\documentclass[reqno]{amsart} \usepackage{hyperref} \AtBeginDocument{{\noindent\small Two nonlinear days in Urbino 2017,\newline \emph{Electronic Journal of Differential Equations}, Conference 25 (2018), pp. 167--178.\newline ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu} \thanks{\copyright 2018 Texas State University.} \vspace{8mm}} \begin{document} \setcounter{page}{167} \title[\hfilneg EJDE-2018/conf/25\hfil $\Delta_\lambda$-laplacians] {Linear and semilinear problems involving $\Delta_\lambda$-laplacians} \author[A. E. Kogoj, E. Lanconelli \hfil EJDE-2018/conf/25\hfilneg] {Alessia E. Kogoj, Ermanno Lanconelli} \address{Alessia E. Kogoj \newline Dipartimento di Scienze Pure e Applicate (DiSPeA), Universit\`a degli Studi di Urbino ``Carlo Bo'', Piazza della Repubblica, 13 - 61029 Urbino (PU), Italy} \email{alessia.kogoj@uniurb.it} \address{Ermanno Lanconelli \newline Dipartimento di Matematica, Universit\`a degli Studi di Bologna, Piazza di Porta San Donato, 5 - 40126 Bologna, Italy} \email{ermanno.lanconelli@unibo.it} \thanks{Published September 15, 2018} \subjclass[2010]{35J70, 35H20, 35K65} \keywords{Degenerate elliptic PDE; semilinear subelliptic PDE; $\Delta_\lambda$-Laplacian} \begin{abstract} In recent years a growing attention has been devoted to $\Delta_\lambda$-Laplacians, linear second-order degenerate elliptic PDO's contained in the general class introduced by Franchi and Lanconelli in some papers dated 1983--84 \cite{franchi_lanconelli_1, franchi_lanconelli_3, franchi_lanconelli_2}. Here we present a survey on several results appeared in literature in the previous decades, mainly regarding: (i) Geometric and functional analysis frameworks for the $\Delta_\lambda$'s; (ii) regularity and pointwise estimates for the solutions to $\Delta_\lambda u =0$; (iii) Liouville theorems for entire solutions; (iv) Pohozaev identities for semilinear equations involving $\Delta_\lambda$-Laplacians; (v) Hardy inequalities; (vi) global attractors for the parabolic and damped hyperbolic counterparts of the $\Delta_\lambda$'s. We also show several typical examples of $\Delta_\lambda$-Laplacians, stressing that their class contains, as very particular examples, the celebrated Baouendi-Grushin operators as well as the $L_{\alpha, \beta}$ and $P_{\alpha, \beta}$ operators respectively introduced by Thuy and Tri in 2002 \cite{thuy_tri_2002} and by Thuy and Tri in 2012 \cite{thuy_tri_2012}. \end{abstract} \dedicatory{Dedicated to Anna Aloe} \maketitle \numberwithin{equation}{section} \newtheorem{theorem}{Theorem}[section] \newtheorem{remark}[theorem]{Remark} \newtheorem{example}[theorem]{Example} \allowdisplaybreaks \section{Introduction} \subsection{$\Delta_\lambda$-operators} In $\mathbb{R}^N$, whose point will be denoted by $x=(x_1, \dots, x_N)$, let us consider a $n$-tuple $\lambda:=(\lambda_1, \dots, \lambda_N)$ of real functions $$ \lambda_j: \mathbb{R}^N\to \mathbb{R}, \quad j=1,\dots, N, $$ such that $\lambda_1=1$ and $\lambda_j(x)=\lambda_j(x_1,\dots,x_{j-1})$ for $j\geq 2$. Define the linear second order partial differential operators $\Delta_\lambda$ as follows: \begin{equation}\label{deltalambda} \Delta_\lambda:=\sum_{j=1}^N \lambda_j \partial_{x_j}(\lambda_j \partial_{x_j}) =\sum_{j=1}^N \lambda^2_j\partial^2_{x_j}. \end{equation} \subsection{}\label{b} If the $\lambda_j$'s are non-identically zero polynomial functions then $\Delta_\lambda$ is hypoelliptic, i.e., every distributional solution $u$ to the equation $$ \Delta_\lambda u= f $$ in an open set $\Omega\subseteq\mathbb{R}^N$, is actually of class $C^\infty$ in $\Omega$ if $f$ is $C^\infty$ in $\Omega$. This is an easy consequence of the celebrated H\"ormander theorem on the hypoellipticity of the ``sum of squares of vector fields'' \cite{hormander}. Indeed, let \begin{equation}\label{liealgebra} \mathfrak{a}:= \operatorname{Lie} \{ \lambda_1 \partial_{x_1}, \dots, \lambda_N \partial_{x_N}\}. \end{equation} Then, $ \lambda_1 \partial_{x_1}= \partial_{x_1} \in \mathfrak{a}$. Moreover, if $j\geq 2$, being $\lambda_j$ a non zero polynomial function, there exists a multi-index $\beta^{(j)}$ such that $D^{\beta^{(j)}}\lambda_j =c_j$, with $c_j$ non zero real constant. This easily implies that $\partial_{x_2}$, $\partial_{x_3}, \dots, \partial_{x_N} \in \mathfrak{a}$. Hence $$ \operatorname{rank}\mathfrak{a}(x)=N \quad \forall x\in \mathbb{R}^N, $$ so that, by the H\"ormander theorem, $\Delta_\lambda$ is hypoelliptic. Celebrated typical examples of $\Delta_\lambda$ hypoelliptic PDO's with polynomial coefficients are the Baouendi-Grushin operators \cite{baouendi_1967, grushin_1970,grushin_1971}, \begin{equation}\label{grushin} \mathcal{L}_{m,p}= \partial^2_{x_1} + \dots + \partial^2_{x_p} + (x_1^2+\dots + x_p^2)^{2m} (\partial^2_{x_{p+1}} + \dots + \partial^2_{x_N}), \end{equation} $m,p\in \mathbb{Z},\ m\geq 0, 1\le p< N$, corresponding to the case $$ \lambda_1(x) =\dots=\lambda_p (x)=1,\quad \lambda_{p+1}(x)=\dots=\lambda_N(x)=(x_1^2+\dots +x_p^2)^m, $$ and the Baouendi-Goulaouic operator \begin{equation}\label{baouendigoulaouic} \mathcal{L}_{2}= \partial^2_{x_1} +\partial^2_{x_2} + x_1^2 \partial^2_{x_3} \quad (\text{in } \mathbb{R}^3), \end{equation} corresponding to the case $$ \lambda_1(x) =\lambda_2(x) =1,\quad \lambda_{3}(x)=x_1. $$ The Baouendi-Goulaouic operator was the first example appeared in literature of $C^\infty$-hypoelliptic ``sum of squares'' operator which is not analytic-hypoelliptic (see \cite{baouendi_goulaouic_1972}). \subsection{} If the $\lambda_j$'s are (merely) smooth functions, a condition making $\Delta_\lambda$ hypoelliptic in $\mathbb{R}^N$ is the following one: \begin{quote} For every $x\in\mathbb{R}^N$ and for every $j\in \{1,\dots, N\}$, there exists a multi-index $\beta$ depending on $x$ and $j$ such that \begin{equation} \label{condizioneipo} D^\beta \lambda_j(x)\neq 0. \end{equation} \end{quote} If we let, for every fixed $x\in\mathbb{R}$, $$ \mathfrak{a}(x)= \{ X(x) \ | \ X \in \mathfrak{a}\}, $$ where $\mathfrak{a}$ is the Lie algebra in \eqref{liealgebra}, then, trivially, $\partial_{x_1} \in \mathfrak{a}(x)$. Moreover, using condition \eqref{condizioneipo}, we can prove that for every $x=(x_1,\dots, x_N)\in \mathbb{R}^N$ there exist smooth functions $$ a_2(x)=a_2(x_1),\ a_3(x)=a_3(x_1,x_2),\dots, a_N(x)=a_N(x_1,\dots, x_{N-1}), $$ such that $a_j(x)\neq 0$ and $a_j\partial_{x_j} \in \mathfrak{a}$, $j=2,\dots, N$. Then $$ N\geq \operatorname{rank} \mathfrak{a}(x)\geq \dim \operatorname{span} \{ \partial_{x_1}, a_1(x)\partial_{x_2}, \dots, a_n(x)\partial_{x_N}\} =N, $$ hence $ \operatorname{rank} \mathfrak{a}(x)=N$. Since $x$ is an arbitrary point of $\mathbb{R}^N$, this proves that $\Delta_\lambda$ satisfies the H\"ormander rank condition, so that is hypoelliptic. If the $\lambda_j$'s are real analytic, condition \eqref{condizioneipo} is equivalent to say that $$ \lambda_j \not\equiv 0\quad \forall j=1,\dots, N. $$ Thus, the hypoellipticity result of subsection \ref{b} can be improved as follows: \begin{quote} If the $\lambda_1, \dots, \lambda_N$ are real analytic functions then $\Delta_\lambda$ is hypoelliptic if (and only if) $$ \lambda_j \not\equiv 0\quad \forall j=1,\dots, N. $$ \end{quote} \subsection{} In some papers dated 1982--1984 (\cite{franchi_lanconelli_1, franchi_lanconelli_3, franchi_lanconelli_2}), Franchi and Lanconelli studied $\Delta_\lambda$-operators only assuming the $\lambda_j$'s locally Lipschitz continuous and of class $C^1$ out of the coordinate axes. Obviously, in such weak regularity assumptions, H\"ormander condition is meaningless. In \cite{franchi_lanconelli_1}, suitable condition only involving the first derivatives of the $\lambda_j$'s, are introduced, allowing to get a kind of De Giorgi-Moser theorem for $\Delta_\lambda$, i.e., the H\"older continuity and the Harnack inequality for the weak solutions. \section{De Giorgi-Moser-type theorem for $\Delta_\lambda$. Liouville-type theorems} Let us assume the $\lambda_j$'s satisfy the following hypotheses. \begin{itemize} \item[(H1)] $\lambda_1,\dots,\lambda_N$ are continuous and of class $C^1$ and strictly positive outside the coordinate hyperplanes; \item[(H2)] $\lambda_1(x)\equiv 1$, $\lambda_j(x)=\lambda_j(x_1,\dots,x_{j-1})$, $ j=2,\dots,N$; \item[(H3)] $\lambda_j(x)=\lambda_j(x^*)$, where $x^*=(|x_1|,\dots,|x_N|)$; \item[(H4)] there exists $\rho\geq 0$ such that $$ 0\leq x_k\partial_{x_k}\lambda_j(x)\leq\rho\lambda_j(x),\quad k=1,\dots,j-1, $$ for every $x\in\mathbb{R}_+^N :=\{(x_1,\dots,x_N)\in\mathbb{R}^N:x_i\geq 0\; \forall i=1,\dots,N\}$. \end{itemize} Under these hypotheses in \cite{franchi_lanconelli_1} a metric $d$ was constructed in $\mathbb{R}^N$ that plays for $\Delta_\lambda$ the same r\^ole as the Euclidean distance plays for the classical Laplacian. This metric, which actually is the Carnot-Carath\'eodory distance related to the vector fields $$ X_1=\lambda_1\partial_{x_1}, \dots, X_N=\lambda_1\partial_{x_N}, $$ is defined as follows. An absolutely continuous path $\gamma:[0,T] \to \mathbb{R}^N,\ T>0$, is $\lambda$-subunit if, letting $e_j=(0,\dots, \underset{j}{1},\dots, 0)$ for every $j=1,\dots, N$, we have \[ { \gamma'(t)= \sum_{j=1}^{N} c_j(t) \lambda_{j}(\gamma(t))e_j }\quad\text{a.e. in } [0,T], \text{ with }\sum_{j=1}^m c_j^2 (t) \le 1. \] In this case we put $l(\gamma):=T$ and for every $x,y\in\mathbb{R}^N$ we define \[ \mathcal{C} (x,y) := \{ \gamma \text{ $\lambda$-subunit path }: \gamma:[0,T]\to X,\, \gamma(0)=x,\, \gamma(T)=y\}. \] Note that Hypotheses (H1) and (H2) imply $\mathcal{C}(x,y)\neq \emptyset$ for all $x,y\in\mathbb{R}^N$. Then, letting \[ d(x,y):= \inf\{ l(\gamma): \gamma \in\mathcal{C} (x,y)\}, \] we have $d(x,y)<\infty$ for every $x,y\in\mathbb{R}^N$. It is easy to see that $(x,y)\mapsto d(x,y)$ is a {\it distance} in $\mathbb{R}^N$, which we call the {\it $\lambda$-distance}. In \cite{franchi_lanconelli_1} and \cite{franchi_lanconelli_2} it is proved that $(\mathbb{R}^N,d)$ is a {\it doubling metric space}, i.e., that there exists a positive constant $c_d$ such that \begin{equation}\label{doubling} |B_d (x,2r)|\le c_d |B_d(x,r)|\quad \forall x\in\mathbb{R}^N,\; \forall r>0, \end{equation} where $|\cdot |$ stands for the Lebesgue measure and $B_d (x,r)$ denotes the $d$-ball of center $x$ and radius $r$, $$ B_d (x,r) = \{ y\in\mathbb{R}^N\ |\,d(x,y)0$, $$ \int_{B_d (x,r)} |u-u_r|^2 \,dy \le Cr^2\int_{B_d(x,\theta r)} |\nabla_\lambda u|^2 \, dy\quad \forall u\in C^1 (\overline{B_d(x,\theta r)}) $$ where $C>0$ and $\theta>1$ are suitable constants independent of $u$, $x$ and $r$, and $u_r$ denotes the average of $u$ on $B_d(x,r)$: $$ u_r =: {1\over{B_d (x,r)}} \int_{B_d(x,r)} u(y)\,dy $$ (see \cite{franchi_lanconelli_2,lanconelli_morbidelli}). To complete the list of the key results needed to show a De Giorgi-type theorem for $\Delta_\lambda$, we recall the existence of global cut-off functions modelled on the geometry of the $d$-balls. More precisely, the following proposition holds: Let $B_d(x,r_1)$ and $B_d(x,r_2)$ be concentric $d$-balls with $00$ is independent of $u,z$ and $r$. \item[(ii)] (Local H\"older continuity) If $B_d(z,2r)\subseteq \Omega$, then \begin{equation}\label{holder} |u(x)-u(y)|\le C\Big( {{d(x,y)}\over r}\Big)^\alpha \sup_{B(z,2r)} |u| \quad \forall x,y \in B_d\Big(z,\frac{r}{2}\Big), \end{equation} where $C>0$ and $\alpha\in ]0,1[$ are independent of $u,z$ and $r$. \end{itemize} \end{theorem} Actually, the conclusions of this theorem hold true for the weak solutions of the {\it $\lambda$-elliptic operators}. A linear second order PDO of the kind $$ \mathcal{L}=\sum_{i,j=1}^N \partial_{x_i}(a_{ij} (x)\partial_{x_j}) = \mathrm{div} (A(x) D) $$ will be called $\lambda$-elliptic in $\mathbb{R}^N$ if the quadratic form related to the symmetric matrix $A(x)=a_{ij}(x))_{i,j=1,\dots,n}$ with measurable entries, satisfies $$ \frac{1}{c}\sum_{j=1}^n(\lambda_j(x)\xi_j)^2 \le \langle A(x)\xi,\xi \rangle \le c \sum_{j=1}^n(\lambda_j(x)\xi_j)^2\quad \forall x,\xi\in\mathbb{R}^N. $$ If $\Omega$ is an open subset of $\mathbb{R}^N$ we say that $u\in W_{\lambda,\mathrm{loc}}^{1,2}(\Omega)$ if, for every $\varphi\in C_0^\infty(\Omega,\mathbb{R})$ one has $u\varphi \in \mathring{W}_{\lambda}^{1,2}(\Omega)$. To define the notion of weak solution to the equation $\mathcal{L} u=0$, we need to introduce the bilinear form $$ \mathcal{L} (u,v)=\int_\Omega \langle A(x) D u (x), D v (x) \rangle\,dx $$ for $u\in C^1(\Omega, \mathbb{R})$ and $v\in C_0^1(\Omega, \mathbb{R})$. $D$ is the Euclidean gradient $D=(\partial_{x_1}, \dots, \partial_{x_N})$. Since $A\geq 0$, we have (because $\mathcal{L}$ is $\lambda$-elliptic) \begin{align*} |\mathcal{L}(u,v)| &\le \int_\Omega \langle A(x) D u(x), D u(x)\rangle^{\frac{1}{2}} \langle A(x) D v(x), D v(x)\rangle^{\frac{1}{2}} \,dx \\ &\le c \int_\Omega |\nabla_\lambda u(x)| \ |\nabla_\lambda v(x)| \,dx\,. \end{align*} Then the bilinear form $a$ is well defined and, if $\Omega$ is bounded, it can be continuously extended to $W_{\lambda,\mathrm{loc}}^{1,2}(\Omega)\times \mathring W_{\lambda}^{1,2}(\Omega)$. A function $u\in W_{\lambda,\mathrm{loc}}^{1,2}(\Omega)$ is a weak solution to $\mathcal{L} u=0$ in $\Omega$ if $$ a(u,v)=0\quad \forall v\in C_0^1(\Omega,\mathbb{R}). $$ The Moser iteration procedure works for $\lambda$-elliptic operators as for $\Delta_\lambda$-operators. Then, De Giorgi-Moser Theorem \ref{degiorgi} extends to the weak solutions to $\mathcal{L} u=0$ for every $\lambda$-elliptic operator $\mathcal{L}$ (for the $\Delta_\lambda$-case, see \cite{franchi_lanconelli_1, franchi_lanconelli_3, franchi_lanconelli_2}, for the $\lambda$-elliptic case see \cite{Lanconelli_Kogoj_2000}). The invariant Harnack inequality \eqref{harnack} immediately leads to the following Liouville-type theorem. Here $\mathcal{L}$ stands for any $\lambda$-elliptic operator. \begin{theorem}\label{liouville} Let $u\in W_{\lambda,\mathrm{loc}}^{1,2}(\mathbb{R}^N)$ be a weak solution to $\mathcal{L} u=0$ in $\mathbb{R}^N$. If $u\geq 0$, then $u$ is identically constant in $\mathbb{R}^N$. \end{theorem} From the H\"older estimates \eqref{holder}, one obtains another Liouville-type theorem. \begin{theorem}\label{liouville2} Let $u\in W_{\lambda,\mathrm{loc}}^{1,2}(\mathbb{R}^N)$ be a weak solution to $\mathcal{L} u=0$ in $\mathbb{R}^N$. Assume that, for a suitable $x_0\in\mathbb{R}^N$, $$ \lim_{r\to \infty} \Big(\frac{1}{r^\alpha} \sup_{B(x_0,r)} |u| \Big)=0, $$ where $\alpha\in ]0,1[$ is the H\"older exponent in \eqref{holder}. Then, $u$ is identically constant in $\mathbb{R}^N$. \end{theorem} As a last theorem we would like to recall is a Colding-Minicozzi-type Liouville theorem for the $\lambda$-elliptic operators $\mathcal{L}$, which is proved in \cite{kogoj_lanconelli_2009}. \begin{theorem} Let $x_0$ be a fixed point of $\mathbb{R}^N$ and denote by $d(x)$ the $\lambda$-distance $d(x_0, x)$. Then, for every $m>0$, the linear space $$ \{ u\in W_{\lambda,\mathrm{loc}}^{1,2}(\mathbb{R}^N) : \mathcal{L} u=0 \text{ in } \mathbb{R}^N, \ u(x)=O(d(x))^m \text{ as } (d(x))\to \infty \} $$ has finite dimension. \end{theorem} We would like to close this subsection by quoting the recent paper \cite{anh_my_liouville_2016} by Anh and My where a Liouville-type theorem for system of semilinear inequalities involving $\Delta_\lambda$-operators is proved. \section{$\Delta_\lambda$-Laplacians} If the functions $\lambda_j$'s, together with hypotheses (H1), (H2), (H3) and (H4), are supposed to be homogeneous with respect to a fixed group of dilations in $\mathbb{R}^N$, the corresponding $\Delta_\lambda$-operators have been called in \cite{kogoj_lanconelli_2012} {\it $\Delta_\lambda$-Laplacians}, since they share some important homogeneity properties with the classical Laplacian. The corresponding geometry of the $\lambda$-distance achieves some crucial analogies with the Euclidean ones. Let $(\delta_r)_{r>0}$ be a group of dilations in $\mathbb{R}^N$ of the kind \begin{equation}\label{dilations} \delta_r:\mathbb{R}^N\to \mathbb{R}^N,\quad \delta_r(x) =\delta_r(x_1,\dots,x_N)=(r^{\varepsilon_1}x_1,\dots,r^{\varepsilon_N}x_N), \end{equation} where $1\leq\varepsilon_1\leq\varepsilon_2\leq\dots\leq\varepsilon_N$. Assume $\lambda_j$ is $\delta_r$-\emph{homogeneous of degree} $\varepsilon_j-1$, i.e., \begin{equation}\label{homogeneity} \lambda_j(\delta_r(x))=r^{\varepsilon_j-1}\lambda_j(x),\quad \forall x\in\mathbb{R}^N,\; r>0,\; j=1,\dots,N. \end{equation} Under this new assumption, $\Delta_\lambda$ becomes $\delta_r$-homogeneous of degree two, i.e., \[ \Delta_\lambda (u(\delta_r(x)))=r^2 (\Delta_\lambda u) (\delta_r(x))\quad \forall x\in\mathbb{R}^N,\quad \forall r>0, \] and for every $u\in C^\infty (\mathbb{R}^N)$. The positive real number $$ Q:=\varepsilon_1+\dots+\varepsilon_N $$ is the {\it homogeneous dimension} of $\mathbb{R}^N$ with respect to the group of dilations $(\delta_r)_{r>0}$. With respect to the Lebesque measure of the $\lambda$-balls and the Sobolev-type embedding Theorems, it plays the r\^ole of the dimension $N$ in the classical Laplacian case. Indeed, it works as the optimal exponent $Q$ in the inequality \eqref{uuu} and in the embedding \eqref{embedding}. In the present homogeneous assumption, precise estimates of both the $\lambda$-distance $d$ and the Lebesque measure of the $d$-balls are showed by Kogoj and Lanconelli in \cite{kogoj_lanconelli_2012}. A deep study of the $\lambda$-geometry for particular form of the $\lambda_j$'s have been recently performed by Wu in \cite{wu}. By crucially exploiting the homogeneity \eqref{homogeneity}, in \cite{kogoj_lanconelli_2012} the following Pohozaev-type identities are proved. We stress that the constant $Q$ which will appear in \eqref{p1}, \eqref{P2} and \eqref{pallino}, is exactly the homogeneous dimension of $\mathbb{R}^N$ with respect to $(\delta_r)_{r>0}$. \\ Let $T$ be linear first order PDO \[ T:\mathbb{R}^N\to\mathbb{R}^N,\quad T(x)=T(x_1,\dots,x_N)=\sum_{j=1}^N \varepsilon_j x_j \partial_{x_j}, \] i.e., the generator of dilation group $(\delta_r)_{r>0}$. Then, if $\Omega$ is a $C^1$ bounded open subset of $\mathbb{R}^N$, we have \begin{equation}\label{p1} \begin{aligned} &\int_{\Omega} T(u)\Delta_{\lambda} u\,dx \\ &= \int_{\partial\Omega} \langle \nabla_{\lambda} u, \nu_{\lambda}\rangle T(u)\,ds -\frac{1}{2} \int_{\partial\Omega} |\nabla_{\lambda} u|^{2} \langle T,\nu\rangle\,ds + \Big({Q\over 2}-1\Big) \int_{\Omega} |\nabla_{\lambda} u|^{2}\,dx \end{aligned} \end{equation} for every $u\in C^1(\overline\Omega),\mathbb{R})$. Here $\langle\cdot,\cdot\rangle$ stands for the Euclidean inner product, $\nu$ is the outward normal to $\Omega$ and $\nu_{\lambda}=(\lambda_{1}\nu_{1},\dots,\lambda_{N}\nu_{N})$. From this identity, we easily obtain an integral identity for the solutions to \begin{equation} \label{solutionto} \Delta_{\lambda} u+f(u)=0, \end{equation} $f:\mathbb{R} \to \mathbb{R}$ is a continuous function. We let $$ F(t):=\int_0^t f(s) ds,\quad t\in\mathbb{R}. $$ Then, if $u\in C^2(\overline\Omega)$ is a solution to \eqref{solutionto} the following identity holds \begin{equation}\label{P2} \begin{aligned} &\int_{\Omega} \Big(F(u) + \Big(\frac{1}{Q} - \frac{1}{2}\Big) u f(u)\Big)\,dx \\ &=\frac{1}{Q} \int_{\partial\Omega} \Big(\langle T,\nu\rangle \big(F(u)- \frac{1}{2} |\nabla_{\lambda}u|^{2}\big) + \langle \nabla_{\lambda}u,\nu\rangle\Big(T(u)+\Big({Q\over 2} -1\Big) u\Big) \Big)ds\,. \end{aligned} \end{equation} Moreover, if $u=0$ on $\partial\Omega$, \[ \int_{\Omega} \Big(F(u) + \Big(\frac{1}{Q} - \frac{1}{2}\Big) u\ f(u)\Big)\,dx =\frac{1}{2Q}\int_{\partial\Omega}\Big({{\partial u}\over{\partial\nu}}\Big)^{2} |\nu_{\lambda}|^{2}\langle T,\nu\rangle\,ds\,. \] Pohozaev-type identities for particular $\Delta_\lambda$-Laplacians were previously proved in \cite{tri_1998_on_grushin,tri_1998_sobolev,thuy_tri_2002, chuong_ke_2004,thuy_tri_2012}. If the domain $\Omega\subseteq \mathbb{R}^N$ is $(\delta_r)_{r>0}$ {\it starlike}, i.e., $$ \langle T,\nu\rangle \geq 0 \text{ at every point of } \partial\Omega, $$ and $C^1$ bounded open set, then the following non-existence result, extending to the $\Delta_\lambda$ a celebrated theorem by Pohozaev, holds. The problem \begin{equation}\label{pallino} \Delta_\lambda u + f(u)=0\quad\text{in } \Omega,\quad u|_{\partial\Omega} =0, \end{equation} has non trivial non-negative solution in $C^2(\overline\Omega)$ if \[ %\label{216} F(t) +\Big( \frac{1}{Q} - \frac{1}{2} \Big) t f(t) <0 \quad \forall t\neq 0. \] Thanks to the properties of the $\Delta_\lambda$'s previously recalled, the techniques of the variational theory of the critical points work equally well for the $\Delta_\lambda$-Laplacian as for the classical Laplacian. Many existence and non-existence results are today present in literature for semilinear $\Delta_\lambda$ boundary value problem, both in subcritical and critical behaviour assumption on the semilinear term $f(u)$ (see, e.g.,\cite{anh_my_2016, anh_my_2017, chen_tan_gao_2017, kogoj_lanconelli_2012, luyen_2017, luyen_tri_2015, luyen_tri_2018, thuy_tri_2002,thuy_tri_2012, tri_2009}). The homogeneity properties of the $\Delta_\lambda$-Laplacians have been also exploited in \cite{kogoj_sonner_hardy} to prove Hardy-type inequalities, which extend previous results by Garofalo and D'Ambrosio for the Baouendi-Grushin case \cite{DAm,garofalo_1993}. Before closing this section, we have to mention that initial value problems for evolution equations modelled on $\Delta_\lambda$-Laplacians have been studied in these last years. In \cite{primo_attractor_grushin} Anh, Hung, Ke and Phong have proved the existence of the global attractor for semilinear parabolic equations involving Baouendi-Grushin-type operators. Kogoj and Sonner have extended this result for $\Delta_\lambda$-Laplacians (and showed the finite fractal dimension of the attractor) in \cite{kogoj_sonner_2013} and for more general degenerate parabolic equations in \cite{kogoj_sonner_2014}. We stress that in this last paper semilinear damped hyperbolic equations involving $\Delta_\lambda$-Laplacians are also considered. Extensions to the critical cases of the results in \cite{kogoj_sonner_2013} and in \cite{kogoj_sonner_2014} have been proved in \cite{li_sun_2016,li_sun_chang_2017}. We also quote the papers \cite{anh_2014, luyen_tri_2016_sibirsk, luyen_tri_2016_grushin, quyet_thuy_nguyen_2017, thao_2016,Thuy_Tri_2013} where evolution equations related to classes of $\Delta_\lambda$ operators are studied. \section{Examples of $\Delta_\lambda$-Laplacians}\label{example} The following examples are taken from \cite{kogoj_sonner_2013}. We split $\mathbb{R}^N$ as $\mathbb{R}^N=\mathbb{R}^{N_1}\times\dots\times \mathbb{R}^{N_k}$, and write \[ x=\big(x^{(1)}, \dots, x^{(k)}\big),\quad x^{(i)}= \big(x^{(i)}_1,\dots,x^{(i)}_{N_i}\big)\in \mathbb{R}^{N_i},\quad i=1,\dots,k. \] We denote the classical Laplace operator in $\mathbb{R}^{N_i}$ by \[ \Delta_{x^{(i)}} = \sum_{j=1}^{N_i} \partial_{x^{(i)}_j}^2, \] and we write $\Delta_\lambda$ operators in the form \[ \Delta_\lambda = (\lambda^{(1)})^2\Delta_{x^{(1)}} +\dots +(\lambda^{(k)})^2 \Delta_{x^{(k)}} \quad \text{in } \mathbb{R}^N=\mathbb{R}^{N_1}\times\dots\times \mathbb{R}^{N_k}, \] where \[ \lambda=\big(\lambda^{(1)},\dots,\lambda^{(k)}\big),\quad \lambda^{(i)}=\big(\lambda^{(i)}_1,\dots,\lambda^{(i)}_{N_i}\big), \] and the functions $\lambda^{(i)}$ are continuous in $\mathbb{R}^{N_i}$, $i=1,\dots,k$. \begin{example} \rm Let $\alpha$ be a real positive constant and $k=2$. We consider at first the Baouendi-Grushin-type operator \[ \Delta_\lambda=\Delta_{x^{(1)}}+ |x^{(1)}|^{2\alpha} \Delta_{x^{(2)}}, \] where $\lambda=(\lambda^{(1)},\lambda^{(2)})$, with $\lambda^{(1)}_j(x)=1, j=1,\dots,N_1$ and $\lambda^{(2)}_j(x) = |x^{(1)}|^{\alpha}, j=1,\dots, N_2$. A group of dilations making $\Delta_\lambda$ homogeneous of degree two is $(\delta_r)_{r>0}$ with \[ \delta_r\Big(x^{(1)},x^{(2)}\Big)=\Big(r x^{(1)}, r^{\alpha+1} x^{(2)}\Big). \] In this case the homogenous dimension of $\mathbb{R}^N$ with respect to $(\delta_r)_{r>0}$ is $$ Q=N_1 + (\alpha+1)N_2. $$ More generally, for a given multi-index $\alpha=(\alpha_1,\dots,\alpha_{k-1})$ with real constants $\alpha_j\geq 0$, $j=1,\dots,k-1$, we define \[ \Delta_\lambda = \Delta_{x^{(1)}} + |x^{(1)}|^{2\alpha_1} \Delta_{x^{(2)}} +\dots+ |x^{(k-1)}|^{2\alpha_{k-1}} \Delta_{x^{(k)}}. \] Then, in our notation $\lambda=\big(\lambda^{(1)},\dots,\lambda^{(k)}\big)$ with \begin{gather*} \lambda_j^{(1)} (x) \equiv 1,\quad j=1,\dots,N_1 \\ \lambda_j^{(i)} (x) = |x^{(i-1)}|^{\alpha_{i-1}} \quad i=2,\dots, k, \quad j=1,\dots,N_i, \end{gather*} and the group of dilations such that $\lambda$ satisfies \eqref{homogeneity} is given by \[ \delta_r\big(x^{(1)}, \dots,x^{(k)}\big) =\big(r^{\varepsilon_1} x^{(1)},\dots, r^{\varepsilon_k} x^{(k)}\big), \] with $\varepsilon_1 =1$ and $\varepsilon_i =\alpha_{i-1} \varepsilon_{i-1} +1$ for $i=2,\dots,k$. In particular, if $\alpha_1=\dots=\alpha_{k-1} =\alpha$, the dilations become \[ \delta_r \Big(x^{(1)}, \dots, x^{(k)}\Big) = \Big( r x^{(1)}, r^{1 +\alpha} x^{(2)},\dots, r^{1+ \alpha + \alpha^2+ \dots + \alpha^{k-1}} x^{(k)}\Big). \] \end{example} \begin{remark} \rm A trivial change of variable makes the operator $$ \Delta_{x^{(1)}} +\frac{1}{4} | x^{(1)}|^2 \Delta_{x^{(2)}} $$ a $\Delta_\lambda$-Laplacian in $\mathbb{R}^{N_1}\times\mathbb{R}^{N_2}$ of the previous type. Moreover, if the dimensions $N_1$ and $N_2$ satisfy the inequality $N_2 <\rho (N_1)$, where $\rho$ is the so called Hurwitz-Radon function, then there exists a composition law $\circ$ in $\mathbb{R}^N$ such that $\mathbb{H}_N := (\mathbb{R}^N, \circ, \delta_\lambda)$ is a group of Heinsenberg type (see \cite[Remark 3.6.7]{BLU}, ) and, denoting by $\Delta_{\mathbb{H}_N}$ the canonical sub-Laplacian on $\mathbb{H}_N$, we have $$ \Big( \Delta_{x^{(1)}} + \frac{1}{4} | x^{(1)}|^2 \Delta_{x^{(2)}}\Big) u =\Delta_{\mathbb{H}_N} u $$ for every smooth function $u:\mathbb{R}^N\to\mathbb{R}$ which is radially symmetric in the variable $x^{(1)}$ (see \cite[p. 251]{BLU}). \end{remark} \begin{example} \rm Let $\alpha, \beta$ and $\gamma$ be nonnegative real constants. We consider the operator \[ \Delta_\lambda =\Delta_{x^{(1)}} + |x^{(1)}|^{2\alpha} \Delta_{x^{(2)}} + |x^{(1)}|^{2\beta} |x^{(2)}|^{2\gamma} \Delta_{x^{(3)}}, \] where $\lambda= (\lambda^{(1)},\lambda^{(2)},\lambda^{(3)})$ with \begin{gather*} \lambda_j^{(1)} (x) \equiv 1,\quad j=1,\dots,N_1 \\ \lambda_j^{(2)} (x) = |x^{(1)}|^{\alpha}, \quad j=1,\dots,N_2, \\ \lambda_j^{(3)}(x) = |x^{(1)}|^{\beta}|x^{(2)}|^{\gamma}, \quad j=1,\dots,N_3. \end{gather*} The dilations become \[ \delta_r\Big(x^{(1)},x^{(2)},x^{(3)}\Big) =\Big( r x^{(1)}, r^{\alpha+1} x^{(2)}, r^{\beta + (\alpha +1)\gamma +1} x^{(3)}\Big). \] Similarly, for operators of the form \begin{align*} \Delta_\lambda &= \Delta_{x^{(1)}} + |x^{(1)}|^{2\alpha_{1,1}} \Delta_{x^{(2)}}+ |x^{(1)}|^{2\alpha_{2,1}}|x^{(2)}|^{2\alpha_{2,2}} \Delta_{x^{(3)}} +\dots \\ &\quad + \Big( \prod_{i=1}^{k-1} |x^{(i)}|^{2\alpha_{k-1,i}}\Big)\Delta_{x^{(k)}}, \end{align*} where $\alpha_{i,j}\geq 0$, $i=1,\dots,k-1, j=1,\dots, i$, are real constants, the group of dilations is given by \[ \delta_r \Big(x^{(1)}, \dots,x^{(k)}\Big) =\Big(r^{\varepsilon_1} x^{(1)},\dots, r^{\varepsilon_k} x^{(k)}\Big) \] with $\varepsilon_1 =1$ and $\varepsilon_j =1+\sum_{i=1}^{j-1}\alpha_{j-1,i} \varepsilon_{i}$, for $i=2,\dots,k$. In particular, if $\alpha_{1,1}=\dots=\alpha_{k-1,k-1}=\alpha$, \[ \delta_r \Big(x^{(1)}, \dots, x^{(k)}\Big) = \Big( r x^{(1)}, r^{\alpha+1} x^{(2)},\dots, r^{(\alpha+1)^{k-1}} x^{(k)}\Big). \] \begin{remark} \rm We would like to remark that this class of operators contains the operators \[ L_{\alpha,\beta}=\Delta_{x^{(1)}} + |x^{(1)}|^{2\alpha} \Delta_{x^{(2)}} + |x^{(1)}|^{2\beta} \Delta_{x^{(3)}}, \] introduced by Thuy and Tri in \cite{thuy_tri_2002}, and the operators \[ P_{\alpha,\beta}=\Delta_{x^{(1)}} + \Delta_{x^{(2)}} + |x^{(1)}|^{2\alpha} |x^{(2)}|^{2\beta} \Delta_{x^{(3)}}, \] introduced by Thuy and Tri in \cite{thuy_tri_2012}. We also want to mention that the class of the Grushin-like operators very recently introduced by Maldonado in \cite[Subsection 4.1]{maldonado} extends the one described above. \end{remark} \end{example} \begin{example} \rm The $\Delta_\lambda$-operators of the following type $$ \Delta_\lambda =\Delta_{x^{(1)}} + \Big(\mu_1(x^{(1)})\Big)^{2} \Delta_{x^{(2)}} + \Big(\mu_2(x^{(1)})\Big)^{2} \Big(\mu_3(x^{(2)})\Big)^{2} \Delta_{x^{(3)}}, $$ where $\mu_1, \mu_2: \mathbb{R}^{N_1}\to\mathbb{R}$ and $\mu_3:\mathbb{R}^{N_2}\to\mathbb{R}$ are continuous functions satisfying (H1)--(H4) and \[ \mu_1(s x^{(1)})=s^\alpha \mu_1(x^{(1)}),\quad \mu_2(s x^{(1)})=s^\beta \mu_2(x^{(1)}), \quad \mu_3(s x^{(2)})=s^\gamma \mu_3(x^{(2)}),\quad \forall s>0, \] with $\alpha, \beta$ and $\gamma$ nonnegative real constants, are $\Delta_\lambda$-Laplacians with the group of dilations $(\delta_r)_{r>0}$, $$ \delta_r\Big(x^{(1)},x^{(2)},x^{(3)}\Big) = \Big( r x^{(1)}, r^{\alpha +1} x^{(2)}, r^{\beta + (\alpha +1)\gamma +1} x^{(3)}\Big). $$ \end{example} \subsection*{Acknowledgements} A. E. Kogoj was partially supported by the Gruppo Nazionale per l' Analisi Matematica, la Probabilit\`a e le loro Applicazioni (GNAMPA) of the Istituto Nazionale di Alta Matematica (INdAM). \begin{thebibliography}{00} \bibitem{anh_2014} C. T. Anh; \emph{Global attractor for a semilinear strongly degenerate parabolic equation on {$\Bbb{R}^N$}}, NoDEA Nonlinear Differential Equations Appl. \textbf{21} (2014), no. 5, 663--678. \MR{3265192} \bibitem{primo_attractor_grushin} C. T. Anh, P. Q. Hung, T. D. Ke, T. T. Phong; \emph{Global attractor for a semilinear parabolic equation involving {G}rushin operator}, Electron. J. Differential Equations (2008), No. 32, 11. \MR{2383395} \bibitem{anh_my_2016} C. T. Anh, B. K. My; \emph{Existence of solutions to {$\Delta_\lambda$}-{L}aplace equations without the {A}mbrosetti-{R}abinowitz condition}, Complex Var. Elliptic Equ. \textbf{61} (2016), no. 1, 137--150. \MR{3428858} \bibitem{anh_my_liouville_2016} C. T. Anh, B. K. My; \emph{Liouville-type theorems for elliptic inequalities involving the {$\Delta_\lambda$}-{L}aplace operator}, Complex Var. Elliptic Equ. \textbf{61} (2016), no. 7, 1002--1013. \MR{3500512} \bibitem{anh_my_2017} C. T. Anh, B. K. My; \emph{Existence and non-existence of solutions to a hamiltonian strongly degenerate elliptic system}, Adv. Nonlinear Anal. (2017). \bibitem{baouendi_1967} M. S. Baouendi; \emph{Sur une classe d'op\'erateurs elliptiques d\'eg\'en\'er\'es}, Bull. Soc. Math. France \textbf{95} (1967), 45--87. \MR{0228819} \bibitem{baouendi_goulaouic_1972} M. S. Baouendi, C. Goulaouic; \emph{Nonanalytic-hypoellipticity for some degenerate elliptic operators}, Bull. Amer. Math. Soc. \textbf{78} (1972), 483--486. \MR{0296507} \bibitem{BLU} A. Bonfiglioli, E. Lanconelli, F. Uguzzoni; \emph{Stratified {L}ie groups and potential theory for their sub-{L}aplacians}, Springer Monographs in Mathematics, Springer, Berlin, 2007. \MR{2363343} \bibitem{chen_tan_gao_2017} J. Chen, X. Tang, Z. Gao; \emph{Infinitely many solutions for semilinear {$\Delta_{\lambda}$}-laplace equations with sign-changing potential and nonlinearity}, Studia Sci. Math. Hungar. \textbf{54} (2017), no. 4, 536--549. \bibitem{chuong_ke_2004} N. M. Chuong, T. D. Ke; \emph{Existence of solutions for a nonlinear degenerate elliptic system}, Electron. J. Differential Equations (2004), No.\ 93, 15. \MR{2075432} \bibitem{DAm} L. D'Ambrosio; \emph{Hardy inequalities related to {G}rushin type operators}, Proc. Amer. Math. Soc. \textbf{132} (2004), no. 3, 725--734. \MR{2019949} \bibitem{franchi_lanconelli_1} L. D'Ambrosio; \emph{Une m\'etrique associ\'ee \`a une classe d'op\'erateurs elliptiques d\'eg\'en\'er\'es}, Rend. Sem. Mat. Univ. Politec. Torino (1983), no. Special Issue, 105--114 (1984), Conference on linear partial and pseudodifferential operators (Torino, 1982). \MR{745979} \bibitem{franchi_lanconelli_3} L. D'Ambrosio; \emph{An embedding theorem for {S}obolev spaces related to nonsmooth vector fields and {H}arnack inequality}, Comm. Partial Differential Equations \textbf{9} (1984), no. 13, 1237--1264. \MR{764663} \bibitem{franchi_lanconelli_2} B. Franchi, E. Lanconelli; \emph{H\"older regularity theorem for a class of linear nonuniformly elliptic operators with measurable coefficients}, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) \textbf{10} (1983), no. 4, 523--541. \MR{753153} \bibitem{garofalo_1993} N. Garofalo; \emph{Unique continuation for a class of elliptic operators which degenerate on a manifold of arbitrary codimension}, J. Differential Equations \textbf{104} (1993), no. 1, 117--146. \MR{1224123} \bibitem{grushin_1970} V. V. Gru\v{s}in; \emph{A certain class of hypoelliptic operators}, Mat. Sb. (N.S.) \textbf{83 (125)} (1970), 456--473. \MR{0279436} \bibitem{grushin_1971} V. V. Gru\v{s}in; \emph{A certain class of elliptic pseudodifferential operators that are degenerate on a submanifold}, Mat. Sb. (N.S.) \textbf{84 (126)} (1971), 163--195. \MR{0283630} \bibitem{hormander} L. H\"ormander; \emph{Hypoelliptic second order differential equations}, Acta Math. \textbf{119} (1967), 147--171. \MR{0222474} \bibitem{kogoj_lanconelli_2009} A. E. Kogoj, E. Lanconelli; \emph{Liouville theorem for {$X$}-elliptic operators}, Nonlinear Anal. \textbf{70} (2009), no. 8, 2974--2985. \MR{2509383} \bibitem{kogoj_lanconelli_2012} A. E. Kogoj, E. Lanconelli; \emph{On semilinear {${\Delta}_{\lambda}$}-{L}aplace equation}, Nonlinear Anal. \textbf{75} (2012), no. 12, 4637--4649. \MR{2927124} \bibitem{kogoj_sonner_2013} A. E. Kogoj, S. Sonner; \emph{Attractors for a class of semi-linear degenerate parabolic equations}, J. Evol. Equ. \textbf{13} (2013), no. 3, 675--691. \MR{3089799} \bibitem{kogoj_sonner_2014} A. E. Kogoj, S. Sonner; \emph{Attractors met {$X$}-elliptic operators}, J. Math. Anal. Appl. \textbf{420} (2014), no. 1, 407--434. \MR{3229832} \bibitem{kogoj_sonner_hardy} A. E. Kogoj, S. Sonner; \emph{Hardy type inequalities for {$\Delta_{\lambda}$}-{L}aplacians}, Complex Var. Elliptic Equ. \textbf{61} (2016), no. 3, 422--442. \MR{3454116} \bibitem{Lanconelli_Kogoj_2000} E. Lanconelli, A. E.. Kogoj; \emph{{$X$}-elliptic operators and {$X$}-control distances}, Ricerche Mat. \textbf{49} (2000), no. suppl., 223--243, Contributions in honor of the memory of Ennio De Giorgi (Italian). \MR{1826225} \bibitem{lanconelli_morbidelli} E. Lanconelli, D. Morbidelli; \emph{On the {P}oincar\'e inequality for vector fields}, Ark. Mat. \textbf{38} (2000), no. 2, 327--342. \MR{1785405} \bibitem{li_sun_2016} D. Li, C. Sun; \emph{Attractors for a class of semi-linear degenerate parabolic equations with critical exponent}, J. Evol. Equ. \textbf{16} (2016), no. 4, 997--1015. \MR{3577407} \bibitem{li_sun_chang_2017} D. Li, C. Sun, Q. Chang; \emph{Global attractor for degenerate damped hyperbolic equations}, J. Math. Anal. Appl. \textbf{453} (2017), no. 1, 1--19. \MR{3641757} \bibitem{luyen_2017} D. T. Luyen; \emph{Two nontrivial solutions of boundary-value problems for semilinear {$\Delta_\gamma$}-differential equations}, Math. Notes \textbf{101} (2017), no. 5-6, 815--823. \MR{3669606} \bibitem{luyen_tri_2015} D. T. Luyen, N. M. Tri; \emph{Existence of solutions to boundary-value problems for semilinear {$\Delta_\gamma$} differential equations}, Math. Notes \textbf{97} (2015), no. 1-2, 73--84. \MR{3394492} \bibitem{luyen_tri_2016_sibirsk} D. T. Luyen, N. M. Tri; \emph{Behavior at large time intervals of solutions of degenerate hyperbolic equations with damping}, Sibirsk. Mat. Zh. \textbf{57} (2016), no. 4, 809--829. \MR{3601331} \bibitem{luyen_tri_2016_grushin} D. T. Luyen, N. M. Tri; \emph{Global attractor of the {C}auchy problem for a semilinear degenerate damped hyperbolic equation involving the {G}rushin operator}, Ann. Polon. Math. \textbf{117} (2016), no. 2, 141--162. \MR{3539074} \bibitem{luyen_tri_2018} D. T. Luyen, N. M. Tri; \emph{Existence of infinitely many solutions for semilinear degenerate {S}chr\"odinger equations}, J. Math. Anal. Appl. \textbf{461} (2018), no. 2, 1271--1286. \MR{3765489} \bibitem{maldonado} D. Maldonado; \emph{On certain degenerate and singular elliptic {PDE}s {I}: nondivergence form operators with unbounded drifts and applications to subelliptic equations}, J. Differential Equations \textbf{264} (2018), no. 2, 624--678. \MR{3720825} \bibitem{quyet_thuy_nguyen_2017} D. T. Quyet, L. T. Thuy, N. X. Tu; \emph{Semilinear strongly degenerate parabolic equations with a new class of nonlinearities}, Vietnam J. Math. \textbf{45} (2017), no. 3, 507--517. \MR{3669155} \bibitem{thao_2016} M. X. Thao; \emph{On the global attractor for a semilinear strongly degenerate parabolic equation}, Acta Math. Vietnam. \textbf{41} (2016), no. 2, 283--297. \MR{3506317} \bibitem{thuy_tri_2002} N. T. C. Thuy, N. M. Tri; \emph{Some existence and nonexistence results for boundary value problems for semilinear elliptic degenerate operators}, Russ. J. Math. Phys. \textbf{9} (2002), no. 3, 365--370. \MR{1965388} \bibitem{thuy_tri_2012} P. T. Thuy, N. M. Tri; \emph{Nontrivial solutions to boundary value problems for semilinear strongly degenerate elliptic differential equations}, NoDEA Nonlinear Differential Equations Appl. \textbf{19} (2012), no. 3, 279--298. \MR{2926298} \bibitem{Thuy_Tri_2013} P. T. Thuy, N. M. Tri; \emph{Long time behavior of solutions to semilinear parabolic equations involving strongly degenerate elliptic differential operators}, NoDEA Nonlinear Differential Equations Appl. \textbf{20} (2013), no. 3, 1213--1224. \MR{3057173} \bibitem{tri_1998_on_grushin} N. M. Tri; \emph{On the {G}rushin equation}, Mat. Zametki \textbf{63} (1998), no. 1, 95--105. \MR{1631852} \bibitem{tri_1998_sobolev} N. M. Tri; \emph{Critical {S}obolev exponent for degenerate elliptic operators}, Acta Math. Vietnam. \textbf{23} (1998), no. 1, 83--94. \MR{1628086} \bibitem{tri_2009} N. M. Tri; \emph{Recent results in the theory of semilinear elliptic degenerate differential equations}, Vietnam J. Math. \textbf{37} (2009), no. 2-3, 387--397. \MR{2568027} \bibitem{wu} J.-M. Wu; \emph{Geometry of {G}rushin spaces}, Illinois J. Math. \textbf{59} (2015), no. 1, 21--41. \MR{3459626} \end{thebibliography} \end{document}