A critical neumann problem with anisotropic p-laplacian

We are concerned with the existence of solution of the problem $-\Delta ^H_pu+|u|^{p-2}u=\lambda|u|^{q-2}u+ |u|^{p^*-2}u\quad \mbox{in}\quad\Omega,$ $u>0\quad \mbox{in}\quad\Omega,$ $a(\nabla u)\cdot \nu =0\quad \mbox{on}\quad\partial \Omega,$ where $\Delta ^H_pu=\mbox{div\,}(a(\nabla u))$, with $a(\xi)=H^{p-1}(\xi)\nabla H(\xi),\, \xi \in \mathbb{R}^N,$ $N\geqslant3,$ is the anisotropic $p$-Laplacian with $1<p<N$, $\lambda>0$ is a parameter, and $p < q<p^*=pN/(N-p)$. Further, $\Omega \subset \Sigma$ is a $C^1$ bounded domain inside a convex open cone $\Sigma$ in $\mathbb{R}^N$ with $\partial \Omega \cap \partial \Sigma$ being a $C^1$-manifold, and $\nu$ is the unit outward normal to $\partial \Omega$. 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 $C^{1,\alpha}(\Omega)$, and is strictly positive in $\Omega$. Such conclusions are achieved from classical elliptic regularity theory and a Harnack inequality, since the solution of $(P)$ is bounded. This in turn is a consequence of a result in this paper which ensures that any $W^{1,p}$-solution of critical Neumann problems with the anisotropic $p$-Laplacian operator on bounded Lipschitz domains in $\mathbb{R}^N$ $(N\geqslant3)$ is bounded.