Partial regularity and Liouville theorems for stable solutions of anisotropic elliptic equations

We study the quasilinear elliptic equation <tex-math id="FE1"> {document}equation* -Qu = eᵘ ~~~~in~~~~ Ω⊂ RN, equation*{document} </tex-math> where the operator <tex-math id="M1">{document}Q{document}</tex-math>, known as the Finsler-Laplacian (or anisotropic Laplacian) operator, is defined by <tex-math id="FE2"> {document}Qu: = ∑i = 1N∂∂ xᵢ(F(∇ u)F_ξᵢ(∇ u)).{document} </tex-math> Here <tex-math id="M2">{document}F_ξᵢ = ∂ F∂ξᵢ{document}</tex-math> and <tex-math id="M3">{document}F: RN→[0, +∞){document}</tex-math> is a convex function of <tex-math id="M4">{document}C²(RN\0\){document}</tex-math> that satisfies certain assumptions. For a bounded domain <tex-math id="M5">{document}Ω{document}</tex-math> and for a stable weak solution of the above equation, we prove that the Hausdorff dimension of singular set does not exceed <tex-math id="M6">{document}$ N-10 ${document}</tex-math>. For the case of entire space, we apply Moser iteration arguments, established by Dancer-Farina and Crandall-Rabinowitz in the context, to prove Liouville theorems for stable solutions and for finite Morse index solutions in dimensions <tex-math id="M7">{document}$ N<10 ${document}</tex-math> and <tex-math id="M8">{document}$ 2<N<10 ${document}</tex-math>, respectively. We also provide an explicit solution that is stable outside a compact set in two dimensions <tex-math id="M9">{document}$ N = 2 ${document}</tex-math>. In addition, we present similar Liouville theorems for the related equations with power-type nonlinearities.