\documentclass[reqno]{amsart} \usepackage{hyperref} \AtBeginDocument{{\noindent\small \emph{Electronic Journal of Differential Equations}, Vol. 2016 (2016), No. 303, pp. 1--8.\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-2016/303\hfil A Fujita-type theorem for a multitime] {A Fujita-type theorem for a multitime evolutionary $p$-Laplace inequality in the Heisenberg group} \author[M. Jleli, M. Kirane, B. Samet \hfil EJDE-2016/303\hfilneg] {Mohamed Jleli, Mokhtar Kirane, Bessem Samet} \address{Mohamed Jleli \newline Department of Mathematics, College of Science, King Saud University, P.O. Box 2455, Riyadh 11451, Saudi Arabia} \email{jleli@ksu.edu.sa} \address {Mokhtar Kirane \newline Laboratoire LaSIE, P\^ole Sciences et Technologies, Universit\'e de La Rochelle, Avenue M. Cr\'epeau, 17042 La Rochelle, France} \email{mkirane@univ-lr.fr} \address{Bessem Samet \newline Department of Mathematics, College of Science, King Saud University, P.O. Box 2455, Riyadh 11451, Saudi Arabia} \email{bsamet@ksu.edu.sa} \thanks{Submitted June 12, 2016. Published November 25, 2016.} \subjclass[2010]{47J35, 35R03} \keywords{Nonexistence; global solution; multitime; differential inequality; \hfill\break\indent Heisenberg group} \begin{abstract} A nonexistence result of global nontrivial positive weak solutions to a multitime evolutionary $p$-Laplace differential inequality in the Heisenberg group is obtained. Our technique of proof is based on the test function method. \end{abstract} \maketitle \numberwithin{equation}{section} \newtheorem{theorem}{Theorem}[section] \newtheorem{lemma}[theorem]{Lemma} \newtheorem{definition}[theorem]{Definition} \newtheorem{corollary}[theorem]{Corollary} \allowdisplaybreaks \section{Introduction} The standard time-dependent partial differential equations of mathematical physics involve evolution in one-dimensional time. Space can be multidimensional, but time stayed one dimensional. The term multitime was introduced in physics by Dirac, Fock and Podolsky, in 1932, considering multi-temporal wave-functions via m-time evolution equation. It was used in mathematics by Friedman and Littman (1962, 1963). Multitime evolution equations arise for example in Brownian motion (diffusion process with inertia) \cite{AK}, transport theory (Fokker-Planck-type equations) \cite{RI}, biology (age-structured population dynamics) \cite{I}, wave and Maxwell's equations \cite{BA,H}, mechanics, physics and cosmology \cite{Re,U}. Some interesting multitime developments of classical, single-time theories and principles from different fields of mathematical research, appeared in the last years (see \cite{AB,AN,CR,KE,MA,UD,Z} and references therein). The study of nonexistence of global solutions to multitime evolutionary problems has begun recently (see \cite{JKS,K}). This paper deals with the nonexistence of global (nontrivial) positive solutions to a multitime evolutionary $p$-Laplace differential inequality in the Heisenberg group. More precisely, we consider the multitime evolutionary $p$-Laplace problem \begin{equation}\label{p} \begin{gathered} \sum_{i=1}^{i=k} u_{t_i}-\operatorname{div}_\mathbb{H} \big( |\nabla_\mathbb{H} u |^{p-2}\nabla_\mathbb{H} u \big)\geq u^q,\quad \text{in } \mathcal{H},\\ u\geq 0,\quad \text{a.e. in } \mathcal{H},\\ u_{|_{t_i=0}}=u_i,\quad \text{in }\mathcal{E}, \end{gathered} \end{equation} where $\mathbb{H}$ is the $(2N+1)$-dimensional Heisenberg group, $k$ is a positive integer ($k\geq 1$), $p,q>1$, $\mathcal{H}=\mathbb{H}\times (0,\infty)^k$, $\mathcal{E}= \mathbb{H}\times (0,\infty)^{k-1}$, and $u_i\in L^1_{\rm loc}(\mathcal{E})$, $i=1,2\dots,k$. Using a duality argument \cite{MP,MP2}, we provide a sufficient condition for the nonexistence of global nontrivial positive weak solutions to the above problem. \section{Preliminaries}\label{S1} For the reader's convenience, we recall some background facts used here. The $(2N+1)$-dimensional Heisenberg group $\mathbb{H}$ is the space $\mathbb{R}^{2N+1}$ equipped with the group operation $$ \vartheta\diamond \vartheta'=(x+x',y+y',\tau+\tau'+2(x\cdot y'-x'\cdot y)), $$ for all $\vartheta=(x,y,\tau),\vartheta'=(x',y',\tau')\in \mathbb{R}^N\times \mathbb{R}^N\times \mathbb{R}$, where $\cdot$ denotes the standard scalar product in $\mathbb{R}^N$. This group operation endows $\mathbb{H}$ with the structure of a Lie group. The distance from un element $\vartheta=(x,y,\tau)\in \mathbb{H}$ to the origin is given by $$ |\vartheta|_\mathbb{H}=\Big(\tau^2+\Big(\sum_{i=1}^N x_i^2+y_i^2\Big)^2\Big)^{1/4}, $$ where $x=(x_1,\dots,x_N)$ and $y=(y_1,\dots,y_N)$. The Gradient $\nabla_{\mathbb{H}}$ over $\mathbb{H}$ is $$ \nabla_{\mathbb{H}}=(X_1,\dots, X_N,Y_1,\dots,Y_N), $$ where for $i=1,\dots,N$, $$ X_i=\partial_{x_i}+2y_i\partial_{\tau}\quad\text{and}\quad Y_i=\partial_{y_i}-2x_i\partial_{\tau}. $$ Let $$ A=\begin{pmatrix} I_N&0&2y\\ 0 &I_N&-2x\\ \end{pmatrix}, $$ where $I_N$ is the identity matrix of size $N\times N$, then $$ \nabla_{\mathbb{H}}=A\nabla_{\mathbb{R}^{2N+1}}. $$ A simple computation gives the expression $$ |\nabla_{\mathbb{H}}u|^2=4(|x|^2+|y|^2) (\partial_{\tau}u)^2+\sum_{i=1}^N \Bigr((\partial_{x_i}u)^2+(\partial_{y_i}u)^2+ 4\partial_\tau u(y_i\partial_{x_i}u-x_i\partial_{y_i}u)\Bigr). $$ The divergence operator in $\mathbb{H}$ is $$ \operatorname{div}_{\mathbb{H}}(u)=\operatorname{div}_{\mathbb{R}^{2N+1}}(Au). $$ For more details on Heisenberg groups and partial differential equations in Heisenberg groups, we refer to \cite{A,B,L,R,T} and references therein. In the proof of our main result, the following inequality will be used several times. \begin{lemma}[$\varepsilon$-Young inequality] \label{Y} Let $a,b,\varepsilon >0$. Then $$ ab\leq \varepsilon a^p+c_\varepsilon b^{p'}, $$ where $p>1$, $p'$ is its corresponding conjugate exponent, i.e., $ \frac{1}{p}+\frac{1}{p'}=1$; and $c_\varepsilon= \big(\frac{1}{\varepsilon p}\big)^{p'/p}\frac{1}{p'}$. \end{lemma} \section{Main result}\label{S2} In this paper, we use the notation: \begin{gather*} d\mathcal{H} = dt_1\dots dt_kd\vartheta,\\ d\mathcal{E}_1 = dt_2\dots dt_kd\vartheta,\\ d\mathcal{E}_i = dt_1\dots dt_{i-1}dt_{i+1}\dots dt_k d\vartheta\quad (i\neq 1). \end{gather*} Moreover, for a given function $\varphi: \mathcal{H}\to \mathbb{R}$, we denote $$ \varphi_i=\varphi\big|_{t_i=0},\quad i=1,2,\dots,k. $$ For $r>1$, we denote by $r'$ its corresponding conjugate exponent. Now, let us define the class of solutions under consideration. \begin{definition}\label{DF} \rm Let $u\in W^{1,p'}_{\rm loc}(\mathcal{H};\mathbb{R}_+)\cap L^q_{\rm loc}(\mathcal{H};\mathbb{R}_+)$ and $u_i\in L^1_{\rm loc}(\mathcal{E};\mathbb{R}_+)$, $i=1,2,\dots,k$, with $p'=\frac{p}{p-1}$. We say that $u$ is a global weak solution to problem \eqref{p} if the following conditions are satisfied: \begin{itemize} \item[(i)] $|\nabla_\mathbb{H} u |^{p-2}\nabla_\mathbb{H} u \in L^{p}_{\rm loc}(\mathcal{H};\mathbb{R}^{2N+1})$; \item[(ii)] For any $\varphi\in W^{1,p'}_{\rm loc}(\mathcal{H};\mathbb{R}_+)$ with compact support, \begin{equation}\label{WS} \int_{\mathcal{H}} u^q\varphi\,d\mathcal{H}\leq \int_{\mathcal{H}} |\nabla_{\mathbb{H}}u|^{p-2}\nabla_{\mathbb{H}}u\cdot \nabla_{\mathbb{H}}\varphi\,d\mathcal{H} -\sum_{i=1}^k \int_{\mathcal{H}} u\varphi_{t_i}\,d\mathcal{H}- \sum_{i=1}^k \int_{\mathcal{E}}u_i\varphi_i\,d\mathcal{E}_i. \end{equation} \end{itemize} \end{definition} Observe that all the integrals in \eqref{WS} are well defined. Our main result is given in the following theorem. \begin{theorem}\label{T1} Let $p>1$. If \begin{equation}\label{SCD} \max\{1,p-1\}1$, $q>\max\{1,p-1\}$ and $\alpha\in (\delta,0)$, where $$ \delta=\max\left\{-1,1-p,\right\}. $$ Let $u$ be a global weak solution to \eqref{p}. Then for any $\varphi\in W^{1,\infty}(\mathcal{H};\mathbb{R}_+)$ with a compact support, we have \begin{equation} \label{SI} \begin{aligned} &\int_{\mathcal{H}}u^{q+\alpha} \varphi \,d\mathcal{H} +\int_{\mathcal{H}} |\nabla_{\mathbb{H}} u|^p u^{\alpha-1}\varphi\,d\mathcal{H} +\sum_{i=1}^k \int_{\mathcal{E}} {u_i}^{\alpha+1}\varphi_i\,d\mathcal{E}_i\\ &\leq C\Big(\sum_{i=1}^k \int_{\mathcal{H}} \big(\frac{|\varphi_{t_i}|^{r}}{\varphi}\big)^{\frac{1}{r-1}}\,d\mathcal{H} +\int_\mathcal{H} \varphi^{1-ps'}|\nabla_\mathbb{H} \varphi|^{ps'}\,d\mathcal{H}\Big), \end{aligned} \end{equation} for some constant $C>0$, where $r=\frac{q+\alpha}{1+\alpha}$, $s=\frac{q+\alpha}{p+\alpha-1}$ and $s'$ is the conjugate exponent of $s$. \end{lemma} \begin{proof} Let $\varepsilon>0$ be fixed and $\alpha \in (\delta,0)$. Suppose that $u$ is a global weak solution to \eqref{p}. Let $$ u_\varepsilon(\vartheta,t_1,\dots,t_k)= u(\vartheta,t_1,\dots,t_k)+\varepsilon, \quad (\vartheta,t_1,\dots,t_k)\in \mathcal{H}. $$ Define $\varphi_\varepsilon$ as $$ \varphi_\varepsilon(\vartheta,t_1,\dots,t_k) =u_\varepsilon^\alpha(\vartheta,t_1,\dots,t_k)\varphi(\vartheta,t_1,\dots,t_k), $$ where $\varphi\in W^{1,\infty}(\mathcal{H};\mathbb{R}_+)$ has a compact support. Observe that $\varphi_\varepsilon$ belongs to the set of admissible test functions in the sense of Definition \ref{DF}. By \eqref{WS}, we have \begin{equation} \label{ES1} \begin{aligned} &\int_{\mathcal{H}}u^q{u_\varepsilon}^\alpha \varphi \,d\mathcal{H} + |\alpha|\int_{\mathcal{H}} |\nabla_{\mathbb{H}} u|^p{u_\varepsilon}^{\alpha-1}\varphi\,d\mathcal{H} +\frac{1}{\alpha+1}\sum_{i=1}^k \int_{\mathcal{E}} {(u_i+\varepsilon)}^{\alpha+1} \varphi_i\,d\mathcal{E}_i \\ &\leq \int_{\mathcal{H}} |\nabla_{\mathbb{H}} u|^{p-1} {u_\varepsilon}^\alpha |\nabla_{\mathbb{H}} \varphi|\,d\mathcal{H}+\frac{1}{\alpha+1} \sum_{i=1}^k \int_{\mathcal{H}} {u_\varepsilon}^{\alpha+1} |\varphi_{t_i}|\,d\mathcal{H}. \end{aligned} \end{equation} Now, using Lemma \ref{Y}, we will estimate the individual terms on the right-hand side of \eqref{ES1}. For some $\varepsilon_1>0$, Lemma \ref{Y} with parameters $r= \frac{q+\alpha}{1+\alpha}$ and $r'= \frac{q+\alpha}{q-1}$ yields $$ \int_{\mathcal{H}} {u_\varepsilon}^{\alpha+1} |\varphi_{t_i}|\,d\mathcal{H} \leq \varepsilon_1 \int_{\mathcal{H}} u_\varepsilon^{q+\alpha}\varphi\,d\mathcal{H} +c_{\varepsilon_1} \int_{\mathcal{H}} \Big(\frac{|\varphi_{t_i}|^{r}}{\varphi} \Big)^{\frac{1}{r-1}}\,d\mathcal{H}, $$ from which follows \begin{equation} \label{esI} \begin{aligned} & \frac{1}{\alpha+1} \sum_{i=1}^k \int_{\mathcal{H}} {u_\varepsilon}^{\alpha+1} |\varphi_{t_i}|\,d\mathcal{H}\\ & \leq \frac{k\varepsilon_1}{\alpha+1} \int_{\mathcal{H}} u_\varepsilon^{q+\alpha}\varphi\,d\mathcal{H} +\frac{c_{\varepsilon_1}}{\alpha+1} \sum_{i=1}^k\int_{\mathcal{H}} \Big(\frac{|\varphi_{t_i}|^{r}}{\varphi}\Big)^{\frac{1}{r-1}}\,d\mathcal{H}. \end{aligned} \end{equation} For some $\varepsilon_2>0$, applying Lemma \ref{Y} with parameters $p$ and $p'=\frac{p}{p-1}$, we obtain \begin{equation} \label{ET1} \begin{aligned} &\int_\mathcal{H} |\nabla_\mathbb{H} u |^{p-1}{u_\varepsilon}^\alpha|\nabla_\mathbb{H} \varphi|\,d\mathcal{H} \\ &\leq \varepsilon_2\int_\mathcal{H} |\nabla_\mathbb{H} u|^p {u_\varepsilon}^{\alpha-1}\varphi\,d\mathcal{H}+c_{\varepsilon_2} \int_\mathcal{H} {u_\varepsilon}^{p+\alpha-1}|\nabla_\mathbb{H} \varphi|^p \varphi^{1-p}\,d\mathcal{H}. \end{aligned} \end{equation} Again, for some $\varepsilon_3>0$, Lemma \ref{Y} with parameters $s=\frac{q+\alpha}{p+\alpha-1}$ and $s'= \frac{q+\alpha}{q-p+1}$, yields \begin{equation}\label{ET2} \int_\mathcal{H} {u_\varepsilon}^{p+\alpha-1}|\nabla_\mathbb{H} \varphi|^p \varphi^{1-p}\,d\mathcal{H} \leq \varepsilon_3 \int_\mathcal{H} {u_\varepsilon}^{q+\alpha}\varphi\,d\mathcal{H} +c_{\varepsilon_3} \int_\mathcal{H} \varphi^{1-ps'}|\nabla_\mathbb{H} \varphi|^{ps'}\,d\mathcal{H}. \end{equation} Combining \eqref{ET1} with \eqref{ET2}, we obtain \begin{equation} \label{ET} \begin{aligned} &\int_\mathcal{H} |\nabla_\mathbb{H} u |^{p-1}{u_\varepsilon}^\alpha|\nabla_\mathbb{H} \varphi|\,d\mathcal{H} \\ & \leq\varepsilon_2\int_\mathcal{H} |\nabla_\mathbb{H} u|^p {u_\varepsilon}^{\alpha-1}\varphi\,d\mathcal{H} +c_{\varepsilon_2} \varepsilon_3 \int_\mathcal{H} {u_\varepsilon}^{q+\alpha}\varphi\,d\mathcal{H}\\ &\quad + c_{\varepsilon_2}c_{\varepsilon_3}\int_\mathcal{H} \varphi^{1-ps'} |\nabla_\mathbb{H} \varphi|^{ps'}\,d\mathcal{H}. \end{aligned} \end{equation} Furthermore, substituting estimates \eqref{esI} and \eqref{ET} into \eqref{ES1}, we obtain \begin{align*} & \int_{\mathcal{H}}u^q{u_\varepsilon}^\alpha \varphi \,d\mathcal{H} + (|\alpha|-\varepsilon_2)\int_{\mathcal{H}} |\nabla_{\mathbb{H}} u|^p{u_\varepsilon}^{\alpha-1} \varphi\,d\mathcal{H} +\frac{1}{\alpha+1}\sum_{i=1}^k \int_{\mathcal{E}} {(u_i+\varepsilon)}^{\alpha+1} \varphi_i\,d\mathcal{E}_i\\ &\leq \big(\frac{k\varepsilon_1}{\alpha+1}+c_{\varepsilon_2}\varepsilon_3\big) \int_{\mathcal{H}}u_\varepsilon^{q+\alpha}\varphi \,d\mathcal{H} +\frac{c_{\varepsilon_1}}{\alpha+1} \sum_{i=1}^k \int_{\mathcal{H}}\big(\frac{|\varphi_{t_i}|^{r}}{\varphi}\big)^{\frac{1}{r-1}}\,d\mathcal{H}\\ &\quad +c_{\varepsilon_2}c_{\varepsilon_3}\int_\mathcal{H} \varphi^{1-ps'} |\nabla_\mathbb{H} \varphi|^{ps'}\,d\mathcal{H}. \end{align*} Passing to the limit inferior as $\varepsilon \to 0$ in the above inequality, and applying the Fatou and Lebesgue theorems, we obtain \begin{align*} & \big(1-\frac{k\varepsilon_1}{\alpha+1}-c_{\varepsilon_2}\varepsilon_3\big) \int_{\mathcal{H}}u^{q+\alpha} \varphi \,d\mathcal{H} + (|\alpha|-\varepsilon_2)\int_{\mathcal{H}} |\nabla_{\mathbb{H}} u|^p u^{\alpha-1}\varphi\,d\mathcal{H}\\ &+\frac{1}{\alpha+1}\sum_{i=1}^k \int_{\mathcal{E}} {u_i}^{\alpha+1}\varphi_i\,d\mathcal{E}_i\\ &\leq \frac{c_{\varepsilon_1}}{\alpha+1} \sum_{i=1}^k \int_{\mathcal{H}} \Big(\frac{|\varphi_{t_i}|^{r}}{\varphi}\Big)^{\frac{1}{r-1}}\,d\mathcal{H} +c_{\varepsilon_2}c_{\varepsilon_3} \int_\mathcal{H} \varphi^{1-ps'}|\nabla_\mathbb{H} \varphi|^{ps'}\,d\mathcal{H}. \end{align*} For $\varepsilon_1, \varepsilon_2, \varepsilon_3$ sufficiently small, we obtain \begin{align*} & \int_{\mathcal{H}}u^{q+\alpha} \varphi \,d\mathcal{H}+\int_{\mathcal{H}} |\nabla_{\mathbb{H}} u|^p u^{\alpha-1} \varphi\,d\mathcal{H}+\sum_{i=1}^k \int_{\mathcal{E}} {u_i}^{\alpha+1}\varphi_i\,d\mathcal{E}_i\\ &\leq C\Big(\sum_{i=1}^k \int_{\mathcal{H}}\Big(\frac{|\varphi_{t_i}|^{r}}{\varphi} \Big)^{\frac{1}{r-1}}\,d\mathcal{H} +\int_\mathcal{H} \varphi^{1-ps'}|\nabla_\mathbb{H} \varphi|^{ps'}\,d\mathcal{H}\Big), \end{align*} for some constant $C>0$, which is the desired result. \end{proof} Now we are ready to prove our main result given by Theorem \ref{T1}. \begin{proof}[Proof of Theorem \ref{T1}] Suppose that $u$ is a nontrivial global weak solution to \eqref{p}. Let us consider the test function \begin{align*} \varphi_R(\vartheta,t) &=\varphi_R(x,y,\tau,t) \\ &=\phi^\omega\Big(\frac{t_1^{2\theta_1}+\dots+t_k^{2\theta_1} +|x|^{4\theta_2}+|y|^{4\theta_2}+\tau^{2\theta_2}}{R^{4\theta_2}}\Big),\quad R>0, \;\omega\gg 1, \end{align*} where $\phi\in C_0^\infty(\mathbb{R}^+)$ is a decreasing function satisfying $$ \phi(z)=\begin{cases} 1 &\text{if } 0\leq z\leq 1\\ 0 &\text{if } z\geq 2, \end{cases} $$ and $\theta_j$, $j=1,2$, are positive parameters, whose exact values will be specified later. Let $$ \rho=\frac{t_1^{2\theta_1}+\dots+t_k^{2\theta_1}+|x|^{4\theta_2}+|y|^{4\theta_2} +\tau^{2\theta_2}}{R^{4\theta_2}}\,. $$ Clearly $\varphi_R$ has support in $$ \Omega_R=\{(\vartheta,t)\in \mathcal{H}:\,0\leq \rho\leq 2\}, $$ while $(\varphi_R)_{t_i}$ ($i=1,2,\dots,k$) and $\nabla_\mathbb{H}\varphi_R$ have support in $$ \Theta_R=\{(\vartheta,t)\in \mathcal H:\, 1\leq \rho\leq 2\}. $$ A simple computation yields \[ \partial_{t_i}\varphi_R(t\vartheta)=2\theta_1\omega t_i^{2\theta_1-1} R^{-4\theta_2} \phi^{\omega-1}(\rho) \phi'(\rho),\quad i=1,2,\dots,k \] while \begin{align*} \nabla_{\mathbb H} \varphi_R(t,\vartheta)|^2 &= 16 \theta_2^2\omega^2 R^{-8\theta_2}(\phi'(\rho))^2\phi^{2\omega-2}(\rho) \Big( (|x|^2+|y|^2)\tau^{4\theta_2-2} \\ &\quad +(|x|^{8\theta_2-2}+|y|^{8\theta_2-2}) +2\tau^{2\theta_2-1}\sum_{i=1}^Nx_iy_i(|x|^{4\theta_2-2}-|y|^{4\theta_2-2})\Big). \end{align*} Then, for all $(t,\vartheta)\in \Omega_R$ and $i=1,2,\dots,k$, we have \begin{equation}\label{V} R|\nabla_{\mathbb H} \varphi_R|+R^{2\theta_2/\theta_1}|\partial_{t_i}\varphi_R| \leq C |\phi'(\rho)|\phi^{\omega-1}(\rho). \end{equation} For simplicity, in the sequel, we will write $\varphi$ instead of $\varphi_R$. Let us consider now the change of variables \[ (t_1,\dots,t_k,x,y,\tau) =(t,\vartheta)\mapsto (\widetilde{t_1},\dots, \widetilde{t_k},\widetilde{x},\widetilde{y},\widetilde{\tau}) =(\widetilde{t},\widetilde{\vartheta}), \] where $$ \widetilde{t}=R^{-2\theta_2/\theta_1}t, \quad \widetilde{x}=R^{-1}x,\quad \widetilde{y}=R^{-1}y,\quad \widetilde{\tau}=R^{-2}\tau, \quad d\widetilde{\mathcal{H}}=d\widetilde{t}\widetilde{d\vartheta}. $$ In the same way, let \begin{gather*} \widetilde{\rho}=\widetilde{t_1}^{2\theta_1}+\dots +\widetilde{t_k}^{2\theta_1}+|\widetilde{x}|^{4\theta_2} +|\widetilde{y}|^{4\theta_2}+\widetilde{\tau}^{2\theta_2},\\ \widetilde{\Omega}=\{(\widetilde{x},\widetilde{y},\widetilde{\tau}, \widetilde{t})\in \mathcal{H}: 0\leq \widetilde{\rho}\leq 2\},\\ \widetilde{\Theta}= \{(\widetilde{x},\widetilde{y},\widetilde{\tau}, \widetilde{t})\in \mathcal{H}: 1\leq \widetilde{\rho}\leq 2\}. \end{gather*} Using the above change of variables and \eqref{V}, we obtain \begin{gather}\label{JJ1} \int_\mathcal{H} \Bigr(\frac{|\varphi_{t_i}|^r}{\varphi}\Bigr)^\frac{1}{r-1} \,d\,\mathcal{H} \leq C R^{Q+2\frac{\theta_2}{\theta_1}(k-\frac{r}{r-1})} \int_\mathcal{H}\phi^{\omega-\frac{r}{r-1}} |\phi'|^{\frac{r}{r-1}} \,d\widetilde{\mathcal{H}},\\ \label{JJ2} \int_\mathcal{H} \varphi^{1-ps'}|\nabla_\mathbb{H} \varphi|^{ps'} \,d\mathcal{H} \leq C R^{Q+2\frac{\theta_2}{\theta_1}k-ps'} \int_\mathcal{H}\phi^{\omega-ps'}|\phi'|^{ps'}\,d\widetilde{\mathcal{H}}. \end{gather} Setting $$ \frac{\theta_2}{\theta_1}=\frac{ps'(r-1)}{2r}, $$ we have \begin{equation}\label{GG} Q+2\frac{\theta_2}{\theta_1}(k-\frac{r}{r-1})= Q+2\frac{\theta_2}{\theta_1}k-ps' =Q-\frac{p(q+\alpha)}{q-p+1}+\frac{p(q-1)k}{q-p+1}. \end{equation} Using \eqref{SI}, \eqref{JJ1}--\eqref{GG}, we obtain \begin{equation}\label{EEG} \int_{\mathcal{H}}u^{q+\alpha} \varphi \,d\mathcal{H}\leq C R^{Q-\frac{p(q+\alpha)}{q-p+1}+\frac{p(q-1)k}{q-p+1}}. \end{equation} Furthermore, noting that $$ Q-\frac{p(q+\alpha)}{q-p+1}+\frac{p(q-1)k}{q-p+1}<0 $$ for $$ q< \frac{pk+Q(p-1)}{Q+(k-1)p} $$ and some $\alpha \in (\delta,0)$ sufficiently small. Under the above condition, letting $R\to \infty$ in \eqref{EEG} and using the monotone convergence theorem, we obtain $$ \int_{\mathcal{H}}u^{q+\alpha} \,d\mathcal{H}\leq 0, $$ which contradicts our assumption about $u$. Finally, the limit case $$ q=\frac{pk+Q(p-1)}{Q+(k-1)p} $$ can be treated by the same way as in \cite{MP3}. \end{proof} We now consider some examples where we can apply Theorem \ref{T1}. Applying Theorem \ref{T1} with $p=2$ and $k=1$, we obtain the following Heisenberg version of Fujita exponent \cite{P}. \begin{corollary} \label{coro3.4} If $1