We are concerned with the existence of solution of the problem where \(\Delta ^H_pu=\hbox {div }(a(\nabla u))\) , with \(a(\xi )=H^{p-1}(\xi )\nabla H(\xi ),\, \xi \in \mathbb {R}^N,\) \(N\geqslant 3,\) is the anisotropic p-Laplacian with \(1<p<N\) , \(\lambda >0\) is a parameter, and \(p< q<p^*=pN/(N-p)\) . Further, \(\Omega \) is a \(C^1\) 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 \(C^{1,\alpha }(\Omega )\) 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 result we prove which assures that any \(W^{1,p}\) -solution of critical Neumann problems with the anisotropic p-Laplacian operator on bounded Lipschitz domains in \(\mathbb {R}^N\) \((N\geqslant 3)\) is bounded.
