Maximum principles, Liouville-type theorems and symmetry results for a general class of quasilinear anisotropic equations (Q350314)

The authors deal with a general class of quasilinear anisotropic equations. The general framework is the following: Let \(F:\mathbb{R}^{N}\to[0,\infty)\), \(N\geq2\), be a positive homogeneous function of degree 1, with the following properties: NEWLINE\[NEWLINE\begin{cases} F\in C_{\mathrm{loc}}^{3,\alpha}(\mathbb{R}^{N}\setminus\{0\}),\;\alpha\in(0,1),\\ F(\xi)>0 \text{ for any } \xi\in \mathbb{R}^{N}\setminus\{0\},\\ F(0)=0. \end{cases}NEWLINE\]NEWLINE Let \(G\in C_{\mathrm{loc}}^{3,\alpha}(0,\infty)\cap C^1[0,\infty)\), be a function such that NEWLINE\[NEWLINE\begin{cases} G(0)=0,\;G'(0)=0,\\ G(t)>0,\;G'(t)>0,\;G''(t)>0 \text{ for any } t>0. \end{cases}NEWLINE\]NEWLINE The authors also assume that either of the following conditions is satisfied: {\parindent=8mm \begin{itemize}\item[(I)] There exists \(p>1\), \(k\in[0,1)\), \(\gamma>0\) and \(\Gamma>0\), such that, for any \(\xi\in \mathbb{R}^{N}\setminus\{0\}\), \(\zeta\in \mathbb{R}^{N}\) NEWLINE\[NEWLINE[\mathrm{Hess}(G\circ F)(\xi)]_{ij}\zeta_{i}\zeta_{j}\geq\gamma(k+|\xi|)^{p-2}|\zeta|^2,NEWLINE\]NEWLINE and NEWLINE\[NEWLINE\sum_{i,j=1}^{N}|[\mathrm{Hess}(G\circ F)(\xi)]_{ij}|\leq\gamma(k+|\xi|)^{p-2}.NEWLINE\]NEWLINE \item[(II)] The composition \(G\circ F\) is of class \(C_{\mathrm{loc}}^{3,\alpha}(\mathbb{R}^{N})\), and for any \(R>0\) there exists a positive constant \(\rho\) such that for any \(\xi,\zeta\in \mathbb{R}^{N}\) with \(|\xi|\leq R\) we have NEWLINE\[NEWLINE[\mathrm{Hess}(G\circ F)(\xi)]_{ij}\zeta_{i}\zeta_{j}\geq\rho|\zeta|^2.NEWLINE\]NEWLINENEWLINENEWLINE\end{itemize}} They also consider the following quasilinear anisotropic operator NEWLINE\[NEWLINEQ(u):=\sum_{i=1}^{N}(\partial/(\partial x_{i}))(G'(F(\nabla u))F_{\xi_{i}}(\nabla u)).NEWLINE\]NEWLINE The first main result of the paper (Theorem 1.1.) is a Liouville-type theorem for the solutions of the equation NEWLINE\[NEWLINEQu+f(u)=0\text{ on }\mathbb{R}^{N},NEWLINE\]NEWLINE where \(f\in C_{\mathrm{loc}}^{1,\alpha}(\mathbb{R})\). The next two results are a Serrin-type symmetry results (Theorem 1.2.) and a uniqueness result (Theorem 1.3.) for the following overdetermined anisotropic boundary value problem NEWLINE\[NEWLINE\begin{cases} Qu=-1 \text{ in }\Omega,\\ u=0 \text{ on }\partial\Omega,\\ F(\nabla u)=c \text{ on }\partial\Omega, \end{cases}NEWLINE\]NEWLINE where \(\Omega\subset \mathbb{R}^{N}\) is a connected, bounded open set, and \(c>0\) is a given constant.NEWLINENEWLINEThe outline of the paper is as follows. The proofs of Theorems 1.1 and 1.2 rely on some key ingredients involving P-functions, which are presented in Section 2. In Section 3, the authors make use of some maximum principle for P-functions, previously established in Section 2, to prove the Liouville-type result stated in Theorem 1.1. In Section 4 they derive a Pohozaev-type integral identity and give the proofs of Theorems 1.2 and 1.3.