Quasilinear Equations in Orlicz–Sobolev Spaces Involving Critical Exponential Growth with Unbounded and Decaying Radial Potentials

This paper concerns with the study of existence of non-trivial solutions to the following quasilinear equation: \(-\text {div} \Big (\Phi '( |\nabla u |)\dfrac{\nabla u}{|\nabla u|} \Big ) +V(|x|)\Phi '(|u|)\dfrac{u}{|u|} = Q(|x|)f(x,u),\quad x \in \mathbb {R}^2,\) $$-\text{div}({\mathrm{\Phi }}^{\prime }(|\mathrm{\nabla }u|){\displaystyle \frac{\mathrm{\nabla }u}{|\mathrm{\nabla }u|}})+V(|x|){\mathrm{\Phi }}^{\prime }(|u|){\displaystyle \frac{u}{|u|}}=Q(|x|)f(x,u),x\in {\mathbb{R}}^2,$$ where \(\Phi \) $\mathrm{\Phi }$ is a \(\mathcal {C}^1\) ${\mathcal{C}}^1$ -Young function, V and Q are positive radial potentials, and the nonlinearity f possesses maximal exponential growth established by Trudinger–Moser inequalities on Orlicz–Sobolev spaces. Using variational methods, we prove the existence of a weak non-trivial solution