Existence of periodic solutions for the Grushin critical problem

Abstract

We study a Grushin critical problem in a strip domain which satisfies the periodic boundary conditions. By applying the finite-dimensional reduction method, we construct a periodic solution when the prescribed curvature function is periodic. Furthermore, we also consider the Grushin critical problem in $\mathbb{R}^{N} (N \geq 5)$. Compared with Billel et al. (Differential Integral Equations 32: 49-90, 2019), we use the method by Guo and Yan (Math. Ann. 388: 795-830, 2024) to construct periodic solutions under some weaker conditions, avoiding the complicated estimates and uniqueness proof. Notably, Guo and Yan (Math. Ann. 388: 795-830, 2024) obtained solutions periodic with respect to some of the first variables, while the solutions in this paper are periodic with respect to some intermediate variables.