0$ for all $x\in \overline \Omega_{\varepsilon_j}$. Then, the following theorem on approximation by bounded domains holds. \begin{theorem}\label{thm3.1} The problem \begin{equation} \label{eP01} -\Delta u = f(x)|u|^{\frac{4}{n-2}}u \quad \text{in }\mathbb{R}^n, \quad n\geq3 \end{equation} has a generalized non-radial nodal $G$-invariant and $\sigma$-antisymmetrical solution $u $ and there is a subsequence $\{ u_j \}$, such that $$ u_ j \rightharpoonup u \quad \text{in $H^2_{1,G}$ as } j\to+\infty. $$ \end{theorem} \begin{proof} According to Theorem \ref{thm2.1}, every problem \eqref{ePej} has at least one non-radial nodal $G$-invariant and $\sigma$-antisymmetrical solution $ u_{\varepsilon_j}$. Let $u_{\varepsilon_j}, j=1,2,\dots$ an arbitrary sequence of such solutions. Since the problem \eqref{ePej} has a nontrivial solution belonging to one of the spaces considered earlier, then for any $\lambda>0$ the function $$ \upsilon_{\varepsilon_j} =\lambda^{\frac{n-2}{4}}u_{\varepsilon_j} \in{\mathaccent"7017 H}^2_{1}(\Omega_{\varepsilon_j} ) $$ is a non trivial solution to the problem \begin{equation} \label{ePejl} \begin{gathered} -\Delta \upsilon_{\varepsilon_j} + \varepsilon_j a(x)\upsilon_{\varepsilon_j} = \lambda f(x)|\upsilon_{\varepsilon_j}|^{\frac{4}{n-2}} \upsilon_{\varepsilon_j},\quad n\geq3, \\ \upsilon_{\varepsilon_j} \not \equiv 0\quad \text{in }\Omega _{\varepsilon_j},\quad \upsilon_{\varepsilon_j} =0\quad\text{on } \partial \Omega _{\varepsilon_j}. \end{gathered} \end{equation} For $$ \lambda=\|u_{\varepsilon_j}\|_{H^2_1(\Omega_{\varepsilon_j})}^{\frac{-4}{n-2}} $$ we conclude that $$ \upsilon_{\varepsilon_j}=\frac{u_{\varepsilon_j}} {\|u_{\varepsilon_j}\|_{H^2_1(\Omega_{\varepsilon_j})}}, $$ which means that the sequence $\{\upsilon_{\varepsilon_j}\}$ is bounded in ${\mathaccent"7017 H}^2_{1}(\Omega_{\varepsilon_j})$ for all $j=1,2,\dots$. Thus, there exists a constant $C$ not dependent on $j$ and such that \begin{equation}\label{E6} \|\upsilon_{\varepsilon_j}\|_{H^2_1(\Omega_{\varepsilon_j})}\leq C. \end{equation} Because of the reflexivity of ${\mathaccent"7017 H}^2_{1}(\mathbb{R}^n )$ and condition \eqref{E6} we may choose a subsequence $\{ \upsilon_j\}$ of the sequence $\{\upsilon_{\varepsilon_j}\}$ such that \begin{equation}\label{E7} \upsilon_j\rightharpoonup \upsilon\quad \text{in }{\mathaccent"7017 H}^2_{1}(\mathbb{R}^n)\quad\text{as }j\to+\infty. \end{equation} We shall show that $\upsilon$ is a nontrivial $G$-invariant generalized solution to the problem \eqref{ePej}. We choose an arbitrary $\varphi\in \mathcal{D}(\mathbb{R}^n)$. Then, according to the definition of $\mathcal{D}(\mathbb{R}^n)$, the support of $\varphi$ is bounded in $\mathbb{R}^n$, which means that there is an $\Omega_{\varepsilon_0}$ such that $\operatorname{supp} \varphi\subset \Omega_{\varepsilon_0}$. Since, by definition, the $\Omega_{\varepsilon_j}$s constitute a family of expanding domains, we can choose the $\Omega_{\varepsilon_0}$ such that $\Omega_{\varepsilon_0} \subset \Omega_{\varepsilon_{_1}}$ and so $\Omega_{\varepsilon_0} \subset \Omega_{\varepsilon_j}$ for all $j=1,2,\dots$. Let $$ g(x,\upsilon_j)=-\varepsilon_ja(x)\upsilon_{\varepsilon_j} + \lambda f(x)|\upsilon_{\varepsilon_j}|^{\frac{4}{n-2}}\upsilon_{\varepsilon_j}. $$ Then, because the $\upsilon_j$ is a generalized solution to \eqref{ePejl}, it holds \begin{equation} \label{E8} \begin{aligned} \int_{\mathbb{R}^n} \nabla\upsilon_j\nabla\varphi\, dx &=\int_{\Omega_{\varepsilon_j}} \nabla\upsilon_j\nabla\varphi dx \\ &=-\int_{\Omega_{\varepsilon_j}} g(x,\upsilon_j)\varphi dx \\ &=-\int_{\Omega_{\varepsilon_0}} g(x,\upsilon_j)\varphi dx \end{aligned} \end{equation} for all $\Omega_{\varepsilon_j} $. By the weak convergence \eqref{E7}, we obtain the following limit relation for the left-hand side of \eqref{E8}: \begin{equation}\label{E9} \lim_{j \to \infty } \int_{\mathbb{R}^n} \nabla\upsilon_j\nabla\varphi dx =\int_{\mathbb{R}^n} \nabla\upsilon\nabla\varphi dx. \end{equation} In addition, it is well known, (see \cite{Heb-Vau1}), that the critical exponent of the Sobolev embedding $ H_{1,G}^2(\Omega_{\varepsilon_0})\hookrightarrow L^{p}(\Omega_{\varepsilon_0}) $ is equal to $$ \frac{2(n-k)}{n-k-2}>\frac{2n}{n-2}=2^*, $$ from which it follows that for any real number $p$, such that: $$ 1

0$ there exists a positive integer $j_{01}$ such that \begin{equation}\label{E14} |\upsilon|<\frac{\varepsilon}{2}\quad\text{for all } j>j_{01}. \end{equation} On the other hand, from \eqref{E10} by the H\"older inequality arises that $\upsilon_j\to \upsilon$ in $L^1(\Omega_{\varepsilon_0})$, which means that for any $\varepsilon >0$ there exists a positive integer $j_{02}$ such that \begin{equation}\label{E15} |\upsilon_j-\upsilon|<\frac{\varepsilon}{2}\quad \text{for all } j>j_{02}. \end{equation} Therefore, by the standard inequality $|\upsilon_j|\leq |\upsilon_j-\upsilon|+|\upsilon|$ by \eqref{E14} and \eqref{E15} we obtain \begin{equation}\label{E16} |\upsilon_j|<\varepsilon\quad\text{for any } j\geq j_0=\max\{j_{o1}, j_{02}\}. \end{equation} We recall now that every solution to the problem \eqref{ePej} belongs to the set $$ \mathcal{H}_\varepsilon^\sigma =\big\{u_\varepsilon\in{\mathaccent"7017 H}^2_{1, G}(\Omega_{\varepsilon_j}) : u_{\varepsilon_j}\circ \sigma =-u_{\varepsilon_j}\text{ and } \int_{\Omega_{\varepsilon_j}} f(x)|u_{\varepsilon_j}|^{\frac{2n}{n-2}} dx=1\big\} $$ Since every $\upsilon_j$ corresponds to an $u_{\varepsilon_j}\in \mathcal{H}_\varepsilon^\sigma$, and $\upsilon_{\varepsilon_j}=\lambda^{\frac{n-2}{4}}u_{\varepsilon_j}$, by definition, we have $$ 1=\int_{\Omega_{\varepsilon_j}} f(x)\lambda^{-n/2}|\upsilon_j|^{\frac{2n}{n-2}}\,dx <\int_{\Omega_{\varepsilon_j}} f(x)\lambda^{-n/2}\varepsilon^{\frac{2n}{n-2}} dx, $$ which is false by \eqref{E16} as the $\varepsilon>0$ can be chosen as small as we want. We have proved that the limit problem \begin{equation} \label{eP0l} -\Delta \upsilon = \lambda f(x)|\upsilon|^{\frac{4}{n-2}} \upsilon \quad \text{in }\mathbb{R}^n,\quad n\geq3 \end{equation} has a generalized non-radial nodal $G$-invariant and $\sigma$-antisymmetrical solution $\upsilon $, which means that the function $u=\lambda^{\frac{2n}{n-2}}\upsilon$ is a generalized non-radial nodal $G$-invariant and $\sigma$-antisymmetrical solution to the limit problem \begin{equation} \label{eP02} -\Delta u = f(x)|u|^{\frac{4}{n-2}}u \quad \text{in }\mathbb{R}^n, \quad n\geq3 . \end{equation} This completes the proof of the theorem. \end{proof} \begin{corollary} \label{coro3.1} The problem \begin{equation} \label{eP22} -\Delta u = |u|^{\frac{4}{n-2}}u ,\quad u\in C^2(\mathbb{R}^n), \quad n\geq3 \end{equation} has a sequence $\{u_k\}$ of non-radial nodal $G$-invariant and $\sigma$-antisymmetrical solutions, such that $$ \lim_{k\to +\infty} \int_{\mathbb{R}^n} |\nabla u_k|^2dx=+\infty $$ \end{corollary} The proof of the above corollary is obtained immediately if we put $$ f(x)=\frac{1}{|\Omega_\varepsilon|}-\varepsilon|x|^\alpha,\quad \alpha>-n $$ and follow the steps of Theorem \ref{thm3.1}. \begin{remark} \label{rmk3.1} \rm The number of the sequences of non-radial nodal $G$-invariant and $\sigma$-antisymmetrical solutions to problem \eqref{eP}, depends on the number of all subgroups of $\,O(n)\,$ of which the cardinal of orbits with minimum volume is infinite, that are on the dimension $n$ of the domain. \end{remark} To formulate our last result which is a direct conclusion from \cite{Wang}, we have to repeat some assumptions about $\Omega$. Suppose that $\Omega$ is a smooth and bounded domain of $\mathbb{R}^n=\mathbb{R}^2\times\mathbb{R}^{n-2}$, $n\geq4$, satisfying the following properties: Let $x=(t_1, t_2, \dots,t_n)=(x_1,x_2)\in\mathbb{R}^2\times\mathbb{R}^{n-2}$, and let $r=|x_1|$. Then: \begin{itemize} \item[(H1)] $x\in \Omega$ if and only if $(t_1, t_2, \dots,-t_j,\dots,t_n) \in \Omega$ for $j=3,4,\dots,n$; \item [(H2)] $(r \cos \theta, r\sin \theta,x_2)\in\Omega$ if $(r,0,x_2)\in\Omega$, for all $\theta\in(0, 2\pi)$; \item[(H3)] There exists a connected component $\Gamma$ of $\partial\Omega\cap\{x_2=0\}$, such that $H(x)\equiv \gamma>0$ for all $x\in \Gamma$, where $H(x)$ is the mean curvature of $\partial\Omega$ at $x\in \partial\Omega$. \end{itemize} \begin{remark} \label{rmk3.2} \rm From (H2) arises that $\Gamma$ is a circle in the plane $t_3=\dots=t_n=0$, and since for $x\in \Gamma$, $H(x)=\frac{{\sum\nolimits_{j = 1}^{n - 1} {k_j ( x)} }} {{n - 1}}$, where $k_j(x)$ are the principal curvatures and $k_1(x)=\frac{1}{\sqrt{t_1^2+t_2^2}}$, implies that $H(x)\equiv\gamma=\frac{1}{\sqrt{t_1^2+t_2^2}}$, which means that a such domain is very common, e.g. a ball or an ellipsoid. \end{remark} \begin{corollary} \label{coro3.2} Suppose that $\Omega$ is a smooth bounded domain satisfying {\rm (H1)--(H3)}. Then the problem \begin{equation} \label{ePp} \begin{gathered} - \Delta u= u^{\frac{{n + 2}}{{n - 2}}}\quad u > 0\quad\text{in } \mathbb{R}^n \backslash \Omega , \\ u( x) \to 0\quad \text{as } | x | \to + \infty , \\ \frac{{\partial u}} {{\partial n}} = 0\quad \text{on } \partial \Omega , \end{gathered} \end{equation} has infinitely many non-radial positive solutions, whose energy can be made arbitrary large. In particular, problem \eqref{ePp} has in $\mathbb{R}^n$, (apart from a set $\Omega$ of finite measure arbitrary small), infinity many non$-$radial positive solutions, whose energy can be made arbitrary large, in the sense that we can choose an $\Omega$ with the above refereed properties and the additional property $|\Omega|<\varepsilon$ for given $\varepsilon>0$. \end{corollary} The proof of the above Corollary follows by \cite[Theorem 1.1]{Wang}. \begin{thebibliography}{99} \bibitem{Adi-Yad} Adimurthi, S. L. Yadava; Elementary proof of the nonexistence of nodal solutions for the semilinear elliptic equations with critical Sobolev exponent, {\it Nonlinear Anal.}, \textbf{14}(9) (1990), 785-787. \bibitem{Amb-Rab} A. Ambrosetti, P. H. Rabinowitz; Dual variational methods in critical point theory and applications, {\it J. Funct. Anal.}, \textbf{14} (1973), 349-381. \bibitem{Amb-Str} A. Ambrosetti, M. Struve; A note on the problem $-\Delta u=\lambda u+|u|^{2^*-2}u$, {\it Manuscripta Math.}, \textbf{54} (1986), 373-379. \bibitem{Act-Bre-Pel} F. V. Atkinson, H. Brezis, L. A. Peletier; Nodal solutions of elliptic equations with critical Sobolev exponents, {\it J. Diff. Eq.}, \textbf{85} (1990), 151-170. \bibitem{Aub1} Th. Aubin; {\it Some non linear problems in Riemannian Geometry}, Springer, Berlin, 1998. \bibitem{Bah-Cor} A. Bahri, J. M. Coron; On a nonlinear elliptic equation involving the limiting Sobolev exponent, {\it Comm. Pure Appl. Math.}, \textbf{41} (1988), 253-294. \bibitem{Ba-Sc} T. Bartsch, M. Schneider; Multiple solutions of a critical polyharmonic equation, {\it J. reine angew. Math.}, \textbf{571} (2004), 131-143. \bibitem{Bre} H. Brezis; Elliptic equations with limiting Sobolev exponent. The impact of topology, Procceding 50th Anniv.Courant Inst. {\it Comm. Pure Appl. Math.}, \textbf{ 39 } (1986), 517-539. \bibitem{Bre-Nir} H. Brezis, L. Nirenberg; Positive solutions of nonlinear elliptic equations involving critical Sobolev exponents. {\it Comm. Pure Appl. Math.}, \textbf{36}(4) (1983), 437-477. \bibitem{Caf-Gid-Spr} L. Caffarelli, B. Gidas, J. Spruck; Asymptotic symmetry and local behavior of semilinear elliptic equations with critical Sobolev growth,{\it Comm. Pure Appl. Math.}, \textbf{42} (1989), 271-297. \bibitem{Car-Com-Lew} A. Carpio Rodriguez, M. Comte, R. Lewandoski; A nonexistence result for nonlinear equation involving critical Sobolev exponent, {\it Ann. Inst. Henri Poincar\'e , Anal. Non Lin\'eaire}, \textbf{9} (1992), 243-261. \bibitem{Cot-Ili} A. Cotsiolis, D. Iliopoulos; Equations elliptiques nonlineaires a croissance de Sobolev sur-critique, {\it Bull. Sci. Math.}, \textbf{119} (1995), 419-431. \bibitem{Cot-Lab3} A. Cotsiolis, N. Labropoulos; Best constants in Sobolev inequalities on manifolds with boundary in the presence of symmetries and applications, {\it Bull. Sci. Math. } \textbf{132} (2008), 562-574. \bibitem{Din} W. Ding; On a conformally invariant elliptic equation on ${\mathbb R}^n$, {\it Commun. Math. Phys.}, \textbf{107} (1986), 331-335. \bibitem{Din1} W. Ding; Positive solutions of $\Delta u+u^{\frac{n+2}{n-2}}=0$ on contactible domains, {\it J. Partial Diff. Eq.}, \textbf{ 2} (1989), 83-88. \bibitem{gherad1} L. Dupaigne, M. Ghergu, V. R\u adulescu; Lane-Emden-Fowler equations with convection and singular potential, {\it J. Math. Pures Appl.}, {\bf 87} (2007), no. 6, 563-581. \bibitem{Fag1} Z. Faget; Best constants in Sobolev inequalities on Riemannian manifolds in the presence of symmetries, {\it Potential Analysis}, \textbf{17} (2002), 105-124. \bibitem{For-Jan} D. Fortunato, E. Jannelli; Infinity many nodal solutions for some nonlineal elliptic problems in symmetrical domains, {\it Proc. R. Soc. Edinb.}, \textbf{105}A (1987), 205-213. \bibitem{gherad2} M. Ghergu, V. R\u adulescu; Ground state solutions for the singular Lane-Emden-Fowler equation with sublinear convection term, {\it J. Math. Anal. Appl.}, {\bf 333} (2007), 265-273. \bibitem{Gi-Ni-Ni} B. Gidas, W.-M. Ni, L. Nirenberg; Symmetry and related properties via the maximum principle, {\it Commun. Math. Phys.}, \textbf{68} (1979), 209-243. \bibitem{Heb-Vau} E. Hebey, M. Vaugon; Existance and multiplicity of nodal solutions for nonlinear elliptic equations with critical Sobolev growth, {\it J. Funct. Anal.}, \textbf{119} (1994), 298-318. \bibitem{Heb-Vau1} E. Hebey, M. Vaugon; Sobolev spaces in the presence of symmetries, {\it J.Math.Pures Appl.}, \textbf{76} (1997), 859-881. \bibitem{Kim-Zhu} J. Kim, M. Zhu; Non-existence results about $-\Delta u = u^ \frac{n+2}{n-2}$ on non-starshaped domains, {\it J. Differential Equations}, \textbf{225} (2006), 737-753. \bibitem{Kra} A. Krasnoselskii; {\it Topological methods in the theory of nonlinear integral equations}, Macmillan, New York and London, 1964. \bibitem{Kuz-Poh} I. Kuzin, S. Pohozaev; {\it Entire solutions of semilinear elliptic equations. Progress in Nonlinear Differential Equations and their Applications}, 33, Birkhauser Verlag, Basel, 1997. \bibitem{Li} Y. Y. Li; Existence of many positive solutions of semilinear elliptic equations on annulus, {\it J. Diff. Eq. }, \textbf{83} (1990), 348-367. \bibitem{Loe-Nir} C. Loewner, L. Nirenberg; Partial differential equations invariant under conformal and projective transformations, In: Contributions to Analysis, 245-272, Academic Press 1974. \bibitem{Maz-Sma} R. Mazzeo, N. Smale; Conformally fiat metrics of constant positive scalar curvature on subdomains of the sphere. {\it J. Differ. Geom.}, \textbf{34} (1991), 581-621. \bibitem{Pas2} D. Passaseo, Multiplicity of positive solutions of nonlinear elliptic equations with critical Sobolev exponent in some contractible domains, {\it Manuscripta Math.}, \textbf{65} (1989), 147-165. \bibitem{Poh} S. I. Pohozaev, Eigenfunctions of the equation $\Delta u+\lambda f(u)=0$, {\it Sov. Math. Dokl.}, \textbf{6} (1965), 1408-1411. \bibitem{Sch1} R. Schoen, Conformal deformation of a Riemannian metric to constant scalar curvature, {\it J. Diff. Geometry} \textbf{20} (1984), 479-495. \bibitem{Sch2} R. Schoen, The existence of weak solutions with prescribed singular behavior for a conformally invariant scalar equation, {\it Commun. Pure Appl. Math.} \textbf{41 } (1988), 317-392. \bibitem{Tar} G. Tarantello, Nodal solutions of semilinar elliptic equations with critical exponent, {\it Diff. Integral Eq.} \textbf{5} (1992), 25-42. \bibitem{Vai} M. Vainberg, The variational method for the study of non-linear operators, Holden-Day, San Francisco, CA, 1964. \bibitem{Wang} L. Wang, Infinitely Many Solutions for an Elliptic Problem with Critical Exponent in Exterior Domain, {\it J. Part. Diff. Eq.}, Vol. 23, No. 1, (2010), 80-104. \end{thebibliography} \end{document}