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

The classical critical Trudinger-Moser inequality in ℝ2 under the constraint $\int_{\mathbb{R}{^2}} {\left( {{{\left| {\nabla u} \right|}^2} + {{\left| u \right|}^2}} \right)dx\;} \leqslant \;1$ was established through the technique of blow-up analysis or the rearrangement-free argument: for any τ > 0, it holds that $$\mathop {\sup }\limits_{\matrix{{u \in {H^1}\left( {{\mathbb{R}^2}} \right)} \cr {\int_{{\mathbb{R}^2}} {\left( {\tau {{\left| u \right|}^2} + {{\left| {{\nabla u}} \right|}^2}} \right)dx} \leqslant 1} \cr } } \int_{\mathbb{R}{^2}} {\left( {{\rm{e}^{4\pi }}^{{{\left| u \right|}^2}} - 1} \right)dx} \;\leqslant \;C\left( \tau \right) < + \infty,$$ and 4π is sharp. However, if we consider the less restrictive constraint $\int_{\mathbb{R}{^2}} {\left( {{{\left| {\nabla u} \right|}^2} + V\left( x \right){u^2}} \right)dx\;} \leqslant \;1$ , where V(x) is nonnegative and vanishes on an open set in ℝ2, it is unknown whether the sharp constant of the Trudinger-Moser inequality is still 4π. 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 $$\mathop {\sup }\limits_{u \in {H^1}\left( {\mathbb{R}{^2}} \right),\;\int_{\mathbb{R}{^2}} {\left( {{{\left| {\nabla u} \right|}^2} + V\left( x \right){u^2}} \right)dx} \leqslant 1} \int_{\mathbb{R}{^2}} {\left( {{\rm{e}^{4\pi {u^2}}} - 1} \right)dx} \;\leqslant \;C\left( V \right) < \infty,$$ when V is nonnegative and vanishes on an open set in ℝ2. As an application, we also prove the existence of ground state solutions to the following Schrodinger equations with critical exponential growth (0.1) $$ - {\rm{\Delta }} u + V\left( x \right)u = f\left( u \right)\;\;\;\;{\rm{in}}\;\mathbb{R}{^2},$$ where V(x) ⩾ 0 and vanishes on an open set of ℝ2 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.