\documentclass[reqno]{amsart} \usepackage{hyperref} \AtBeginDocument{{\noindent\small \emph{Electronic Journal of Differential Equations}, Vol. 2016 (2016), No. 220, pp. 1--10.\newline ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu} \thanks{\copyright 2016 Texas State University.} \vspace{8mm}} \begin{document} \title[\hfilneg EJDE-2016/220\hfil Multiple solutions for possibly degenerate equations] {Multiple solutions for possibly degenerate equations in divergence form} \author[A. Pinamonti \hfil EJDE-2016/220\hfilneg] {Andrea Pinamonti} \address{Andrea Pinamonti \newline Universit\'a di Padova, Dipartimento di Matematica Pura ed Applicata, via Trieste 63, 35121 Padova, Italy} \email{Andrea.Pinamonti@gmail.com} \thanks{Submitted June 22, 2016. Published August 16, 2016.} \subjclass[2010]{35A01, 35A15} \keywords{Carnot groups; multiplicity results; variational methods} \begin{abstract} Via variational methods, we establish the existence of at least two distinct weak solutions for the Dirichlet problem associated to a possibly degenerate equation in divergence form. \end{abstract} \maketitle \numberwithin{equation}{section} \newtheorem{theorem}{Theorem}[section] \newtheorem{lemma}[theorem]{Lemma} \newtheorem{proposition}[theorem]{Proposition} \allowdisplaybreaks \section{Introduction} Let $\Omega$ be a bounded open subset of $\mathbb{R}^n$, $1\leq m\leq n$ and $X=(X_1,\ldots, X_m)$ be a family of locally Lipschitz vector fields in $\mathbb{R}^n$. We prove a multiplicity result for the problem \begin{equation} \label{prob} \begin{gathered} \operatorname{div}_{X} a(x,Xu)=\lambda f(x,u) \quad \text{in }\Omega \\ u=0 \quad \text{on }\partial\Omega \end{gathered} \end{equation} where $f:\Omega\times \mathbb{R}\to \mathbb{R}$ and $a:\Omega\times \mathbb{R}^m\to \mathbb{R}^m$ are continuous maps satisfying suitable growth assumptions and $\lambda\in\mathbb{R}$. We denote by $\operatorname{div}_{X} u= -\sum_{i=1}^m X^{*}_i u_i$ the so-called $X$-divergence where $X^{*}_i$ denotes the operator formally adjoint to $X_i$, that is the operator for which \[ \int_{\mathbb{R}^n} \psi X_i \varphi dx = \int_{\mathbb{R}^n} \varphi X^{*}_i\psi dx\quad \forall \varphi,\psi\in C^{\infty}_0(\mathbb{R}^n). \] We prove that under suitable assumptions, there is an explicit interval of values for $\lambda$ for which \eqref{prob} has at least two distinct weak solutions. This type of problem has been studied when $m=n$ and $X$ is the standard Euclidean gradient, we refer for instance to \cite{APV, MB,CPV,PM,DV,KLV,KRV,YGY} and references therein. If $m0$ and $p>1$ such that $|a(x,\xi)|\leq c(1+|\xi|^{p-1})$ for all $x\in\Omega$, $\xi\in\mathbb{R}^m$. \item[(A4)] $\mathcal{A}$ is uniformly convex, i.e. there is $k>0$ such that \[ \mathcal{A}\Big(x,\frac{\xi+\eta}{2}\Big) \leq \frac{1}{2}\mathcal{A}(x,\xi)+\frac{1}{2}\mathcal{A}(x,\eta)-k|\xi-\eta|^p \] for all $x\in\Omega$ and $\xi,\eta\in\mathbb{R}^m$. \item[(A5)] $0\leq a(x,\xi)\xi\leq p\mathcal{A}(x,\xi)$ for all $x\in\overline{\Omega}$, $\xi\in \mathbb{R}^m$. \item[(A6)] There are $C_1,C_2>0$ such that \[ C_1|\xi|^p\leq \mathcal{A}(x,\xi)\leq C_2 |\xi|^p \] for all $x\in\overline{\Omega}$ and $\xi\in\mathbb{R}^m$. \item[(A7)] The control distance $d$ (see \cite[Definition 5.2.2]{BLU}) associated to the family $X$ is defined, moreover $(\mathbb{R}^n, d)$ is complete and the topology generated by $d$ is equivalent to the one generated by the Euclidean distance. For every compact set $K$ of $\mathbb{R}^n$, there exists $C>1$ and $R_0>0$ such that denoted by $B_r(x)$ the $d-$ball centered at $x\in\mathbb{R}^n$ with radius $r>0$ the following condition holds: \[ 0<|B_{2r}(x)|\leq C|B_r(x)|\quad \forall x\in K, 00$ such that \[ \frac{1}{|B_r|}\int_{B_r(x)} |u-u_r| dx\leq \frac{\theta r}{|B_{\nu r}|}\int_{B_{\nu r}(x)}|Xu| dx\quad \forall u\in C^1(\overline{\Omega}) \] for every $x\in K$ and $0p$ and $S_p>0$ such that \begin{align}\label{Sobolev} \|u\|_{L^{p^{*}}(\Omega)}\leq S_p \|X u\|_{L^p(\Omega)}\quad \forall u\in C^1_0(\Omega). \end{align} \item[(A10)] There are $a_1,a_2>0$ and $q\in (p,p^{*})$ such that $|f(x,t)|\leq a_1+a_2|t|^{q-1}$ for every $x\in\Omega$ and $t\in\mathbb{R}$. \item[(A11)] There are $\alpha> \frac{C_2}{C_1}p$ and $r_0>0$ such that $0<\alpha\int_0^t f(x,\tau) d\tau\leq tf(x,t)$ for every $x\in\overline{\Omega}$ and $|t|\geq r_0$. Here $C_1$ and $C_2$ are as in $[A6]$. \end{itemize} By \eqref{Sobolev}, the function $\|u\|_{\mathcal{X}}:=\|Xu\|_{L^{p}(\Omega)}$ is a norm in $C^1_0(\Omega)$. Consequently, we define \begin{align}\label{Sobolev_appro} W^{1,p}_{0}(\Omega; X):= \overline{C^1_0(\Omega)}^{\|\cdot\|_{\mathcal{X}}}. \end{align} As pointed out in \cite{LG}, if $u\in W^{1,p}_{0}(\Omega; X)$ then $X_j u$ exists in the sense of distributions and $X_j u\in L^p(\Omega)$ for $j=1,\ldots, m$. Consequently, the gradient $X u$ is well-defined for any $u\in W^{1,p}_{0}(\Omega; X)$. If follows from \eqref{Sobolev} that for every $1\leq q\leq p^{*}$, there exits $c_q>0$ such that \begin{align}\label{Sobp} \|u\|_{L^{q}(\Omega)}\leq c_q \|u\|_{\mathcal{X}}\quad \forall u\in W^{1,p}_{0}(\Omega; X). \end{align} We are now in a position to state our main result. \begin{theorem}\label{mainth} Assume {\rm(A1)--(A11)} are satisfied. Let $\Omega$ be an open bounded subset of $\mathbb{R}^n$. Then for every $\rho>0$ and \[ 0<\lambda<\Lambda(\rho):=\Big(a_1c_{1} C_1^{-1/p} \rho^{\frac{1}{p}-1}+a_2 c_{q}^q q^{-1} C_1^{-\frac{q}{p}} \rho^{\frac{q}{p}-1}\Big)^{-1}, \] problem \eqref{prob} has at least two weak solutions one of which satisfies \[ \|u\|_{\mathcal{X}}^p<\frac{\rho}{C_1}. \] \end{theorem} The plan of the paper is the following. In Section \ref{frame} we introduce and describe our variational framework. In Section \ref{str} we collect some results that we will use in the proof of Theorem \ref{mainth}. Section \ref{Main} is entirely devoted to the proof of Theorem \ref{mainth}. Finally, in Section \ref{exem} we provide some interesting examples satisfying our assumptions. \section{Variational framework}\label{frame} For describing the variational framework, we need the following theorem that has been proved in \cite{HK, GN}, see also \cite{APS,FSSC,KS}. \begin{theorem}\label{imm} Assume {\rm (A7)--(A9)}. If $\Omega\subset\mathbb{R}^n$ is a bounded open set and $p>1$ then $W^{1,p}_0(\Omega; X)$ is reflexive and the embedding \[ W^{1,p}_0(\Omega; X)\hookrightarrow L^q(\Omega) \] is compact for every $1\leq q< p^*$. \end{theorem} In the sequel we will use the following interesting result, see \cite[Theorem 6]{R1} for a proof. \begin{theorem}\label{ricceri} Let $Y$ be a reflexive real Banach space, and let $\Phi, \Psi:Y\to \mathbb{R}$ be two continuously G\^ateaux differentiable functionals such that $\Phi$ is sequentially weakly lower semicontinuous and coercive. Further, assume that $\Psi$ is sequentially weakly continuous. In addition, assume that, for each $\mu>0$, the functional \[ J_{\mu}:=\mu\Phi-\Psi \] satisfies the Palais-Smale condition. Then, for every $\rho>\inf_{Y}\Phi$ and every \[ \mu>\inf_{u\in\Phi^{-1}((-\infty,\rho))} \frac{\sup_{v\in \Phi^{-1}((-\infty,\rho))}\Psi(v)-\Psi(u)}{\rho-\Phi(u)}, \] the following holds: \\ either the functional $J_{\mu}$ has a strict global minimum in $\Phi^{-1}((-\infty,\rho))$, or $J_{\mu}$ has at least two critical points one of which lies in $\Phi^{-1}((-\infty,\rho))$. \end{theorem} For the sake of completeness, we recall that given a Banach space $Y$ with topological dual $Y^{*}$, a $C^1$-functional $\mathcal{I}: Y\to \mathbb{R}$ is said to satisfy the Palais-Smale condition if for every $\eta\in\mathbb{R}$, every sequence $\{x_n\}_{n\in\mathbb{N}}\subset Y$ such that \[ \mathcal{I}(x_n)\to \eta,\quad \|\mathcal{I}'(x_n)\|_{Y^{*}}\to 0\quad \text{as}\ n\to \infty \] admits a convergent subsequence in $Y$. As usual, \begin{equation} \label{normdual} \|\mathcal{I}'(u)\|_{Y^{*}}:=\sup\big\{|\mathcal{I}'(u)[\varphi]|: \varphi\in Y, \|\varphi\|_{Y}=1\big\}. \end{equation} Let us define the functional $\mathcal{I}_{\lambda}\colon W^{1,p}_0(\Omega; X)\to \mathbb{R}$ by \[ \mathcal{I}_{\lambda}(u):=\frac{1}{\lambda}\Phi(u)-\Psi(u), \] where \[ \Phi(u):=\int_{\Omega}\mathcal{A}(x, Xu(x)) dx\quad \text{and}\quad \Psi(u):=\int_{\Omega} F(x, u(x)) dx \] where $\lambda\in \mathbb{R}\setminus\{0\}$ and $F(x,t):=\int_0^t f(x, \tau) d\tau$. It is easy to see that $\mathcal{I}_{\lambda}\in C^1(W^{1,p}_0(\Omega; X), \mathbb{R})$ and \[ \mathcal{I}'_{\lambda}(u)[\varphi] =\frac{1}{\lambda}\int_{\Omega}\langle a(x, Xu), X\varphi\rangle dx -\int_{\Omega} f(x, u)\varphi dx\quad \varphi\in W^{1,p}_0(\Omega; X). \] We say that $u\in W^{1,p}_0(\Omega; X)$ is a weak solution of \eqref{prob} if \[ \mathcal{I}'_{\lambda}(u)[\varphi]=0\quad \forall \varphi\in W^{1,p}_0(\Omega; X). \] \section{Structural properties}\label{str} In this section we collect some interesting consequences of (A1)--(A11). The following Lemma corresponds to \cite[Remark 3.2]{MF}. \begin{lemma}\label{sk} If $f$ satisfies {\rm (A11)} (also known as Ambrosetti-Rabinowitz condition) then \[ F(x, t)\geq F(x, v) t^{\alpha} \] for every $x\in \overline{\Omega}$ and every $(t,v)\in \mathbb{R}^2$ with $t\geq 1$ and $|v|\geq r_0$. \end{lemma} Let $Y$ be a Banach space and $Y^*$ its dual. We recall \cite{BR,PM} that an operator $a: Y\to Y^*$ verifies the $(S_{+})$-condition if for any sequence $(x_n)_{n\in\mathbb{N}}\subset Y$, $x_n\rightharpoonup x$ and \[ \limsup_{n\to \infty} \langle a(x_n), x_n-x\rangle\leq 0 \] it holds $x_n\to x$ strongly in $Y$. We also recall \cite{PM} that a convex functional $A:Y\to \mathbb{R}$ is uniformly convex if for any $\varepsilon>0$ there is $\delta>0$ such that \[ A\Big(\frac{x+y}{2}\Big)\leq \frac{1}{2}A(x)+\frac{1}{2}A(y)-\delta \] for all $x,y\in Y$ with $\|x-y\|>\varepsilon$. If $A$ is uniformly convex on every ball, $A$ is called locally uniformly convex. The following result corresponds to \cite[Proposition 2.1]{PM}. \begin{proposition}\label{conv} Suppose $A:Y\to \mathbb{R}$ is a $C^1$ locally uniformly convex functional that is locally bounded. Then $a=DA:Y\to Y^*$ verifies the $(S+)-$condition. \end{proposition} By \cite[Remark 3.3]{MD}, the functional $\Phi(u)=\int_{\Omega} \mathcal{A}(x, Xu) dx$ is locally bounded and locally uniformly convex. Proposition \ref{conv} gives that, \[ \Phi'(u)[\varphi]=\int_{\Omega} \langle a(x, Xu), X\varphi \rangle dx \] satisfies the $(S_{+})-$condition. We conclude this section with some comments about assumptions (A7)--(A9). In \cite{GN}, it is proved that (A7) and (A8) imply (A9) for every $\Omega$ with sufficiently small diameter, $\overline{\Omega}\subset K^{\circ}$ and $p^*=pQ/(Q-p)$ with $Q=\log_2(C)$. Moreover, as pointed out in \cite{LG}, if the family $X$ has the additional property \begin{itemize} \item [(A12)] Let $\alpha_1,\ldots, \alpha_n\in\mathbb{N}$ and $R>0$ we define the map $\delta_R\colon \mathbb{R}^n\to \mathbb{R}^n$ as \[ \delta_R(x)=\big(R^{\alpha_1} x_1,\ldots, R^{\alpha_n} x_n\big). \] Then \[ X_j(\delta_R u)(x)=R(X_ju)(\delta_R x)\quad \forall u\in C^{\infty}(\mathbb{R}^n) \] where $\delta_R u(x)=u(\delta_R(x))$. \end{itemize} Then \eqref{Sobolev} holds for every open bounded subset of $\mathbb{R}^n$. As proved in \cite[Remark 9]{KL2} the same conclusion holds if (A7) and (A8) are satisfied for every $r>0$. We also point out that in general (A9) does not imply (A7), \cite[Section 6.2]{LG}. \section{Main theorem}\label{Main} In this Section we prove Theorem \ref{mainth}. We start with two preliminary lemmas. \begin{lemma}\label{bbb} Every Palais-Smale sequence $\{u_i\}_{i\in\mathbb{N}}\subset W^{1,p}_0(\Omega; X)$ for $\mathcal{I}_{\lambda}$ is bounded. \end{lemma} \begin{proof} We proceed by contradiction. Possibly passing to a subsequence we can assume $\|u_i\|_{\mathcal{X}}\to \infty$ as $i\to \infty$. Let $\alpha> p\frac{C_2}{C_1}$, by definition \begin{equation} \label{stima1} \begin{aligned} \mathcal{I}_{\lambda}(u_i)-\frac{\mathcal{I}'_{\lambda}(u_i)[u_i]}{\alpha} &=\frac{1}{\lambda}\int_{\Omega}\mathcal{A}(x, Xu_i) dx -\frac{1}{\alpha\lambda}\int_{\Omega} \langle a(x, Xu_i), Xu_i\rangle dx\\ &\quad +\int_{\Omega} \frac{f(x, u_i(x))u_i(x)}{\alpha}-F(x, u_i(x))dx. \end{aligned} \end{equation} Recalling that $f$ is continuous and denoting \[ \upsilon:=\sup\big\{\big|\frac{f(\xi,t)t}{\alpha}-F(\xi,t)\big|: \xi\in\overline{\Omega},\, |t|\leq r_0\big\}<\infty, \] we obtain \begin{equation} \label{stima2} \int_{|u_i(x)|\leq r_0}\frac{f(x, u_i(x))u_i(x)}{\alpha}-F(x, u_i(x)) dx \geq -|\Omega| \upsilon \end{equation} and by (A11), \begin{align}\label{stima3} \int_{|u_i(x)|> r_0}\frac{f(x, u_i(x))u_i(x)}{\alpha}-F(x, u_i(x)) dx\geq 0. \end{align} Assumptions (A5) and (A6) give \begin{align}\label{stima4} \frac{1}{\lambda}\Big(C_1-\frac{C_2p}{\alpha}\Big) \|u\|_{\mathcal{X}}^p\leq \frac{1}{\lambda}\int_{\Omega}\mathcal{A}(x, Xu_i) dx -\frac{1}{\alpha\lambda}\int_{\Omega} \langle a(x, Xu_i), Xu_i\rangle dx\,. \end{align} Using \eqref{stima1}, \eqref{stima2}, \eqref{stima3} and \eqref{stima4} \[ \frac{1}{\lambda}\Big(C_1-\frac{C_2p}{\alpha}\Big)\|u\|_{\mathcal{X}}^p\leq \mathcal{I}_{\lambda}(u_i)-\frac{\mathcal{I}'_{\lambda}(u_i)[u_i]}{\alpha}+|\Omega|\upsilon, \] note that $\alpha> p\frac{C_2}{C_1}$ implies $C_1-\frac{C_2p}{\alpha}>0$. Therefore, \[ \frac{1}{\lambda}\Big(C_1-\frac{C_2p}{\alpha}\Big)\|u\|_{\mathcal{X}}^p \leq \mathcal{I}_{\lambda}(u_i) +\frac{\|\mathcal{I}_{\lambda}'(u_i)\|_{\mathcal{X}^{-1}} \|u_i\|_{\mathcal{X}}}{\alpha}+|\Omega|\upsilon. \] Let $i_0\in\mathbb{N}$ be such that $\| u_i\|_{\mathcal{X}}\geq 1$ for every $i\geq i_0$. Since $p>1$ then for every $i\geq i_0$ \begin{align}\label{ineqfon} 0<\frac{1}{\lambda}\Big(C_1-\frac{C_2p}{\alpha}\Big) \leq \frac{\mathcal{I}_{\lambda}(u_i)}{\|u_i\|_{\mathcal{X}}} +\frac{\|\mathcal{I}_{\lambda}'(u_i)\|_{\mathcal{X}^{-1}}}{\alpha} +\frac{|\Omega|\upsilon}{\|u_i\|_{\mathcal{X}}}. \end{align} Letting $i\to \infty$ in \eqref{ineqfon} and recalling that $\{u_i\}_{i\in\mathbb{N}}$ is a Palais-Smale sequence we obtain a contradiction. \end{proof} \begin{lemma}\label{PScond} The functional $\mathcal{I}_{\lambda}$ satisfies the Palais-Smale condition. \end{lemma} \begin{proof} Let $\{u_i\}_{i\in\mathbb{N}}\subset W^{1,p}_0(\Omega; X)$ be a Palais-Smale sequence for $\mathcal{I}_{\lambda}$. By Lemma \ref{bbb}, $\{u_i\}_{i\in\mathbb{N}}$ is bounded and since $W^{1,p}_0(\Omega; X)$ is reflexive, there is a subsequence, that we still denote by $\{u_i\}_{i\in\mathbb{N}}$, and $\hat{u}\in W^{1,p}_0(\Omega; X)$ such that $u_i\rightharpoonup \hat{u}$ in $W^{1,p}_0(\Omega; X)$. We have to prove that $u_i \to \hat{u}$ in $W^{1,p}_0(\Omega; X)$. By definition, \begin{equation} \label{eq} \begin{aligned} &\int_{\Omega}\langle a(x,X u_i), X (u_i(x)-\hat{u}(x))\rangle dx \\ &=\lambda\mathcal{I}_{\lambda}'(u_i)[u_i-\hat{u}]+\lambda\int_{\Omega} f(x, u_i(x)) (u_i(x)-\hat{u}(x)) dx. \end{aligned} \end{equation} Since $\|\mathcal{I}'_{\lambda}(u_i)\|_{\mathcal{X}^{-1}}\to 0$, and $\{u_i-\hat{u}\}_{i\in\mathbb{N}}$ is bounded in $W^{1,p}_0(\Omega; X)$, and recalling that \[ |\mathcal{I}_{\lambda}'(u_i)[u_i-\hat{u}]| \leq \|\mathcal{I}'_{\lambda}(u_i)\|_{\mathcal{X}^{-1}}\|u_i-\hat{u}\|_{\mathcal{X}}, \] we obtain \begin{equation} \label{convI} \mathcal{I}_{\lambda}'(u_i)[u_i-\hat{u}]\to 0\quad \text{as } i\to\infty. \end{equation} By Theorem \ref{imm}, $u_i\to \hat{u}$ in $L^{q}(\Omega)$. By (A10), \begin{align*} 0&<\int_{\Omega} f(x, u_i(x))(u_i(x)-\hat{u}(x)) dx\\ &\leq a_1\int_{\Omega}|u_i(x)-\hat{u}(x)| dx +a_2\int_{\Omega} |u_i(x)|^{q-1}|u_i(x)-\hat{u}(x)| dx\\ &\leq Ca_1 \|u-u_i\|_{L^q(\Omega)}+a_2\|u_i\|_{L^q(\Omega)}\|u-u_i\|_{L^q(\Omega)} \end{align*} hence \begin{equation} \label{convf} \int_{\Omega} f(x, u_i(x))(u_i(x)-\hat{u}(x)) dx\to 0\quad \text{as } i\to\infty. \end{equation} Putting together \eqref{eq}, \eqref{convI} and \eqref{convf} we conclude that \begin{equation} \label{cv} \Phi'(u_i)[u_i-\hat{u}]=\int_{\Omega}\langle a(x,X u_i), X (u_i(x) -\hat{u}(x))\rangle dx \to 0\quad \text{as } i\to\infty. \end{equation} Since $\Phi'$ has the $(S_{+})$-property, $u_i\to \hat{u}$ in $W^{1,p}_0(\Omega; X)$. \end{proof} \begin{proof}[Proof of Theorem \ref{mainth}] By Lemma \ref{PScond}, $\mathcal{I}_{\lambda}$ satisfies the Palais-Smale condition and by (A4) and (A6), $\Phi$ is coercive and sequentially weakly lower semicontinuous. Since $f$ is continuous and using Theorem \ref{Sobolev} then $\Psi$ is sequentially weakly continuous. We claim that for every $\rho>0$ and $0<\lambda<\Lambda(\rho)$ \[ \frac{1}{\lambda}>\Theta(\rho):=\inf_{u\in \Phi^{-1}((-\infty,\rho))} \frac{\sup_{v\in \Phi^{-1}((-\infty,\rho))}\Psi(v)-\Psi(u)}{\rho-\Phi(u)}. \] By (A2) we obtain $\Phi(0)=0$ and by definition $\Psi(0)=0$. Hence, \[ \Theta(\rho)\leq \frac{\sup_{v\in \Phi^{-1}((-\infty,\rho))}\Psi(v)}{\rho}. \] By (A6), \begin{equation} \label{disk} \Phi^{-1}((-\infty,\rho))\subseteq \{v\in X\ |\ \|v\|_{\mathcal{X}}\leq C_1^{-1/p} \rho^{\frac{1}{p}}\}; \end{equation} therefore \[ \Theta(\rho)\leq \frac{\sup_{\{v\in X\ |\ \|v\|_{\mathcal{X}}\leq C_1^{-1/p} \rho^{\frac{1}{p}}\}}\Psi(v)}{\rho}. \] Using (A10), \eqref{disk} and \eqref{Sobp} we easily obtain \[ \Theta(\rho)\leq \frac{a_1c_{1} }{C_1^{\frac{1}{p}}}\rho^{\frac{1}{p}-1} +\frac{a_2 c_{q}^q}{qC_1^{\frac{q}{p}}} \rho^{\frac{q}{p}-1} \] and the conclusion follows. Now we prove that $J_{\lambda}$ cannot have a strict global minimum in $\Phi^{-1}((-\infty,\rho))$. By Lemma \ref{sk} and $(A6)$ it follows \begin{align*} J_{\lambda}(tu_0) &=\frac{1}{\lambda} \Phi(t u_0)-\Psi(t u_0)\\ &\leq \frac{C_2}{\lambda} t^p \int_{\Omega} |Xu_0|^p dx-t^{\alpha}\int_{\{\xi\in\Omega\ |\ |u_0(x)|\geq r_0\}} F(x, u_0(x)) dx + \nu|\Omega|, \end{align*} for every $u_0\in W^{1,p}_0(\Omega; X)$, where $\nu=\sup\{|F(x,t)|,\ x\in\overline{\Omega}, |t|\leq r_0\}$. Choosing $u_0$ such that $|\{x\in\Omega: |u_0(x)|\geq r_0\}|>0$, recalling that $\alpha>\frac{C_2}{C_1}p>p$ and $F(x,t)>0$ for $|t|\geq r_0$ we obtain \[ \lim_{t\to +\infty}J_{\lambda}(tu_0)=-\infty. \] Applying Theorem \ref{ricceri} we complete the proof. \end{proof} \section{Examples}\label{exem} In this section we collect some interesting examples of vector fields satisfying (A7), (A8) and (A9). \subsection{Euclidean space} Let $m=n$ and $X$ be the standard Euclidean gradient. It is well known that (A7) and (A8) are satisfied for every $r>0$, therefore also (A9) holds. In this case a result similar to Theorem \ref{mainth} has been proved in \cite{MD}. We invite the reader to have also a look at \cite{MB} where the case $\mathcal{A}(x,\xi)=|\xi|^2$ is investigated and \cite{PM,CPV,DV,KLV,YGY} for the case $\mathcal{A}(x,\xi)=|\xi|^p$. \subsection{Carnot Groups} We recall that a Carnot group $\mathbb{G}$ is a connected Lie groups whose Lie algebra $\mathcal{G}$ is finite dimensional and stratified of step $s\in\mathbb{N}$. Precisely, there exist linear subspaces $V_1,\dots,V_s$ of $\mathcal{G}$ such that \[ \mathcal{G}=V_1\oplus \cdots \oplus V_s \] with \[ [V_1,V_{i-1}]=V_{i} \text{ if }2\leq i\leq s, \quad\text{and}\quad [V_1,V_s]=\{0\}. \] Here $[V_1,V_i]:=\operatorname{span}\{[a,b]: a\in V_1,\ b\in V_i\}$. Since $\mathcal{G}$ is stratified then every element in $\mathcal{G}$ is the linear combination of commutators of elements $V_1$. We refer to \cite{BLU} for a complete introduction to the subject. Let $\dim (V_1)=m$ and $X=(X_1,\ldots, X_m)$ be a basis of $V_1$. In \cite{HK}, it is proved that (A7) and (A8) are satisfied for every $r>0$ therefore also (A9) holds. We point out that when $p=2$ and $\mathcal{A}(x,\xi)=|\xi|^2$, Theorem \ref{mainth} boils down to \cite[Theorem 3.1]{MF}. \subsection{H\"ormander vector fields} Let $X=(X_1,\ldots, X_m)$ be a family of smooth vector fields in $\mathbb{R}^n$. We say that $X$ satisfies the H\"ormander condition if \[ \operatorname{rank}\big(\operatorname{Lie}\{X_1,\ldots, X_m\}\big)(x) =n\quad \forall x\in\mathbb{R}^n \] where $\operatorname{Lie}\{X_1,\ldots, X_m\}$ denotes the Lie algebra generated by $X$. Clearly, Carnot groups satisfy the H\"ormander condition, on the other hand there are plenty of examples of vector fields satisfying the H\"ormander condition whose generated Lie algebra is not stratified. For instance, we can consider in $\mathbb{R}^2$ the family $X=(X_1, X_2)$ where \[ X_1=\partial_{x},\quad X_2=x^2 \partial_y \] then $\operatorname{rank}\big(\operatorname{Lie}\{X_1,X_2\}\big)(x,y)=2$ for every $(x,y)\in\mathbb{R}^2$ and $\operatorname{Lie}\{X_1,X_2\}$ is not stratified. In \cite{NS} and \cite{J} it is proved that (A7) and (A8) hold respectively with $R_0>0$. \subsection{Vector fields not satisfying the H\"ormander condition but satisfying (A7) and (A8)} The following example is contained in \cite{LG}. Let us consider the family $X=(X_1,X_2)$ in $\mathbb{R}^3$ where \[ X_1=\partial_x,\quad X_2=|x|^m \partial_y+\partial_z,\quad m\in [1,\infty), \] in \cite{LG} is it proved that $X$ satisfies (A7) and (A8). The family $X$ satisfies (A12) with $\delta_R(x,y,z)=(Rx, R^{m+1}y, Rz)$, therefore it also satisfies (A9).\\ The following family of vector fields has been studied in \cite{FL1,FL2}, \[ X_j=\lambda_j \partial_{x_j}\quad j=1,\ldots, m \] where each $\lambda_j$ is a real-valued function and the family $(\lambda_j)$ satisfies suitable conditions, see \cite{FL1,FL2}. As proved in \cite{FL1,FL2}, these conditions ensure the validity of (A7) and (A8). We conclude with an explicit application of Theorem \ref{mainth} to Carnot groups, note that the following result generalizes \cite[Theorem 3.1]{MF}. Let $X=(X_1,\ldots, X_m)$ be a basis of $V_1$ and $p>1$. We define \[ \Delta_p u=\operatorname{div}_{X} (|Xu|^{p-2} Xu). \] \begin{theorem} \label{thm8} Let $\mathbb{G}$ be a Carnot group, $\Omega$ be a bounded open subset of $\mathbb{G}$ and $p\geq 2$. 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