A critical neumann problem with anisotropic p-laplacian
We are concerned with the existence of solution of the problem $ -Δ^H_pu+|u|^{p-2}u=λ|u|^{q-2}u+ |u|^{p^*-2}u\quad \mbox{in}\quadΩ,$ $u>0\quad \mbox{in}\quadΩ,$ $a(\nabla u)\cdot ν=0\quad \mbox{on}\quad\partial Ω,$ where $Δ^H_pu=\mbox{div\,}(a(\nabla u))$, with $a(ξ)=H^{p-1}(ξ)\nabla H(ξ),\, ξ\in \mathbb{R}^N,$ $N\geqslant3,$ is the anisotropic $p$-Laplacian with $1 0$ is a parameter, and $p < q<p^*=pN/(N-p)$. Further, $Ω\subset Σ$ is a $C^1$ bounded domain inside a convex open cone $Σ$ in $\mathbb{R}^N$ with $\partial Ω\cap \partial Σ$ being a $C^1$-manifold, and $ν$ is the unit outward normal to $\partial Ω$. 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 $C^{1,α}(Ω)$, and is strictly positive in $Ω$. 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.