A Critical Neumann problem with anisotropic $$p-$$Laplacian

We are concerned with the existence of solution of the problem where Δ Hₚu=div (a(∇ u)) Δ p H u = div ( a ( ∇ u ) ) , with a(ξ )=Hᵖ⁻¹(ξ )∇ H(ξ ),\, ξ ∈ RN, a ( ξ ) = H p - 1 ( ξ ) ∇ H ( ξ ) , ξ ∈ R N , N 3, N ⩾ 3 , is the anisotropic p -Laplacian with $$1<p<N$$ 1 < p < N , λ >0 λ > 0 is a parameter, and p< q<p^*=pN/(N-p) p < q < p ∗ = p N / ( N - p ) . Further, Ω 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 C1,α(Ω ) 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