Liouville Theorems for Stable Solutions to Anisotropic Quasilinear Equations

Source abstract

We consider possibly unbounded and sign-changing stable weak solutions of −ΔmHu+|x|b|u|q−1u=|x|a|u|p−1u in Rn, where m≥2, 1≤q≤m−1<p, and H is a uniformly convex, positively one-homogeneous function. We prove Liouville theorems under explicit dimensional conditions and under integral, pointwise, or gradient growth assumptions. When H(ξ)=|ξ|, the results reduce to the corresponding weighted m-Laplace conclusions.