In this paper we study the following critical anisotropic p-Laplace equation on infinite strip-like domains \(\begin{aligned} -\Delta _{p}^{H}u=\lambda |u|^{q-2}u+|u|^{p^{*}-2}u \ \ \ {\text{ in }} \ W^{1,p}_{0}(\Omega ), \end{aligned}\) where \(\Omega =\omega \times \mathbb {R}^{N-m}\) with \(1\le m<N\) , \(\omega \subset \mathbb {R}^{m}\) is an open bounded Lipschitz set, \(N\ge p^{2}\) , \(\lambda >0\) , \(1<p\le q<p^{*}\) , \(p^{*}=\frac{Np}{N-p}\) denotes the Sobolev critical exponent, H is a Finsler norm. The purpose of this paper are twofold: first using the anisotropic Sobolev inequality proved by Figalli et al. [Symmetry results for critical anisotropic p-Laplacian equations in convex cones, Geom. Funct. Anal., 2020], we establish the existence of nonnegative least energy solutions to the above equation through variational methods. Then by exploiting the Moser iteration technique and Brézis-Kato approach, we prove that for a suitable range of exponent q, these nonnegative weak solutions are in \(L^{\infty }(\Omega )\) .
