Existence of periodic solutions for the Grushin critical problem
Wenju Wu, Fulin Zhong
TL;DR
This work studies a Grushin-type critical elliptic problem with periodic boundary conditions on a strip, and its related critical problem in $\mathbb{R}^N$ for $N\ge 5$. It employs a finite-dimensional reduction framework, building an approximate multi-peak profile $P W_{\hat{x},\lambda}$ and a fixed-point correction $\omega_L$ to obtain genuine, positive solutions $u$ that concentrate around the peak as $\lambda$ grows. A carefully constructed linear theory around the peak, via the operator $\tilde{L}$ and the contraction mapping, ensures invertibility and enables precise control of the correction. The main results prove the existence of $L$-periodic bubbling solutions in the strip and, in the Euclidean setting, infinitely many $L$-periodic solutions under weaker curvature conditions than previous work, including periodicity in intermediate variables rather than only the first few coordinates. These findings advance the understanding of Grushin-type problems with critical growth and periodic structures.
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.