Global Sobolev inequalities and degenerate p-Laplacian equations

We prove that a local, weak Sobolev inequality implies a global Sobolev estimate using existence and regularity results for a family of p -Laplacian equations. Given Q C Rn, let p be a quasi-metric on Q, and let Q be an n x n positive semi-definite matrix function defined on Q. For an open set 0 C Q, we give sufficient conditions to show that if the local weak Sobolev inequality (integral vertical bar f vertical bar pq dx)1/p sigma <= C[r(B)integral(B) vertical bar root A del f|pdx + integral|f|p dx]1/p holds for some a > 1, all balls B C 0, and functions f E Lip0(0), then the global Sobolev inequality also holds. Central to our proof is showing the existence and boundedness of solutions of the Dirichlet problem where Xp,r is a degenerate p -Laplacian operator with a zero order term: Xp,tau u = div(|rQV|(P-2)QVu) u. (C) 2019 Elsevier Inc. All rights reserved.