\documentclass[reqno]{amsart} \usepackage{hyperref} \usepackage{tikz} \usetikzlibrary{arrows} \AtBeginDocument{{\noindent\small Two nonlinear days in Urbino 2017\newline \emph{Electronic Journal of Differential Equations}, Conference 25 (2018), pp. 179--196.\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}{179} \title[\hfilneg EJDE-2018/conf/25\hfil Critical Dirichlet problems] {Critical Dirichlet problems on $\mathcal H$ domains\\ of Carnot groups} \author[G. Molica Bisci \& P. Pucci \hfil EJDE-2018/conf/25\hfilneg] {Giovanni Molica Bisci, Patrizia Pucci} \address{Giovanni Molica Bisci \newline Department Patrimonio, Architettura, Urbanistica (PAU), Universit\`a `Mediterranea' di Reggio Calabria, Via Graziella, Feo di Vito 89124, Reggio Calabria, Italy} \email{gmolica@unirc.it} \address{Patrizia Pucci \newline Department of Mathematics and Informatics, University of Perugia, Via Vanvitelli, 1, 06123 Perugia, Italy} \email{patrizia.pucci@unipg.it} \dedicatory{Dedicated to the memory of our beloved friend Anna} \thanks{Published September 15, 2018} \subjclass[2010]{35R03, 35A15} \keywords{Carnot groups; compactness results; subelliptic critical equations} \begin{abstract} The paper deals with the existence of at least one (weak) solution for a wide class of one-parameter subelliptic critical problems in unbounded domains $\Omega$ of a Carnot group $\mathbb{G}$, which present several difficulties, due to the intrinsic lack of compactness. More precisely, when the real parameter is sufficiently small, thanks to the celebrated symmetric criticality principle of Palais, we are able to show the existence of at least one nontrivial solution. The proof techniques are based on variational arguments and on a recent compactness result, due to Balogh and Krist\'aly in \cite{bk}. In contrast with a persisting assumption in the current literature we do not require any longer the strongly asymptotically contractive condition on the domain $\Omega$. A direct application of the main result in the meaningful subcase of the Heisenberg group is also presented. \end{abstract} \maketitle \numberwithin{equation}{section} \newtheorem{theorem}{Theorem}[section] \newtheorem{corollary}[theorem]{Corollary} \newtheorem{proposition}[theorem]{Proposition} \newtheorem{remark}[theorem]{Remark} \newtheorem{lemma}[theorem]{Lemma} \newtheorem{definition}[theorem]{Definition} \allowdisplaybreaks \section{Introduction} This paper constitutes the initial part of a project devoted to the study of nonlinear equations defined on possibly unbounded domains of Carnot groups. Differential problems involving a subelliptic operator on an unbounded domain $\Omega$ of stratified groups have been intensively studied in recent years by many authors, see, among others, the papers of Garofalo and Lanconelli \cite{GL}, Maad \cite{M1,M2}, Schindler and Tintarev \cite{ST}, Tintarev \cite{T} and references therein. On the contrary, once a domain is not bounded the Folland-Stein space $HW^{1,2}_0(\Omega)$ maybe not be compactly embedded into a Lebesgue space. This lack of compactness produces several difficulties exploiting variational methods. To recover compactness on the unbounded case a persisting hypothesis in the above cited results was the \emph{strongly asymptotically contractive} condition on $\Omega$, introduced by Maad, see \cite{M1} for details. Indeed, every bounded domain is strongly asymptotically contractive. In the Euclidean setting unbounded domains were covered in the pioneering paper \cite{dPF}. Now, we observe that a strongly asymptotically contractive domain $\Omega$ is geometrically thin at infinity. In presence of symmetries, by replacing the contractive assumption on $\Omega$ with a geometrical hypothesis, see condition $(\mathcal{H})$ below, introduced recently by Balogh and Krist\'aly in \cite{bk}, we are able to treat here subelliptic critical equations, in which the domain is possibly large at infinity. The purpose of the present paper is to establish the existence of (weak) solutions of the one-parameter problem \begin{equation}\label{prob} \begin{gathered} -\Delta_{\mathbb G}u+u= h(q)f(u)+\lambda|u|^{2^*-2}u \quad\text{in }\Omega,\\ \,u=0 \quad \text{on }\partial\Omega. \end{gathered} \end{equation} More precisely, our strategy is to find a topological group $T$, acting continuously on $HW^{1,2}_0(\Omega)$, such that the $T$-invariant closed subspace $HW^{1,2}_{0,T}(\Omega)$ can be compactly embedded in suitable Lebesgue spaces. Successively, assuming the left invariance of the standard Haar measure $\mu$ of the Carnot group $\mathbb{G}$, with respect to the action of the group $*:T\times HW^{1,2}_0(\Omega)\to HW^{1,2}_0(\Omega)$, see Bourbaki \cite[Chapter III \S 2 No 4]{Btop} and Bourbaki \cite[Chapter 7 \S 1 No 1]{BLie}, the principle of symmetric criticality of Palais, see Lemma~\ref{4.3} below, can be applied to the associated energy Euler-Lagrange functional $I_\lambda$, allowing a variational approach of problem~\eqref{prob}. Moreover, as usual, when dealing with critical equations, one of the main difficulties appears since the Palais-Smale condition for the Euler-Lagrange functional $I_\lambda$ does not hold at any level, but just under a suitable threshold. Along this paper we overcome these difficulties, using some strategies considered in the literature also in context different than the one treated here, see, for instance, papers \cite{BP,Loi,MBR,Pucci}. Let us briefly introduce the structural setting of problem \eqref{prob}. Let $\mathbb{G}=(\mathbb{G}, \circ)$ be a Carnot group of step $r$ and homogeneous dimension $Q>2$, with neutral element denoted by $e$. Let $T=(T, \cdot)$ be a closed infinite topological group acting continuously and left-distributively on $\mathbb{G}$ by the map $*:T\times\mathbb G\to\mathbb G$. Assume that $T$ acts isometrically on the horizontal Folland-Stein space $HW^{1,2}_0(\mathbb{G})$, where the action $\sharp:T\times HW^{1,2}_0(\mathbb{G})\to HW^{1,2}_0(\mathbb{G})$ is defined for every $({\tau},u)\in T\times HW^{1,2}_0(\mathbb{G})$ by \begin{equation*}\label{action} ({\tau}\sharp u)(q)=u({\tau}^{-1}*q)\quad\text{for all } q\in\mathbb G. \end{equation*} In what follows $d_{CC}:\mathbb{G}\times\mathbb{G}\to\mathbb R^+_0$ denotes the Carnot-Carath\'{e}odory distance on $\mathbb{G}$, while $\mu$ is the natural Haar measure on $\mathbb{G}$ and $``\liminf"$ is the Kuratowski lower limit of sets. Let $\Omega$ be a nonempty open $T$-invariant subset of $\mathbb{G}$, with boundary $\partial\Omega$, and assume that \begin{itemize} \item[(H1)] for every $(q_k)_k\subset \mathbb{G}$ such that \[ \lim_{k\to \infty}d_{CC}(e,q_k)=\infty\quad\text{and}\quad \mu\Big(\liminf_{k\to \infty}(q_k\circ \Omega)\Big)>0, \] where $q_k\circ \Omega=\{q_k\circ q:q\in \Omega\}$, then there exist a subsequence $(q_{k_{j}})_j$ of $(q_k)_k$ and a sequence of subgroups $(T_{q_{k_{j}}})_{j}$ of $T$, with cardinality $\operatorname{card}(T_{q_{k_{j}}})=\infty$, having the property that for all ${\tau}_1$, ${\tau}_2\in T_{q_{k_{j}}}$, with ${\tau}_1\neq{\tau}_2$, it results $$ \lim_{j\to \infty}\inf_{q\in \mathbb{G}}d_{CC}(({\tau}_1*q_{k_j})\circ q, ({\tau}_2*q_{k_j})\circ q)=\infty. $$ \end{itemize} A domain $\Omega$ of $\mathbb{G}$, for which condition (H1) holds, is simply called \emph{$\mathcal H$ domain}. In \eqref{prob} the \emph{subelliptic Laplacian operator} $\Delta_{\mathbb G}$ on $\mathbb{G}$ is the second-order differential operator $$ \Delta_{\mathbb{G}}=\sum_{k=1}^{m_1}X_k^2, $$ where $\mathcal{B}=\{X_1,\dots,X_{m_1}\}$ is a basis of the first graduated component $\mathfrak{G}_1$ of the stratified Lie algebra $\mathfrak{G}=\oplus_{k=1}^{r}\mathfrak{G}_k$ associated to $\mathbb{G}$; see Section~\ref{sec2}. The \emph{critical Sobolev exponent} $2^*$ in the Carnot group $\mathbb{G}$ is $2^*=2Q/(Q-2)$. The parameter $\lambda$ is a real number. The nonlinearity $f:\mathbb{R}\to\mathbb{R}$ is a continuous function, with associated primitive $$ F(t)=\int_0^{t}f(\xi)d\xi\quad\text{for every }t\in\mathbb{R},$$ and satisfies \begin{itemize} \item[(H2)] $F>0$ in $\mathbb{R}\setminus\{0\}$, and there exist $C>0$ and $s\in (1, 2)$ such that $$ |f(t)|\leq C|t|^{s-1}\quad \text{for all } t\in\mathbb{R}; $$ \item[(H3)] there exist $a_0>0, \delta>0$ and $s_1\in (1,2)$ such that \[ F(t) \geq a_0|t|^{s_1}\quad \text{for all } t\in\mathbb{R}, \text{ with } |t|\leq \delta. \] \end{itemize} Since $Q>2$, by \cite{GV} we know that for all $\varphi \in C_0^\infty(\Omega)$ \begin{equation}\label{3.1} \| \varphi\|_{2^*}\le C_{Q,2} \|D_{\mathbb G} \varphi \|_2, \end{equation} where $ C_{Q,2}$ is a positive constant depending on the dimension $Q$ and $$ D_\mathbb{G}=(X_1,\dots,X_{m_1}) $$ denotes the horizontal gradient. Concerning the function $h$ in \eqref{prob}, we assume that $h$ satisfies \begin{itemize} \item[(H4)] $0\leq h\in L^{\frac{2^*}{2^*-s}}(\Omega)$ and there exists a nonempty open set $\Omega_0\subset\Omega$ such that $$ \inf_{q\in\Omega_0}h(q)>0. $$ \end{itemize} Clearly, condition (H4) simply requires that $h$ be nontrivial and belong to a suitable Lebesgue space. Finally, suppose that \begin{itemize} \item[(H5)] the functional $\Psi:HW^{1,2}_0(\Omega)\to\mathbb{R}$ given by $$ \Psi(u)=\int_\Omega h(q)f(u)d\mu(q)\quad\text{for all } u\in HW^{1,2}_0(\Omega) $$ is $T$-invariant, that is $\Psi(\tau\sharp u)=\Psi(u)$ for all $({\tau},u)\in T\times HW^{1,2}_0(\mathbb{G})$. \end{itemize} In Section~\ref{sec2} we present the useful criterion Lemma~\ref{lem4} on the validity of assumption (H5). We are now able to state the main existence result for~\eqref{prob}. \begin{theorem}\label{th2} Let $\Omega$ be a $\mathcal H$ domain of $\mathbb{G}$. Assume that $f$ and $h$ fulfil {\rm (H2)--(H5)}. Then \eqref{prob} admits at least one nontrivial solution $u_{\lambda}$ in the Folland-Stein space $HW_0^{1,2}(\Omega)$ for all $\lambda\le0$. Furthermore, if $\lambda>0$ and $h$ satisfies \begin{equation}\label{h2} \|h\|_{\frac{2^*}{2^*-s}}<\frac{1}{C}\Big(\frac{1}{C_{Q,2}^{(2-s)Q+2s}} \Big(\frac{2^{\alpha}}Q\Big)^{2-s}\Big)^{1/2}, \end{equation} where $C$ and $s$ are introduced in {\rm (H2)}, $C_{Q,2}>0$ in~\eqref{3.1}, and $$ \alpha=\frac{2^*(6-s)-8}{(2^*-2)(2-s)}, $$ then there exists $\lambda^*>0$ such that problem \eqref{prob} admits at least one nontrivial solution $u_{\lambda}$ in $HW_0^{1,2}(\Omega)$ for all $\lambda\in(0, \lambda^{*})$. \end{theorem} Thanks to \cite[Theorem 1.1]{bk} and Lemma \ref{lem4} below, a direct application of Theorem~\ref{th2} gives the existence of at least one solution for subelliptic equations defined on a special class of (unbounded) domains of the Heisenberg group $\mathbb{H}^n=\mathbb{C}^n\times \mathbb{R}$, $n\geq 1$. More precisely, let $\psi_1$, $\psi_2:\mathbb R^+_0\to \mathbb{R}$, $\mathbb R^+_0=[0,\infty)$, be two functions that are bounded on bounded sets, with $\psi_1(t)<\psi_2(t)$ for every $t\in\mathbb R^+_0$. Define $$ \Omega_\psi=\big\{q\in \mathbb{H}^n: q=(z,t)\text{ with } \psi_1(|z|),>=triangle 45] (-2,0) -- (5,0) node [right] {$|z|$}; \draw [->,>=triangle 45] (0,-2.2) -- (0,3.2) node [above] {$t$}; \draw[very thick] (-1.5,.5) to [out=300,in=180] (-.5,-.5) to [out=0,in=180] (3,2.5) to [out=360,in=140] (4.5,2) ; \draw[very thick] (-1.5,-1.5) to [out=350,in=240] (4.5,1); \node at (-0.2,0) [above] {$O$}; \node at (1.7,2.1) [above] {$\psi_2$}; \node at (2,-1.5) [right] {$\psi_1$}; \node at (5.3,1.5) [left] {$\Omega_\psi$}; \end{tikzpicture} \vspace{-3truecm} \caption{A simple prototype of $\Omega_\psi$}\label{Fig} \end{figure} } Then the subelliptic problem~\eqref{prob} becomes \begin{equation}\label{pH} \begin{gathered} -\Delta_{\mathbb H^n}u+u= h(q)f(u)+\lambda|u|^{2^*-2}u\quad \text{in }\phantom{\partial}\Omega_\psi\\ u=0 \quad \text{on }\partial\Omega_\psi, \end{gathered} \end{equation} where $\Delta_{\mathbb H^n}$ the subelliptic Kohn-Laplace operator. Let $\mathbb{U}(n)=U(n)\times \{1\}$, where $$ U(n)=U(n,\mathbb{C})=\big\{\tau\in GL(n;\mathbb{C}): \langle\tau z,\tau z'\rangle_{\mathbb C^n}= \langle z,z'\rangle_{\mathbb C^n} \text{ for all } z,z'\in \mathbb C^n\big\}, $$ that is $U(n)$ is the usual unitary group. Here $\langle\cdot,\cdot\rangle_{\mathbb C^n}$ denotes the standard Hermitian product on $\mathbb C^n$, in other words $\langle z,z'\rangle_{\mathbb C^n}=\sum_{k=1}^nz_k\cdot\overline{z'_k}$. Hence, $\mathbb{U}(n)$ is the unitary group endowed with the natural multiplication law $\cdot:\mathbb{U}(n)\times \mathbb{U}(n)\to \mathbb{U}(n)$, which acts continuously and left-distributively on $\mathbb H^n$ by the map $* : \mathbb{U}(n)\times \mathbb{H}^n \to \mathbb{H}^n$, defined by $$ \widehat{\tau}*q=(\tau z,t)\quad\text{for all $\widehat{\tau}=(\tau,1)\in \mathbb{U}(n)$ and all }q=(z,t)\in\mathbb H^n, $$ thanks to \cite[Lemma 3.1]{bk}. Taking $T=\mathbb{U}(n)$, then $\Omega_\psi$ is $\mathbb{U}(n)$-invariant and a $\mathcal H$ domain, as shown in the proof of Theorem~1.1 of~\cite{bk}. Moreover, \begin{equation*} HW^{1,2}_{0,\mathbb{U}(n)}(\Omega_\psi) =\{u\in HW^{1,2}_0(\Omega_\psi): u(z,t)=u(|z|,t)\text{ for all } q=(z,t)\in\Omega_\psi\}, \end{equation*} that is $HW^{1,2}_{0,\mathbb{U}(n)}(\Omega_\psi)=HW^{1,2}_{\rm cyl}(\Omega_\psi)$ is the space of cylindrically symmetric functions of $HW^{1,2}_0(\Omega_\psi)$. Finally, $\mathbb{U}(n)$ acts isometrically on the horizontal Folland-Stein space $HW^{1,2}_0(\mathbb H^n)$, where the action $\sharp:\mathbb{U}(n)\times HW^{1,2}_0(\mathbb H^n)\to HW^{1,2}_0(\mathbb H^n)$ is defined for every $(\widehat{\tau},u)$ in $\mathbb{U}(n)\times HW^{1,2}_0(\mathbb H^n)$ by \begin{equation*}\label{action} (\widehat{\tau}\sharp u)(q)=u({\tau}^{-1}z,t)\quad\text{for all } q=(z,t)\in\mathbb H^n, \end{equation*} in view of \cite[Lemma 3.2]{bk} A special case of Theorem \ref{th2} reads as follows. \begin{corollary}\label{th3} Let $\Omega_\psi$ be defined as above. Assume that $f$ and $h$ fulfil {\rm (H2)--(H4)}, and $h$ is cylindrically symmetric, that is $h(q)=h(z,t)=h(|z|,t)$ for every $q=(z,t)\in\Omega_\psi$. Then \eqref{prob} admits at least one nontrivial solution $u_{\lambda}$ in $HW^{1,2}_{0,\mathbb{U}(n)}(\Omega_\psi)$ for all $\lambda\le0$. Furthermore, if $\lambda>0$ and $h$ satisfies also \eqref{h2}, then there exists $\lambda^{*}>0$ such that problem \eqref{prob} admits at least one nontrivial solution $u_{\lambda}$ in $HW^{1,2}_{0,\mathbb{U}(n)}(\Omega_\psi)$ for all $\lambda\in(0, \lambda^{*})$. \end{corollary} If the functions $\psi_1$ and $\psi_2$ are bounded, the domain $\Omega_\psi$ is strongly asymptotically contractive and the whole space $HW^{1,2}_{0}(\Omega_\psi)$ is compactly embedded in $L^{\nu}(\Omega_\psi)$ for every $\nu\in (2,2^*)$. We refer to~\cite{bk,M2} for further details. In such a case Corollary~\ref{th3} follows by using the embedding result proved by Garofalo and Lanconelli in~\cite{GL}. See also Schindler and Tintarev \cite{ST}. On the Heisenberg setting, a Rubik-cube technique, see \cite{bk}, applied to subgroups of $\mathbb{U}(n)$ and suitable variational arguments allow us to obtain further multiplicity results that will be presented in the forthcoming paper \cite{MBP}. The manuscript is organized as follows. In Section~\ref{sec2} we present the notations and recall some properties of the functional solution space of~\eqref{prob}. In particular, in order to apply critical point methods to problem \eqref{prob}, we need to exploit some analytic properties of the closed subspace $HW^{1,2}_{0,T}(\Omega_\psi)$, introduced above. Then, in the same section, we give the key Lemmas~\ref{lem1} and \ref{th2.1} which are particularly useful for the proof of Theorem~\ref{th2}. Finally, in Section~\ref{sec3} we describe the geometrical profile of the underlying functional in Lemmas~\ref{lemma4.1} and~\ref{lemma4.3} and we prove the existence result stated in Theorem~\ref{th2}. For general references on the subject and on methods treated along the paper we refer to the monographs \cite{BLU,KRV} as well as \cite{dAM,MZZ,MBF,V} and the references therein. \section{Notation and preliminaries}\label{sec2} In this section we briefly recall some basic facts on Carnot groups and the functional Folland-Stein space~$HW^{1,2}_0(\Omega)$. A Carnot group $\mathbb G=(\mathbb{G},\circ)$ is a connected, simply connected, nilpotent Lie group, whose Lie algebra $\mathfrak{G}$ admits a stratification, i.e. $$ \mathfrak{G}=\oplus_{k=1}^{r}\mathfrak{G}_k, $$ where the integer $r$ is called the \emph{step} of $\mathbb{G}$, while $\mathfrak{G}_k$ is the linear subspace of finite dimension $m_k$ of $\mathfrak{G}$ for every $k\in \{1,\dots,r\}$, and $$ [\mathfrak{G}_1,\mathfrak{G}_k]=\mathfrak{G}_{k+1}\quad\text{for all $k$, with } 1\leq k] (R) -- (GK); \draw [->] (GK) -- (G); \draw [dashed,->] (R) -- (G); \end{tikzpicture} \end{center} where $\pi^{-1}$ is the inverse of the canonical projection $\pi:\mathfrak{G}\to \mathbb{R}^m$ such that \begin{center} \begin{tikzpicture} \node (R) at (0,1.4) {$\mathbb{R}^{m} \ni(x_1,\dots,x_{m_1},\dots,x_m)$}; \node (GK) at (6.3,1.4) {$X=\sum_{k=1}^{m_1}x_kX_k+\sum_{k=m_1+1}^{m}x_kX_k'\in \mathfrak{G}$}; \node (G) at (6.3,0) {$q\in \mathbb{G}$}; \node at (2.8,1.6) {$\pi^{-1}$}; \node at (3,0.4) {$\varrho$}; \node at (6.8,0.65) {$\exp_G$}; \draw [->] (R) -- (GK); \draw [->] (GK) -- (G); \draw [dashed,->] (R) -- (G); \end{tikzpicture} \end{center} Thus, we often identify every element $q\in \mathbb{G}$ with its \emph{exponential coordinates} $(x_1,\dots,x_{m_1},x_{m_1+1},\dots,x_m)\in \mathbb{R}^{m}$ respect to the basis $\mathcal{B}^{*}$ in $\mathfrak{G}$. More precisely, it is possible to identify the Carnot group $(\mathbb{G},\circ)$ with $(\mathbb{R}^{m},\star)$, where the expression of the group operation $\star$ is given by $$ x\star y=\varrho^{-1}(\varrho(x)\circ \varrho(y))\quad\text{for all } x,\,y\in \mathbb{R}^{m} $$ and is explicitly determined by the Baker-Campbell-Hausdorff formula. Whenever we are in presence of a stratification, it is possible to define a one-parameter group $\{\Delta_\eta\}_{\eta>0}$ of dilatations of the algebra. More precisely, for a fixed real number $\eta> 0$ and all $X\in \mathfrak{G}_k$, we set $\Delta_\eta (X)=\eta^k X$ and extend the map $\Delta_\eta$ to the whole $\mathfrak{G}$ by linearity. Furthermore, the family $\{\Delta_\eta\}_{\eta>0}$ induces a family $\{\delta_\eta\}_{\eta>0}$ of the group automorphisms on $\mathbb{G}$ by the exponential map such that the following diagram is commutative \begin{center} \begin{tikzpicture} \node (G) at (0,1.5) {$\mathbb{G}$}; \node (GK) at (3,1.5) {$\mathfrak{G}=\oplus_{k=1}^{r}\mathfrak{G}_k$}; \node (GG) at (0,0) {$\mathbb{G}$}; \node (GKK) at (3,0) {$\mathfrak{G}=\oplus_{k=1}^{r}\mathfrak{G}_k$}; \node at (1.2,1.8) {$\exp_G^{-1}$}; \node at (1.1,0.3) {$\exp_G$}; \node at (0.3,.8) {$\delta_\eta$}; \node at (3.3,.8) {$\Delta_\eta$}; \draw [->] (G) -- (GK); \draw [->] (GK) -- (GKK); \draw [dashed,->] (G) -- (GG); \draw [->] (GKK) -- (GG); \end{tikzpicture} \end{center} that is $$ \delta_\eta(q)= \exp _{\mathbb{G}}(\Delta_\eta(\exp _{\mathbb{G}}^{-1}(q))) $$ for every $q\in \mathbb{G}$. The \emph{homogeneous dimension} $Q$ of $\mathbb{G}$, attached to the automorphisms $\{\delta_\eta\}_{\eta>0}$, is defined by $$ Q=\sum_{k=1}^{r}k\,{\rm dim\,} \mathfrak{G}_k=m_1+2m_2+\cdots+rm_r. $$ In particular, the above definition of $Q$ and the fact that $\{\delta_\eta\}_{\eta>0}$ is a family of automorphisms on $\mathbb{G}$ imply that the Jacobian determinant of the dilation $\delta_\eta$ is constant in $q$ and given by $\eta^Q$. Moreover, let $\mu$ denote the push-forward of the $m$-dimensional Lebesgue measure $\mathfrak{L}_m$ on $\mathfrak{G}$ via the exponential map. Then, $d\mu$ defines a biinvariant Haar measure on $\mathbb{G}$ and $$ d\mu(q\circ \delta_\eta)=\eta^{Q}d\mu(q). $$ Since $\mathbb{G}$ can be identified with $(\mathbb{R}^m,\star)$ by using the exponential map, if $E\subset\mathbb{G}$ is a measurable subset, its Haar measure is explicitly given by $\mu(E)=\mathfrak{L}_m(\rho^{-1}(E))$. Therefore, the same notation will be used for both measures. Take $q_1$, $q_2\in \mathbb{G}$ and let $H\Gamma_{q_1,q_2}(\mathbb{G})$ be the set of piecewise smooth curves $\gamma$, such that $\gamma: [0,1]\to \mathbb{G}$, $\dot\gamma(t)\in \mathfrak{G}_1$ a.e. $t\in [0,1]$, $(\gamma(0),\gamma(1))=(q_1,q_2)$ and $$ \int_0^{1}\|\dot\gamma(t)\|_0dt<\infty. $$ Since $H\Gamma_{q_1,q_2}(\mathbb{G})\neq \emptyset$ by the celebrated Chow-Rashevski\u\i\ theorem in~\cite{C}, it is possible to define the \emph{Carnot-Carath\'{e}odory distance} on $\mathbb{G}$, as follows $$ d_{CC}(q_1,q_2)=\inf_{\gamma\in H\Gamma_{q_1,q_2}(\mathbb{G})} \int_0^{1}\|\dot\gamma(t)\|_0dt. $$ The metric $d_{CC}$ is left invariant on $\mathbb{G}$ and for every $\eta>0$ it results $$ d_{CC}(\delta_\eta(q_1),\delta_{\eta}(q_2))=\eta\, d_{CC}(q_1,q_2), $$ for every $q_1,q_2\in \mathbb{G}$. The Euclidean distance to the origin $|\cdot|$ on $\mathfrak{G}$ induces a homogeneous pseudo-norm $|\cdot|_{\mathfrak{G}}$ on $\mathfrak{G}$ and (via the exponential map) one on the group $\mathbb{G}$. Indeed, for $X\in \mathfrak{G}$, with $X=\sum_{k=1}^{r}X_k$, where $X_k\in \mathfrak{G}_k$, define a pseudo-norm on $\mathfrak{G}$ as follows $$ |X|_{\mathfrak{G}}=\Big(\sum_{k=1}^{r}|X_k|^{2r!/k}\Big)^{2r!}. $$ The induced pseudo-norm on $\mathbb{G}$ has the form $$ |q|_{\mathbb{G}}=|\exp _{\mathbb{G}}^{-1}(q)|_{\mathfrak{G}}\quad \text{for all } q\in \mathbb{G}. $$ The function $|\cdot|_{\mathbb{G}}$ is usually known as the \emph{non-isotropic gauge}. It defines a pseudo-distance on $\mathbb{G}$ given by $$ d(p,q)=|p^{-1}\circ q|_{\mathbb{G}}\quad\text{for all } p,q\in \mathbb{G}, $$ that is equivalent to the \emph{Carnot-Carath\'{e}odory distance} $d_{CC}$ on $\mathbb{G}$. Thus, Carnot groups are endowed with the intrinsic Carnot-Carath\'{e}odory geometry. The adjective intrinsic is meant to emphasize a privileged role played by the horizontal layer and by group translations and dilations. It is worth stressing that the Carnot-Carath\'{e}odory geometry is not Riemannian at any scale. In fact, Carnot groups can be seen as a particular case of more general structures, the so-called \emph{sub-Riemannian spaces}. The most basic second-order partial differential operator in a Carnot group $\mathbb{G}$ is the \emph{sub-Laplacian}, or equivalently the \emph{horizontal Laplacian in} $\mathbb{G}$, given by $$ \Delta_{\mathbb{G}}=\sum_{k=1}^{m_1}X_k^2. $$ We shall denote by $D_\mathbb{G}=(X_1,\dots,X_{m_1})$ the related \emph{horizontal gradient} and set $\|D_{\mathbb G}u\|_0=\big(\sum_{k=1}^{m_1}(X_ku)^2\big)^{1/2}$. Obviously, Euclidean spaces are commutative Carnot groups, and, more precisely, the only commutative Carnot groups. The simplest example of Carnot group of step two is provided by the Heisenberg group $\mathbb H^n$ of topological dimension $m=2n+1$ and homogeneous dimension $Q=2n+2$, that is the Lie group whose underlying manifold is $\mathbb R^{2n+1}$, endowed with the non-Abelian group law $$ q\circ q'=\Big(z+z',t+t'+2\sum_{i=1}^n (y_ix_i'-x_iy_i')\Big) $$ for all $q$, $q'\in \mathbb H^n$, with $$ q=(z,t)=(x_1,\dots,x_n,y_1,\dots,y_n,t),\quad q'=(z',t')=(x_1',\dots,x_n',y_1',\dots,y_n',t'). $$ The vector fields for $j=1,\dots,n$ \begin{equation}\label{eq:1} X_j=\frac {\partial} {\partial x_j}+2y_j\frac {\partial} {\partial t}, \quad Y_j=\frac {\partial} {\partial y_j}-2x_j\frac {\partial} {\partial t}, \quad \frac {\partial} {\partial t}, \end{equation} constitute a basis $\mathcal B^*$ for the real Lie algebra $\mathfrak{H}=\mathfrak{G}$ of left invariant vector fields on $\mathbb H^n$. The basis $\mathcal B^*$ satisfies the Heisenberg canonical commutation relations for position and momentum $[X_j,Y_k]=-4 \delta_{jk}{\partial}/ {\partial t}$, all other commutators being zero. If $u\in C^2(\mathbb H^n)$, then the \emph{horizontal Laplacian in} $\mathbb H^n$ of $u$, called the \emph{Kohn-Spencer Laplacian}, is defined as follows \begin{align*} \Delta_{\mathbb H^n}u &=\sum_{j=1}^n (X_j^2+Y_j^2)u\\ &=\sum_{j=1}^n \Big(\frac{\partial^2} {\partial x_j^2} +\frac{\partial^2} {\partial y_j^2} +4y_j\frac{\partial^2} {\partial x_j\partial t} -4x_j\frac{\partial^2} {\partial y_j\partial t}\Big)u +4|z|^2\frac{\partial^2 u} {\partial t^2}, \end{align*} and $\Delta_{\mathbb H^n}$ is \emph{hypoelliptic} according to the celebrated Theorem~1.1 due to \emph{H\"ormander} in~\cite{hor}. Turning back to \eqref{prob}, we need to introduce the suitable solution space. Let $\Omega$ be a nontrivial open subset of $\mathbb G$. The Folland-Stein horizontal Sobolev space $HW^{1,2}_0(\Omega)$ is the completition of $C^{\infty}_0(\Omega)$, with respect to the Hilbertian norm \begin{equation}\label{Norma} \begin{gathered} \|u\|= \Big( \int_{\Omega} \|D_{\mathbb G}u\|_{0}^2 d\mu(q) + \int_{\Omega} |u|^2 d\mu(q)\Big)^{1/2},\\ \langle u,\varphi\rangle = \int_{\Omega}\langle D_{\mathbb G}u, D_{\mathbb G} \varphi\rangle_0\,d\mu(q)+ \int_{\Omega}u\varphi d\mu(q). \end{gathered} \end{equation} Of course, if $\Omega=\mathbb G$, then $HW^{1,2}(\mathbb G)=HW^{1,2}_0(\mathbb G)$, where $HW^{1,2}(\mathbb G) $ denotes the horizontal Sobolev space of the functions $u \in L^2(\mathbb G)$ such that $D_{\mathbb G}u$ exists in the sense of distributions and $\|D_{\mathbb G}u\|_0$ is in $L^2(\mathbb G)$, endowed with the Hilbertian norm~\eqref{Norma}. In particular, the embedding \begin{equation} \label{emb}HW^{1,2}_0(\Omega) \hookrightarrow L^\nu (\Omega) \end{equation} is continuous for any $\nu \in [2,2^*]$; see Folland and Stein \cite{folStein}. Furthermore, by \cite{GarN,IV, Vas} we know that, if $\mathcal O$ is a bounded open set of $\mathbb G$, the embedding \begin{equation}\label{eq:25} HW^{1,2}_0(\mathcal O) \hookrightarrow \hookrightarrow L^\nu(\mathcal O) \end{equation} is compact for all $\nu$, with $1\le \nu<2^*$. Let $(\mathbb{G}, \circ)$ be a Carnot group, and $(T, \cdot)$ be a closed topological group, with neutral element $\jmath$. The group $T$ is said to \emph{act continuously} on $\mathbb{G}$, if there exists a map $*:T\times\mathbb{G}\to \mathbb{G}$ such that the following conditions \begin{itemize} \item[(H6)] $j*q=q$ for every $q\in \mathbb{G}$; \item[(H7)] ${\tau}_1*({\tau}_2*q)=({\tau}_1\cdot{\tau}_2)*q$ for every $\tau_1,\tau_2\in T$ and $q\in \mathbb{G}$ \end{itemize} hold. In addition, the action $*$ is \emph{left distributed} if \begin{itemize} \item[(H8)]${\tau}*(p\circ q)=({\tau}*p)\circ({\tau}*q)$ for every ${\tau}\in T$ and $p,q\in \mathbb{G}$. \end{itemize} A set $\Omega\subset \mathbb{G}$ is $T$-\emph{invariant}, with respect to $*$, if $T*\Omega=\Omega$. We assume that $T$ induces an action $\sharp:T\times HW^{1,2}_0(\mathbb G)\to HW^{1,2}_0(\mathbb G)$, defined for every $(\tau,u)\in T\times HW^{1,2}_0(\mathbb{G})$ by \begin{equation}\label{action} ({\tau}\sharp u)(q)=u({\tau}^{-1}*q)\quad\text{for all } q\in\mathbb G. \end{equation} The group $T$ acts \emph{isometrically} on $HW^{1,2}_{0}(\Omega)$ if \begin{equation}\label{invariance} \|{\tau}\sharp u\|=\|u\|\quad\text{for all } (\tau,u)\in T\times HW^{1,2}_0(\mathbb{G}). \end{equation} Let \begin{equation*} HW^{1,2}_{0,T}(\Omega) =\{u\in HW^{1,2}_0(\Omega): {\tau}\sharp u=u\text{ for all } \tau\in T\} \end{equation*} be the $T$-invariant subspace of $HW^{1,2}_0(\Omega)$. Clearly, $HW^{1,2}_{0,T}(\Omega)$ is closed, since the action $\sharp$ of $T$ on $HW^{1,2}_{0}(\Omega)$ is continuous by (H6) and (H7). The following compactness result is due to Balog and Krist\'aly and given in \cite[Theorem 3.1]{bk}. \begin{lemma}\label{lem1} Let $\mathbb{G}=(\mathbb{G}, \circ)$ be a Carnot group of step $r$ and homogeneous dimension $Q>2$, with neutral element denoted by $e$. Let $T=(T, \cdot)$ be a closed infinite topological group acting continuously and left distributively on $\mathbb{G}$ by the map $*:T\times \mathbb G\to\mathbb G$. Assume furthermore that $T$ acts isometrically on $HW^{1,2}_0(\mathbb{G})$, where the action $\sharp: T\times HW^{1,2}_0(\mathbb{G})\to HW^{1,2}_0(\mathbb{G})$ is defined in \eqref{action}. Let $\Omega$ be a nonempty $T$-invariant open subset of $\mathbb{G}$, satisfying condition {\rm (H1)}. Then the embedding $$ HW^{1,2}_{0,T}(\Omega) \hookrightarrow\hookrightarrow L^\nu (\Omega) $$ is compact for every $\nu \in (2,2^*)$. \end{lemma} \begin{remark}\rm By \eqref{emb} the embeddings $$ HW^{1,2}_{0,T}(\Omega) \hookrightarrow L^\nu (\Omega) $$ are continuous for every $\nu\in [2,2^*]$. In particular, there exists a constant $C_\nu$ such that \begin{equation}\label{eq3} \|u\|_\nu\le C_\nu \|u\|\quad\text{for all } u \in HW^{1,p}_{0,T}(\Omega), \end{equation} where $C_\nu$ depends on $\nu$ and $Q$. \end{remark} We also notice that inequality \eqref{3.1} yields \begin{equation}\begin{aligned}\label{embedding2} \|u\|_{2^*}\leq C_{Q,2}\|D_{\mathbb H^n} u \|_2 \end{aligned}\end{equation} for all $u\in HW^{1,2}_{0,T}(\Omega)$. \begin{lemma}\label{th2.1} Let $(u_k)_k$ be in $HW^{1,2}_{0,T}(\Omega)$, such that $u_k\rightharpoonup u$ weakly in $HW^{1,2}_{0,T}(\Omega)$, and $u_k\to u$ a.e. in $\Omega$. Then \begin{equation} \begin{gathered}\label{eq3.199} \lim_{k\to\infty}\int_{\Omega} |u_k-u|^{2^*}d\mu(q) =\lim_{k\to\infty}\int_{\Omega}|u_k|^{2^*}d\mu(q)-\int_{\Omega}|u|^{2^*}d\mu(q),\\ \lim_{k\to\infty}\int_{\Omega}|u|^{2^*-2}u(u_k-u)d\mu(q)=0,\\ \lim_{k\to\infty}\int_{\Omega}|u_k|^{2^*-2}u_kud\mu(q)=\int_{\Omega}|u|^{2^*}d\mu(q). \end{gathered} \end{equation} \end{lemma} \begin{proof} The first part of \eqref{eq3.199} is just the celebrated Brezis-Lieb lemma in \cite{brezislieb}. For the second part of~\eqref{eq3.199}, it is enough to observe that $u_k\rightharpoonup u$ in $L^{2^*}(\Omega)$ by Lemma~\ref{lem1} and that $\varphi\mapsto \int_{\Omega}|u|^{2^*-2}u\varphi d\mu(q)$ is a linear continuous functional on $L^{2^*}(\Omega)$. While the third limit is a consequence of \cite[Proposition A.8]{Autuori}. \end{proof} A function $u\in HW^{1,2}_0(\Omega)$ is said to be a (weak) \emph{solution} of problem~\eqref{prob} if \begin{equation}\label{weaksol} \langle u,\varphi\rangle=\int_{\Omega}h(q)f(u)\varphi d\mu(q) +\lambda\int_{\Omega}|u|^{2^*-2}u\varphi d\mu(q) \end{equation} for any $\varphi\in HW^{1,2}_0(\Omega)$. Problem \eqref{prob} has a variational nature and the Euler-Lagrange functional $I_\lambda$ associated to \eqref{prob} is $$ {I}_\lambda(u)=\frac{1}{2}\|u\|^2-\int_{\Omega}h(q)F(u)d\mu(q) -\frac{\lambda}{2^*}\int_{\Omega} |u|^{2^*} d\mu(q). $$ Clearly, the functional $I_\lambda$ is well-defined in $HW^{1,2}_0(\Omega)$ and, thanks to (H2) and (H4), it is of class $C^1(HW^{1,2}_0(\Omega))$. Moreover, for every $u\in HW^{1,2}_0(\Omega)$ \begin{equation}\label{weakform} \langle I'_\lambda(u),\varphi\rangle =\langle u,\varphi\rangle-\int_{\Omega}h(q)f(u)\varphi d\mu(q) -\lambda\int_{\Omega}|u|^{2^*-2}u\varphi d\mu(q) \end{equation} for all $\varphi\in HW^{1,2}_0(\Omega)$. Hence, the critical points of $I_\lambda$ in~$HW^{1,2}_0(\Omega)$ are exactly the (weak) solutions of~\eqref{prob}. Let $u\in HW^{1,2}_{0,T}(\Omega)$ be a solution of problem~\eqref{prob} only in the $HW^{1,2}_{0,T}(\Omega)$ sense, that is \begin{align}\label{weaksol2} \langle u,\varphi\rangle=\int_{\Omega}h(q)f(u)\varphi d\mu(q) +\lambda\int_{\Omega}|u|^{2^*-2}u\varphi d\mu(q) \end{align} for any $\varphi\in HW^{1,2}_{0,T}(\Omega)$. Then, $u\in HW^{1,2}_{0,T}(\Omega)$ is a solution of \eqref{prob} in the whole space $HW^{1,2}_0(\Omega)$, that is in sense of definition \eqref{weaksol}, if the \emph{principle of symmetric criticality} of Palais given in~\cite{pala} holds. To prove this let us recall the well known principle of symmetric criticality of Palais stated in the general form proved in~\cite{demo} for reflexive strictly convex Banach spaces. For details and comments we refer to \cite[Section 5]{cp}. More precisely, let $X=(X,\|\cdot\|_X)$ be a reflexive strictly convex Banach space. Suppose that $\mathcal G$ is a subgroup of isometries $g: X\to X$, that is $g$ is linear and $\|{gu}\|_X=\|{u}\|_X$ for all $u\in X$. Consider the $\mathcal G$-invariant closed subspace of $X$, \begin{equation*} \Sigma_{\mathcal G}=\{u\in X: gu=u\text{ for all }g\in\mathcal G\}. \end{equation*} By \cite[Proposition 3.1]{demo} we have \begin{lemma}\label{lem3} Let $X$, $\mathcal G$ and $\Sigma$ be as before and let $I$ be a $C^1$ functional defined on $X$ such that $I\circ g=I$ for all $g\in \mathcal{G}$. Then $u\in\Sigma_{\mathcal G}$ is a critical point of $I$ if and only if $u$ is a critical point of $\mathcal J=I\vert_{\Sigma_{\mathcal G}}$. \end{lemma} From now on we assume that $T$ satisfies the main structural conditions of Theorem~\ref{th2} and that $\Omega$ is a nonempty open subset of $\mathbb{G}$, which is $T$-invariant. We apply the principle of symmetric criticality to the Sobolev space $HW^{1,2}_{0,T}(\Omega)$ under the action $\sharp:T\times HW^{1,2}_0(\mathbb{G})\to HW^{1,2}_0(\mathbb{G})$ defined in \eqref{action}. Clearly, \begin{equation}\label{invariance} \|{\tau}\sharp u\|=\|u\|\quad\text{for all } (\tau, u)\in T\times HW^{1,2}_{0}(\Omega), \end{equation} since $T$ acts isometrically on $HW^{1,2}_{0}(\Omega)$ by assumption. Moreover, the functional $\Psi:HW^{1,2}_0(\Omega)\to\mathbb{R}$ is $T$-invariant by assumption (H5). Thus, $I_\lambda$ is $T$-invariant in $HW^{1,2}_{0}(\Omega)$. Hence, the principle of symmetric criticality of Palais ensures that $u\in HW^{1,2}_{0,T}(\Omega)$ is a solution of problem \eqref{prob} if and only if $u$ is a critical point of the functional $\mathcal{J}_\lambda:HW^{1,2}_{0,T}(\Omega)\to \mathbb{R}$, where $\mathcal{J}_\lambda=I_\lambda\vert_{HW^{1,2}_{0,T}(\Omega)}$. We end the section by an essential lemma which shows when the key assumption (H5) is satisfied. To this aim, we need to introduce some facts well known in abstract group measure theory. \begin{lemma}\label{lem4} Suppose that the action $*$ of the group $T$ on the Carnot group $\mathbb G$ satisfies conditions {\rm (H6)--(H8)}. Assume furthermore that the natural Haar measure $\mu$, defined on $\mathbb G$, is left $*$ invariant, that is for all measurable subset $E$ of $\mathbb G$ and for all $\tau\in T$ $$ \mu(\tau*E)=\mu(E), $$ where $\tau*E=\{\tau*q:q\in E\}$. If $h$ is $T$-invariant, that is $h(\tau*q)=h(q)$ for all $\tau\in T$ and $q\in\mathbb G$, and $f:\mathbb R\to \mathbb R$ is a continuous function, then {\rm (H5)} holds. \end{lemma} \begin{proof} Fix $\tau\in T$ and $u\in HW^{1,2}_0(\Omega)$. Then, putting $\tau^{-1}*q=p$, we get by (H6)--(H8) \begin{align*} \Psi(\tau\sharp u) &=\int_\Omega h(q)f((\tau\sharp u)(q))d\mu(q) =\int_\Omega h(q)f(u(\tau^{-1}*q))d\mu(q)\\ &=\int_{\tau*\Omega} h(\tau*p)f(u(p))d\mu(\tau*p)\\ &=\int_{\Omega}h(p)f(u(p))d\mu(p)=\Psi(u), \end{align*} since $\Omega$ and $h$ are $T$-invariant by assumption, and the left $*$ invariance of the measure $\mu$ implies $$d\mu(\tau*p)=d\mu(p)\quad\text{for all }p\in\mathbb G,$$ which is exactly \cite[formula (10)]{BLie}, being 1 the multiplier of $\mu$. See also \cite[Chapter 4]{BBE}. This shows that $\Psi$ is $T$-invariant, that is $\Psi$ satisfies (H5), and concludes the proof. \end{proof} \section{Proof of Theorem \ref{th2}}\label{sec3} In this section we suppose that the assumptions of Theorem~\ref{th2} are satisfied, without further mentioning. Thus, problem~\eqref{prob} has a variational structure and, as explained in Section~\ref{sec2}, it is enough to study the critical points of the functional $\mathcal{J}_\lambda:HW^{1,2}_{0,T}(\Omega)\to\mathbb R$, defined by \begin{equation}\label{istorto} {\mathcal{J}}_\lambda(u)=\frac{1}{2}\|u\|^2-\int_{\Omega}h(q)F(u)d\mu(q) -\frac{\lambda}{2^*}\int_{\Omega} |u|^{2^*} d\mu(q) \end{equation} for all $u\in HW^{1,2}_{0,T}(\Omega)$. We first show that $\mathcal{J}_\lambda$ has a useful geometrical profile, and recall that, when $\lambda>0$, we require also \eqref{h2} on $h$. \begin{lemma}\label{lemma4.1} For any parameter $\lambda\le1$ there exist positive numbers $\rho_0$ and $\mathfrak j$ such that ${\mathcal J}_{\lambda}(u)\geq\mathfrak j$ for any $u\in HW^{1,2}_{0,T}(\Omega)$, with $\|u\|=\rho_0$, and for any function $h$ of the type stated in Theorem \ref{th2}. Moreover, $$ m_{\lambda}=\inf_{u\in \overline{B}_{\rho_0}} {\mathcal J}_{\lambda}(u)<0, $$ where $B_{\rho_0}=\{u\in HW^{1,2}_{0,T}(\Omega): \|u\|<\rho_0\}$, and there exist a sequence $(u_k)_k$ in $B_{\rho_0}$ and a function $u_{\lambda}$ in $\overline{B}_{\rho_0}$ such that for all $k$, \begin{equation}\label{ekeland} \begin{gathered} \|u_k\|<\rho_0,\quad m_{\lambda}\leq {\mathcal J}_{\lambda}(u_k) \leq m_{\lambda}+\frac{1}{k},\\ u_k\rightharpoonup u_{\lambda}\text{ in }HW^{1,2}_{0,T}(\Omega), \quad u_k\to u_{\lambda}\text{ a.e. in } \Omega,\\ {\mathcal J}_{\lambda}'(u_k)\to0\quad \text{in }[HW^{1,2}_{0,T}(\Omega)]'. \end{gathered} \end{equation} \end{lemma} \begin{proof} Fix $\lambda\le1$. By (H2), Lemma~\ref{lem1} and \eqref{embedding2} we obtain \begin{equation}\label{new}\begin{aligned} {\mathcal J}_{\lambda}(u) &\geq\frac{1}{2}\|u\|^2-C\int_{\Omega}h(q)|u|^{s}d\mu(q) -\frac{\lambda}{2^*}\|u\|_{2^*}^{2^*}\\ &\geq\frac{1}{2}\|u\|^2 -CC_{Q,2}^s\|h\|_{\frac{2^*}{2^*-s}}\|u\|^{s} -\frac{\lambda^+}{2^*}C_{Q,2}^{2^*}\|u\|^{2^*}, \end{aligned} \end{equation} for all $u\in HW^{1,2}_{0,T}(\Omega)$. Therefore, if $\lambda\le0$, for $\rho_0>0$ sufficiently large we have $$ {\mathcal J}_{\lambda}(u) \geq \rho_0^s\Big[\frac{1}{2} \rho_0^{2-s}-CC_{Q,2}^s\|h\|_{\frac{2^*}{2^*-s}} \Big] =\jmath>0 $$ for all $u\in HW^{1,2}_{0,T}(\Omega)$, with $\|u\|=\rho_0$, since $10$ \begin{align*} CC_{Q,2}^s\|h\|_{\frac{2^*}{2^*-s}}\|u\|^s \leq \varepsilon\|u\|^{2} +\varepsilon^{-\frac{s}{2-s}} \Big(CC_{Q,2}^s\|h\|_{\frac{2^*}{2^*-s}}\Big)^{\frac{2}{2-s}}, \end{align*} being $10$ is such that $$ \eta(\rho_0)=\max_{t\geq 0}\eta(t)=\frac{1}{2} \Big(\frac{1}{2}-\frac{1}{2^*}\Big)\big(2C_{Q,2}^{2^*}\big)^{2/(2-2^*)}>0 $$ because $2<2^*$. Therefore, for any function $h$, satisfying \eqref{h2}, and for any $u$ in $HW^{1,2}_{0,T}(\Omega)$, with $\|u\|=\rho_0$, we obtain $$ {\mathcal J}_{\lambda}(u)\geq \eta(\rho_0)- \Big(2^sCC_{Q,2}^s\|h\|_{\frac{2^*}{2^*-s}}\Big)^{2/(2-s)}=\mathfrak j>0, $$ which concludes the proof of the first part. Let $q_0\in \Omega_0$ and $R>0$ be so small that $B\subset\Omega_0$, where $B=B(q_0,2R)$ is the open ball of center $q_0$ and radius $R$ and $\Omega_0$ is given in (H4). Choose $\varphi \in C_0^\infty(B)$ such that $0\le\varphi\leq 1$, with $\|\varphi\|\leq\rho_0$, and $\int_{B}\varphi^{s_1}d\mu(q)>0$. Let $\delta>0$ be the number given in (H3). For all $t\in(0,\delta)$, then (H3) and (H4) yield \begin{align*} {\mathcal J}_{\lambda}(t\varphi) &\leq\frac{1}{2}\|t\varphi\|^2 -\int_{\Omega}h(q)F(t\varphi)d\mu(q)-\lambda\, \frac{t^{2^*}}{2^*}\int_{\Omega}\varphi^{2^*}d\mu(q)\\ &\leq \frac{t^2}{2}\|\varphi\|^{2}-\int_{\Omega}h(q)F(t\varphi)d\mu(q) +\lambda^-\frac{t^{2^*}}{2^*}\int_{B}\varphi^{2^*}d\mu(q)\\ &\leq\frac{t^2}{2}\rho_0^2 -t^{s_1}a_0\inf_{q\in\Omega_0}h(q)\int_{B}\varphi^{s_1}d\mu(q) +\lambda^-\frac{t^{2^*}}{2^*}\int_{B}\varphi^{2^*}d\mu(q). \end{align*} Hence, ${\mathcal J}_{\lambda}(t\varphi)<0$ for for $t\in(0,\delta)$ sufficiently small, since $10$ there exists a measurable set $E\subset \Omega$ such that \begin{align*} &\int_{\Omega\setminus E} h(q)|u_k-u_{\lambda}|^sd\mu(q)\\ &\leq \Big(\int_{\Omega\setminus E}|h(q)|^{2^*/(2^*-s)}d\mu(q) \Big)^{(2^*-s)/2^*} \|u_k-u_{\lambda}\|_{2^*}^2 \leq \frac{\varepsilon}{2}. \end{align*} Furthermore, for any measurable subset $U\subset E$, by the H\"older inequality $$ \int_U h(q)|u_k-u_{\lambda}|^sd\mu(q) \leq c\Big(\int_U|h(q)|^{2^*/(2^*-s)}d\mu(q)\Big)^{(2^*-s)/2^*}, $$ where $c=\sup_k\|u_k-u_{\lambda}\|_{2^*}^2$. Hence, $\{h(q)|u_k-u_{\lambda}|^s\}_k$ is equi-integrable and uniformly bounded in $L^1(E)$, thanks to (H4). Thus by \eqref{eq3.14} and the Vitali convergence theorem, for all $\varepsilon>0$ there exists $k_0>0$ such that $$ \int_{E}h(q)|u_k-u_{\lambda}|^sd\mu(q)\leq \frac{\varepsilon}{2} $$ for all $k\geq k_0$. Therefore, \begin{align*} \int_{\Omega}h(q)|u_k-u_{\lambda}|^sd\mu(q) \leq \int_{\Omega\setminus E}h(q)|u_k-u_{\lambda}|^sd\mu(q) +\int_{E}h(q)|u_k-u_{\lambda}|^sd\mu(q) \leq\varepsilon \end{align*} for all $k\geq k_0$. This proves the claim and \eqref{4.3}. Now (H2) and the H\"older inequality give \begin{align*} \big|\int_{\Omega}h(q)f(u_k)(u_k-u_{\lambda})d\mu(q)\big| &\leq C\int_{\Omega}h(q)|u_k|^{s-1}|u_k-u_{\lambda}|d\mu(q)\\ &\leq \tilde C\Big(\int_{\Omega}h(q)|u_k-u_{\lambda}|^sd\mu(q)\Big)^{1/s}, \end{align*} for a suitable constant $\tilde C>0$. Thus, by \eqref{4.3} it follows that \begin{align}\label{eq4.3} \lim_{k\to\infty}\int_{\Omega}h(q)f(u_k)(u_k-u_{\lambda})d\mu(q)=0. \end{align} Similarly, by using again (H4) and (H2) we have as $k\to\infty$ \begin{equation}\label{due2} \int_{\Omega}h(q)f(u_k)\varphi d\mu(q)\to\int_{\Omega}h(q)f(u_{\lambda}) \varphi d\mu(q), \end{equation} for any $\varphi\in HW^{1,2}_{0,T}(\Omega)$. By \eqref{ekeland}, \eqref{eq3.14}--\eqref{due2} we see that $u_\lambda$ is a solution of \eqref{prob}, that is $u_\lambda$ is a critical point of ${\mathcal J}_\lambda$ in $HW^{1,2}_{0,T}(\Omega)$. In particular, as $k\to\infty$ \begin{align*} o(1)=\langle {\mathcal J}_\lambda'(u_k)-{\mathcal J}_\lambda'(u_\lambda), u_k-u_\lambda\rangle =(\kappa_{\lambda}^2-\|u_{\lambda}\|^2) -\|u_k\|_{2^*}^{2^*}+\|u_\lambda\|_{2^*}^{2^*}+o(1). \end{align*} Consequently, by \eqref{eq3.14} and the Br\'ezis-Lieb lemma \cite{brezislieb} we get the main formula \begin{align}\label{eq4.6} \lim_{k\to\infty}\|u_k-u_{\lambda}\|^2= \lambda\lim_{k\to\infty}\|u_k-u_{\lambda}\|_{2^*}^{2^*}=\lambda\,{c}_{\lambda}. \end{align} Let us first consider the case $\lambda\le0$. Then, \eqref{eq4.6} gives at once that $\|u_k-u_{\lambda}\|=o(1)$ as $k\to\infty$, that is $(u_k)_k$ strongly converges to $u_{\lambda}$ in $HW^{1,2}_{0,T}(\Omega)$, as stated. Let us now consider the case $\lambda\in(0,1]$. By using \eqref{embedding2}, with $u=u_k-u_{\lambda}$, we get \begin{equation}\label{jnew} \lambda\,{c}_{\lambda} \ge C_{Q,2}^{2^*}{c}_{\lambda}^{2/2^*} \end{equation} for all $\lambda\in(0,1]$. Let us define $$ \lambda^{*}= \begin{cases} \inf\{\lambda\in(0,1]:{c}_{\lambda}>0\},&\text{if there exists $\lambda\in(0,1]$ such that } {c}_{\lambda}>0,\\ 1,&\text{if ${c}_{\lambda}= 0$ for all } \lambda\in(0,1]. \end{cases} $$ We claim that $\lambda^{*}>0$ if there exists $\lambda>0$ such that ${c}_{\lambda}>0$. Otherwise, there exists a sequence $(\lambda_k)_k$, with $c_{\lambda_k}>0$, such that $\lambda_k\to 0$ as $k\to\infty$. Thus, \eqref{jnew} implies that $$ \lambda_kc_{\lambda_k}^{1-2/2^*}\ge C_{Q,2}^{2^*}>0. $$ This is an obvious contradiction since $\{{c}_{\lambda}\}_{\lambda\in(0,1]}$ is uniformly bounded above by~\eqref{embedding2}. Indeed, $(u_k)_k\subset B_{\rho_0}$, $u_{\lambda}\in\overline{B}_{\rho_0}$ and $\rho_0$, given in Lemma~\ref{lemma4.1}, is independent of $\lambda$. Hence, ${c}_{\lambda}=0$ for any $\lambda\in (0, \lambda^{*})$. Therefore, for all $\lambda\in (0, \lambda^{*})$, $$ \lim_{k\to\infty}\|u_k-u_{\lambda}\|_{2^*}=0. $$ Now \eqref{eq4.6} implies $$ \lim_{k\to\infty}\|u_k-u_{\lambda}\|=0. $$ In conclusion, $u_k\to u_{\lambda}$ as $k\to\infty$ in $HW^{1,2}_{0,T}(\Omega)$ for all $\lambda<\lambda^{*}$, as stated. \end{proof} \begin{proof}[Proof of Theorem \ref{th2}] Let $\mathcal{J}_\lambda$ be the restriction of the energy functional $I_\lambda$ to the subspace $HW^{1,2}_{0,T}(\Omega)$. For any $\lambda\le1$ Lemma~\ref{lemma4.1} and the Ekeland variational principle give the existence of a Palais-Smale sequence $(u_k)_k$ in $HW^{1,2}_{0,T}(\Omega)$ of $\mathcal{J}_\lambda$ at level $m_{\lambda}<0$. Moreover, by Lemma~\ref{lemma4.3} there exists $\lambda^{*}>0$ such that, up to a subsequence, $(u_k)_k$ strongly converges to some $u_{\lambda}$ in $HW^{1,2}_{0,T}(\Omega)$ for all $\lambda<\lambda^{*}$. Furthermore, $m_{\lambda}=\mathcal{J}_\lambda(u_\lambda)<0$ and $\mathcal{J}_\lambda'(u_\lambda)=0$ for all $\lambda<\lambda^{*}$. Consequently, the function $u_\lambda\in HW^{1,2}_{0,T}(\Omega)$ is a nontrivial critical point of the functional $\mathcal{J}_\lambda$. Now, as observed in Section~\ref{sec2}, since the action $\sharp:T\times HW^{1,2}_0(\Omega)\to HW^{1,2}_0(\Omega)$ given in \eqref{action} is supposed to be isometric, the functional $I_\lambda$ is $T$-invariant by assumption (H5). Hence, the principle of symmetric criticality of Palais, recalled in \eqref{lem3}, implies that $u_\lambda\in HW^{1,2}_{0,T}(\Omega)$ is a nontrivial critical point also for $I_\lambda$ in $HW^{1,2}_0(\Omega)$, that is a nontrivial solution for \eqref{prob} in the sense of definition \eqref{weaksol}. This completes the proof. \end{proof} \subsection*{Acknowledgments} The authors were partly supported by the Italian MIUR project \emph{Variational methods, with applications to problems in mathematical physics and geometry} (2015KB9WPT\_009), and are members of the \emph{Gruppo Nazionale per l'Analisi Matematica, la Probabilit\`a e le loro Applicazioni} (GNAMPA) of the \emph{Istituto Nazionale di Alta Matematica} (INdAM). The manuscript was realized within the auspices of the INdAM-GNAMPA Project 2018 denominated {\em Problemi non lineari alle derivate parziali} Prot\_U-UFMBAZ-2018-000384), and of the {\em Fondo Ricerca di Base di Ateneo - Esercizio 2015} of the University of Perugia, named {\em PDEs e Analisi Nonlineare}. \begin{thebibliography}{00} \bibitem{Autuori} G. Autuori, P. Pucci; \emph{Existence of entire solutions for a class of quasilinear elliptic equations}, NoDEA Nonlinear Differential Equations Appl., \textbf{20} (2013), 977--1009. \bibitem{bk} Z.M. Balogh, A. 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