A critical neumann problem with anisotropic p-laplacian

We are concerned with the existence of solution of the problem $-{\mathrm{\Delta }}_p^{H}u+\mathrm{\mid }u{\mathrm{\mid }}^{p-2}u=\lambda \mathrm{\mid }u{\mathrm{\mid }}^{q-2}u+\mathrm{\mid }u{\mathrm{\mid }}^{p^{\ast }-2}uin\mathrm{\Omega },$ $u>0in\mathrm{\Omega },$ $a(\mathrm{\nabla }u)\cdot \nu =0on\mathrm{\partial }\mathrm{\Omega },$ where ${\mathrm{\Delta }}_p^{H}u=div(a(\mathrm{\nabla }u))$, with $a(\xi )=H^{p-1}(\xi )\mathrm{\nabla }H(\xi ),\xi \in {\mathbb{R}}^N,$ $N⩾3,$ is the anisotropic $p$-Laplacian with $1<p<N$, $\lambda >0$ is a parameter, and $p<q<p^{\ast }=pN\mathrm{/}(N-p)$. Further, $\mathrm{\Omega }\subset \mathrm{\Sigma }$ is a $C^1$ bounded domain inside a convex open cone $\mathrm{\Sigma }$ in ${\mathbb{R}}^N$ with $\mathrm{\partial }\mathrm{\Omega }\cap \mathrm{\partial }\mathrm{\Sigma }$ being a $C^1$-manifold, and $\nu$ is the unit outward normal to $\mathrm{\partial }\mathrm{\Omega }$. 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 $C^{1,\alpha }(\mathrm{\Omega })$, and is strictly positive in $\mathrm{\Omega }$. 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 ${\mathbb{R}}^N$ $(N⩾3)$ is bounded.