Pohozaev identity for Finsler anisotropic problems

Abstract

In this paper we derive the Pohozaev identity for quasilinear equations -div(B′(H(∇u))∇H(∇u))=g(x,u)inΩ,(E)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\begin{aligned} -{\ ext {div}}(B'(H(\ abla u))\ abla H(\ abla u))=g(x, u) \\quad \ ext{ in }\\,\\, \\Omega , \\quad \\quad {(E)} \\end{aligned}$$\\end{document}involving the anisotropic Finsler operator -{text {div}}(B'(H(nabla u))nabla H(nabla u)). In particular, by means of fine regularity results on the vectorial field B'(H(nabla u))nabla H(nabla u), we prove the identity for weak solutions and in a direct way.