A Critical Neumann problem with anisotropic p-Laplacian

Abstract We are concerned with the existence of solution of the problem where Hₚu= div (a (u) ) Δ p H u = div (a (∇ u) ), with a () =H^p-1 () H (), \, RN, a (ξ) = H p - 1 (ξ) ∇ H (ξ), ξ ∈ R N, N 3, N ⩾ 3, is the anisotropic p -Laplacian with 1 1 p N, >0 λ > 0 is a parameter, and p p q p ∗ = p N / (N - p). Further, Ω is a C¹ C 1 bounded domain inside a convex open cone. To succeed with a variational approach, where the strong convergence of a bounded (PS) subsequence needs to be proved, one has to deal with anisotropic norms in the absence of a Tartar’s type inequality, unlike the isotropic p -Laplace case. This is overcome by proving the a. e. convergence of its gradients. Furthermore, the solution of (P) is shown to belong to C^1, () C 1, α (Ω) from classical elliptic regularity theory, and is positive from a Harnack inequality, since any solution of (P) is bounded. This in turn is a consequence of a resu