On a Schrödinger equation with periodic potential and critical growth in $\mathbb{R}^{2}$

Abstract.

The main purpose of this paper is to establish the existence of a solution of the semilinear Schrödinger equation

$$ - \Delta u + V{\left( x \right)}u = f{\left( u \right)},\;{\text{in}}\;\mathbb{R}^{2} $$

where V is a 1-periodic function with respect to x, 0 lies in a gap of the spectrum of  − Δ  +  V, and f(s) behaves like  ±  exp(α s²) when s →  ±  ∞.