Abstract
This paper focuses on the following scalar field equation $$ \begin{cases} -\Delta u=f(u)&\mbox{in }\Omega,\\ u=0&\mbox{on }\partial\Omega, \end{cases} $$ 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 $$f$$ .