertical bar 2+V(x)u2)dx <= 1)integral(R2)(e(4 pi u2) - 1)dx <= C(V) < infinity, when V is nonnegative and vanishes on an open set in R-2. As an application, we also prove the existence of ground state solutions to the following Schrodinger equations with critical exponential growth -Delta u + V(x)u = f(u) in R-2, (0.1) where V(x) >= 0 and vanishes on an open set of R-2 and f has critical exponential growth. Having the positive constant lower bound for the potential V(x) (e.g., the Rabinowitz type potential) has been the standard assumption when one deals with the existence of solutions to the above Schroodinger equations when the nonlinear term has the exponential growth. Our existence result seems to be the first one without this standard assumption.