Zero Mass Case for a Nonlinear Elliptic Equation in Striplike Domains

This paper focuses on the following scalar field equation
\begin{equation*} \begin{cases} -\Delta u=f(u)&\text{in }\Omega,\\ u=0&\text{on }\partial\Omega, \end{cases} \end{equation*}
where $\Omega=\mathbb{R}^k\times D$, $k\geq2$ and $D$ is a bounded domain in $\mathbb{R}^\ell$ with $\ell\geq1$. By using the variational method, we extend the results of Byeon and Tanaka in [6] and prove, in the zero mass case, the existence of a positive axially symmetric solution under suitable growth conditions on $f$.