On existence of solutions for some classes of elliptic problems with supercritical exponential growth

In the present paper, we study the existence of solutions for the following classes of elliptic problems where \(\Omega \subset \mathbb {R}^2\) $\mathrm{\Omega }\subset {\mathbb{R}}^2$ is a smooth bounded domain and where \(V\in C^0(\mathbb {R}^2)\) $V\in C^0({\mathbb{R}}^2)$ is periodic in \(\mathbb {Z}^2\) ${\mathbb{Z}}^2$ with \(0\not \in \sigma (-\Delta +V)\) $0\notin \sigma (-\mathrm{\Delta }+V)$ . In the both problems above, f is a continuous function of the form \(\begin{aligned} f(t)=h(t)e^{\alpha _0 |t|^\tau }, \quad t \in \mathbb {R}\end{aligned}\) $$\begin{array}{c}\hfill f(t)=h(t)e^{{\alpha }_0{|t|}^{\tau }},t\in \mathbb{R}\end{array}$$ for some \(\alpha _0>0\) ${\alpha }_0>0$ and \(\tau \ge 2\) $\tau \ge 2$ and h satisfying some technical conditions. By using variational methods, we show that problems (P) and \((P_V)\) $(P_V)$ have a nontrivial solution for different types of \(\alpha _0>0\) ${\alpha }_0>0$ and \(\tau \ge 2\) $\tau \ge 2$ .