\documentclass[reqno]{amsart}
\usepackage{hyperref}

\AtBeginDocument{{\noindent\small
\emph{Electronic Journal of Differential Equations},
Vol. 2017 (2017), No. 150, pp. 1--12.\newline
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu}
\thanks{\copyright 2017 Texas State University.}
\vspace{8mm}}

\begin{document}
\title[\hfilneg EJDE-2017/150\hfil An alternative approach to critical PDEs]
{An alternative approach to critical PDEs}

\author[N. Labropoulos \hfil EJDE-2017/150\hfilneg]
{Nikos Labropoulos}

\address{Nikos Labropoulos \newline
Department of Mathematics,
University of Patras,
Patras 26110, Greece}
\email{nal@upatras.gr}

\dedicatory{Communicated by Vicentiu Radulescu}

\thanks{Submitted March 8, 2017. Published June 23, 2017.}
\subjclass[2010]{35J60, 35B33, 35J20}
\keywords{Laplacian; non-radial solution; critical exponent}

\begin{abstract}
 In this article, we use an alternative method to prove the
 existence of an infinite sequence of distinct non-radial nodal
 $G$-invariant solutions for critical nonlinear elliptic problems
 defined in the whole the  Euclidean space. Our proof is via
 approximation of the problem on symmetric bounded domains. The
 base model  problem of interest  originating from Physics  is
 stated below:
 $$
   -\Delta  u  = |u|^{\frac{4}{n-2}}u ,\quad u\in C^2(\mathbb{R}^n), \quad
   n\geq3.
 $$
\end{abstract}

\maketitle
\numberwithin{equation}{section}
\newtheorem{theorem}{Theorem}[section]
\newtheorem{remark}[theorem]{Remark}
\newtheorem{definition}[theorem]{Definition}
\newtheorem{corollary}[theorem]{Corollary}
\allowdisplaybreaks

\section{Introduction}

 In this article, our main motivation is based on the work by
Ding \cite{Din} which proved the existence of
non-radial solutions of the above problem. In this work we find
both the type and the number of such solutions. The pleasant
surprise is the fact that in order to answer these two questions
it needed to use a new method of solving critical, (or supercritical),
 PDEs, the method itself seems to have particular
value in that it can be used and in other types of PDEs. However,
the main objective here is to prove the  existence of non-radial
nodal (sign-changing) solutions, for the following critical
nonlinear elliptic  problem:
\begin{equation} \label{eP}
-\Delta  u  = |u|^{\frac{4}{n-2}}u ,\quad u\in C^2(\mathbb{R}^n), \quad
   n\geq3.
\end{equation}
As mentioned above the problem \eqref{eP} owns its origin in
many astrophysical and physical contexts and more precisely in the
the Lane-Emden-Fowler problem,
 \begin{gather*}
   - \Delta u = u^p ,\quad u > 0\quad \text{in }\Omega,  \\
  u = 0\quad\text{on }\partial \Omega,
\end{gather*}
where $\Omega$ is a domain with smooth boundary in $\mathbb{R}^N$
and $p > 1$.  But its greatest interest lies in its relation to
the Yamabe problem (for a complete and detailed study we refer to
\cite{Aub1}, nevertheless it has an autonomous presence holding an
important place among the most famous nonlinear partial
differential equations). Indicatively, we refer to the classical
papers \cite{Gi-Ni-Ni,Din,Maz-Sma}, which are
some of the large number of very good papers that are devoted to
the study of this problem.

Concerning the resolution of the problem \eqref{eP}, our
proof is via approximation of the problem on symmetric bounded
domains. This method is different from that previously used by
other authors referred within this paper and can be used to solve
polyharmonic equations with supercritical exponent and even in the
critical of supercritical case, providing an alternative way of
utilizing the best constants of  the Sobolev inequalities.
Furthermore, it enables us to determine the kind and the number of
solutions of the problem.


In  problem \eqref{eP} the main difficulty comes from the
double lack of compactness. By lack of compactness, we mean that
the functionals that we consider do not satisfy the Palais-Smale
condition, (i.e. there exists a sequence along which the
functional remains bounded, its gradient goes to zero, and does
not converge). The first main difficulty comes from the fact that
the exponent $2^*=\frac{2n}{n-2}=\frac{4}{n-2}+1$ is critical, and
the second is some extra difficulty because of the lack of
compactness in unbounded domains.

The first obstacle can be overcome  by obtaining the solutions of
the following corresponding  problem
\begin{equation} \label{ePe}
\begin{aligned}
-\Delta   u_\varepsilon + \varepsilon a(x)u_\varepsilon
  =  f(x)|u_\varepsilon|^{\frac{4}{n-2}}u_\varepsilon,\quad n\geq 3, \\
 u_\varepsilon \not  \equiv  0\quad\text{in }\Omega _\varepsilon ,\quad
u_\varepsilon = 0\quad\text{on }\partial \Omega _\varepsilon,
\end{aligned}
\end{equation}
where $\Omega_\varepsilon, \varepsilon>0 $  is an expanding
domain in $\mathbb{R}^n$, $n\geq3$, invariant under the action of
a subgroup $G$ of the isometry group $O(n)$ and
$a,f\in C^\infty(\overline \Omega_\varepsilon)$ are two smooth
$G-$invariant functions on $\overline\Omega_\varepsilon$.

The main idea to overcome the second difficulty is to solve the
problem \eqref{ePe} in a sequence of
$\Omega_\varepsilon $s and henceforth to obtain the solutions of
the limit problem \eqref{eP} as the limits of the solutions,
as $\frac{1}{\varepsilon}$ tends to $\infty$, of the sequence of
the problem \eqref{ePe}.


 Problem \eqref{ePe} has been
studied by many authors. We refer to
\cite{Amb-Str,Act-Bre-Pel,Bre-Nir,Din,For-Jan,Heb-Vau}
and the references therein for a further discussion
of both the problem itself and several variations of it. Some
special cases also have been studied. For example, no solution can
exist if $\Omega$ is starshaped, as a consequence of the Pohozaev
identity (see in \cite{Poh}). Furthermore, if $\Omega$ is an
annulus, there are infinite solutions (see in \cite{Li}). Also, a
general result of Bahri and Coron  guarantees the existence of
positive solutions in domains $\Omega$ having nontrivial topology
(i.e. certain homology groups of $\Omega$ are non trivial) (see in
\cite{Bah-Cor}). The existence and multiplicity of positive or
nodal solutions of critical equations on bounded domains or in
some contractible domains have been determined by other authors
(see for example in \cite{Din1,For-Jan,Heb-Vau,Pas2,Tar}).
Some more nonexistence results in this
case are available, (see in \cite{Adi-Yad,Act-Bre-Pel,Car-Com-Lew,Kim-Zhu}).

 As we have mentioned above, in problem
\eqref{ePe} the main difficulty comes because the
exponent $2^*$ is the critical exponent for the Sobolev imbedding
$H_1^2(\Omega)\hookrightarrow L^{p}(\Omega) $. Because the Sobolev
embedding $H_1^2(\Omega)\hookrightarrow L^{p}(\Omega) $ is compact
for any real $p$ such that $ 1\leq p < 2^*$ while if $ 1\leq p\leq
2^* $ is only continuous (see in \cite{Aub1}), in our case we have
to solve  a variational problem with lack of compactness. The
symmetry property of the domain allows us to improve the Sobolev
embedding in higher $L^p$ spaces and to overcome this obstruction.
More precisely, it is well known that if $(M,g)$ is a smooth
compact Riemannian $n-$manifold invariant under the action of an
arbitrary compact subgroup $G$ of $\mathrm{Isom}_g(M)$,
$O^x_G=\{\sigma(x), \sigma\in G\}$, $\mathrm{Card}O^x_G=\infty$
and $k=\mathrm{min}_{x\in M}\,\mathrm{dim}O^x_G$, then $k\geq1$
and the Sobolev embedding $ H_{1, G}^2(\Omega)\hookrightarrow
L^{p}(\Omega) $ is compact for any $ 1\leq p <\frac{2(n-k)}{n-k-2}
$ but if $ 1\leq p \leq\frac{2(n-k)}{n-k-2} $ is only continuous
(see in \cite{Heb-Vau1}, \cite{Fag1}, \cite{Cot-Lab3}). Thus, in
our case the symmetry property of $\Omega_\varepsilon$s allows
us to solve problems with subcritical exponent using the classical
variation method.

 As a small overview on the history and progress of the
study of our problem we mention the following: Loewner and
Nirenberg \cite{Loe-Nir} studied  problem \eqref{eP}  for
$n=4$. Gidas, Ni and Nirenberg  in their celebrated paper
\cite{Gi-Ni-Ni} proved symmetry and some related properties of
positive solutions of a larger class of second order elliptic
equations. Concerning  problem \eqref{eP} they proved
that any positive solution,  which has
finite energy, namely $ \int_{\mathbb{R}^n}|\nabla u|^2 dx<\infty$,
is necessarily of the form
\[
u(x)=\Big(\frac{\lambda\sqrt{n(n-2)}}{\lambda^2+|x-x_0|^2}\Big)^{(n-2)/2},
\]
  where $\lambda>0, x_0\in \mathbb{R}^n$. These solutions yield
the well-known one-instanton solutions in a regular gauge of the
Yang-Mills equation. In addition, since the equation
$$
-\Delta  u = |u|^{\frac{4}{n-2}}u, \quad n\geq3,
$$
 is invariant under the conformal transformations of $\mathbb{R}^n$,
 if $u(x)$ is a solution, then for any $\lambda > 0$ and $x_0\in \mathbb{R}^n$,
$\lambda^{\frac{n - 2}{2}} u (\frac{x-x_0}{\lambda} ) $  is also a
solution. Moreover, all solutions obtained in this way have the
same energy and we will say that these solutions are equivalent.
In particular, all these solutions are equivalent. Ding   in
\cite{Din}  used Ambrosetti and Rabinowitz analysis (see in
\cite{Amb-Rab})   to prove that this problem has infinite distinct
solutions $u\in C^2(\mathbb{R}^n)$, with finite energy and which
changes sign, but he did not specify the type of these solutions.
Caffarelli, Gidas and Spruck in their classical paper
\cite{Caf-Gid-Spr} studied non-negative smooth solutions of the
conformally invariant equation
$$
 -\Delta  u  = u^{\frac{n+2}{n-2}}, \quad u\geq 0,\quad  n\geq3,
$$
 in a punctured ball $B_1(0)\backslash \{0\}\subseteq
\mathbb{R}^n$, with an isolated singularity at the origin. In this
paper, the authors introduced a heuristic idea of asymptotic
symmetry technique  which can roughly be described as follows:
After an inversion, the function $u$ becomes defined in the
complement of $B_1$, is strictly positive of $\partial B_1$, and
in some sense `goes to zero' at infinity. If the function $u$ can
be extended to $B_1$ as a super solution of our problem, then it
can start the reflection process at infinity and moved all the way
to $\partial B_1$. This would imply asymptotic radial symmetry at
infinity. With this comprehensive report on this issue we would
like, on the one hand, to emphasize the important contribution of
this great article of Caffarelli, Gidas and Spruck  on the study
on the direction of finding the radial solutions  of our problem
and on the other hand, we wish to make clear that in our
procedural paper we do not care about the radial solutions but we
do care about the existence of non-radial solutions. Schoen in
\cite{Sch1}  built solutions of \eqref{eP} with prescribed
isolated singularities. In another paper \cite{Sch2}, Schoen and
Yau have used the geometrical meaning of problem \eqref{eP} in
order to derive, through ideas of conformal geometry, the
existence of weak solutions having a singular set whose Hausdorff
dimension is less than or equal to $\frac{n-2}{2}$. Let us notice
that in this paper the authors explain how to build solutions of
\eqref{eP} with a singular set whose Hausdofff dimension is not
necessarily an integer. Mazzeo and Smale have proved
\cite{Maz-Sma} the existence of singular solutions of \eqref{eP} for a
very large variety of singular sets. Bartsch and
Schneider in \cite{Ba-Sc} proved that for $N>2m$ the  equation
$$
(-\Delta)^m=|u|^{\frac{4m}{N-2m}}u
$$
on $\mathbb{R}^N$ has a sequence of nodal, finite energy solutions
which is unbounded in $\mathcal{D}^{m,2}(\mathbb{R}^N)$, the
completion of $\mathcal{D}(\mathbb{R}^N)$ with respect to the
scalar product
$$
( {u,\upsilon }) = \begin{cases}
  \int_{\mathbb{R}^N } \Delta ^{m/2} u \cdot \Delta ^{m/2} \upsilon ,
 &\text{if $m$ is even}\\[4pt]
 \int_{\mathbb{R}^N } \nabla \Delta ^{(m-1)/2} u \cdot \nabla
   \Delta ^{(m-1)/2} \upsilon, &\text{if $m$ is odd}.
\end{cases}
$$
This generalizes the result of Ding for $m=1$, and provides
interesting information  concerning the number and the kind of the
solutions of the equation (see Remark \ref{rmk3.1}). Finally, for reasons
of completeness, we refer in this point to the paper of Wang
\cite{Wang} where the following nonlinear Neumann elliptic problem
is studied:
\begin{equation} \label{ePN}
\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}
where $n$ denotes interior unit normal vector and $\Omega$ is a
smooth bounded domain in $\mathbb{R}^N$, $N\geq4$. In this paper,
it is proved that if $N\geq4$, (Wang believes that the results
will also hold in the case of $N=3$), and $\Omega$ is a smooth and
bounded domain then the problem \eqref{ePN}
has infinity many non-radial positive solutions, whose energy can
be made arbitrarily large when $\Omega$ is convex as seen from inside
(with some symmetries). We refer to the Wang's problem
\eqref{ePN} due to its close relationship with our problem
and as we will see later if we choose suitable $\Omega$ we can
have a result on this problem in almost all the space. In
particular, in  both problems we have to solve the same
non-linear differential equation with critical exponent with
boundary conditions Dirichlet and Neumann respectively. In
addition, in  both cases the domain $\Omega$ presents some
symmetries. However, a subsequent process in each case is
completely different from that of another. In our case, our goal
is to solve the problem in the whole space, starting from an open
symmetric domain $\Omega$ of $n$-dimensional space and we extend
$\Omega$ so that it remains symmetrical to fill almost all the
space. In the other case is considered the corresponding Neumann
problem  in ${\mathbb R}^N\backslash \Omega$ where $\Omega$ is convex seen
from inside with some symmetries. If we choose appropriate a such
$\Omega$ with a small volume as much as we can say that the
solutions of Wang  satisfy the conditions of the problem in almost
all the space. Finally, in both problems we take infinity many 
non-radial solutions, whose energy can be made arbitrary large,
however in  the first problem we  find nodal solutions while in
the second are founded positive solutions.

 In this research  our goal is to specify the kind and
the number of solutions of the problem \eqref{eP}. We prove
the existence of a sequence $\{u_k\}$ of non-radial, inequivalent,
nodal $G$-invariant solutions of $\;(\mathrm{P})$, such that: $
\lim_{k\to\infty}\int_{\mathbb{R}^n}|\nabla u_k|^2 dx=\infty$.


This article is arranged as follows: Section 2 is devoted to
notation and some necessary preliminary results. In addition,
in this section two  examples  are presented. Furthermore, in
Section 2, we introduce our main tool, meaning the process through
which an open symmetric domain of $n-$ dimensional space can be
extended in an appropriate manner to \emph{`fill'} eventually the
entire space \emph{`almost everywhere'}, remaining symmetric, and
subsequently we solve the auxiliary problem
\eqref{ePe}. Section 3 is devoted to some basic
definitions  and to the proof of the main theorem.

\section{Some notation and preliminary results}

 Let $C^\infty(\Omega)$ be the space of smooth functions on
$\Omega$ and $\mathcal{D}(\Omega)$ be the set of infinitely
differentiable functions whose support is compact in $\Omega$.
We define, also,  the Sobolev space $H^2_1(\Omega)$ as the
completion of $C^\infty(\Omega)$ with respect to the norm:
$$
 \| u\|_{H_1^2  ( \Omega  )}  = \Big( \int_\Omega (|\nabla u|^2
+ | u |^2) dx \Big)^{1/2} \,.
$$
The Sobolev space ${\mathaccent"7017 H}^2_1(\Omega)$ as the
closure of $\mathcal{D}(\Omega)$ in $H^2_1(\Omega)$.

  In the following, we suppose that $\Omega$ is a
bounded, smooth, domain of $\;\mathbb{R}^n$, $n\geq3$,
$G$-invariant under the action of a compact subgroup $G$ of the
isometry group $O(n)$, without finite subgroup. Such $\Omega$s
in $\mathbb{R}^n$ can be constructed as follows:

 Let $\Omega$ be a bounded, smooth, domain of
$\mathbb{R}^n=\mathbb{R}^k\times\mathbb{R}^{n-k} $,
$k\geq 2$, $n-k\geq 1$ such that
$ \overline{\Omega}\subset (\mathbb{R}^k
\backslash \{0\} )\times \mathbb{R}^{n-k} $. Suppose that
$\overline{\Omega}$ is invariant under the action of $G_{k,n-k}$,
that is $ \tau (\overline{\Omega})=\overline{\Omega}$
for  all $\tau \in G_{k,n-k}$, where
$G_{k,n-k}=O(k)\times Id_{n-k}$ is the subgroup of the isometry
group $O(n)$ of the type:
$$
(x_1, x_2)\to (\sigma (x_1),x_2),\quad \sigma\in O(k),\quad
x_1 \in \mathbb{R}^k ,\; x_2 \in \mathbb{R}^{n-k} .
$$
Then $\overline \Omega$ is a bounded,
smooth, domain of $\;\mathbb{R}^n$, invariant under the action of
the subgroup $G_{k,n-k}$ of the isometry group $O(n)$.
We denote by $H^2_{1,G}(\Omega)$ and
${\mathaccent"7017 H}^2_{1,G}(\Omega)$ the subspaces of
$H^2_1(\Omega)$ and ${\mathaccent"7017 H}^2_1(\Omega)$  of all
$G$-invariant functions, respectively.

We consider the functional
 $$
 J(u)=\int_{\Omega} (|\nabla
 u|^2+ a(x)u^2 )dx,
$$
and suppose that the operator $L (u)=-\Delta u+a(x)u$ is
\emph{coercive}. That is, there exists a real number $\lambda>0$,
such that for all $u\in{\mathaccent"7017 H}^2_1(\Omega)$:
 $$
 J(u)\geq\lambda\int_{\Omega}(|\nabla  u|^2+ u^2 )dx.
$$
For example, the  operator $L$ is coercive if $a(x)\geq 0$, for all
$x\in\Omega$, and more generally when $a(x)$ is greater than minus
the best Poincar\'e constant of
${\mathaccent"7017 H}^2_1(\Omega)$.

We consider now the problem
\begin{equation} \label{eP0}
\begin{gathered}
-\Delta  u + a(x)u  = f(x)|u|^{\frac{4}{n-2}}u , \quad n\geq 3, \\
u \not  \equiv 0\quad \text{in }\Omega  ,\quad
u = 0\quad \text{on }\partial \Omega  \,,
\end{gathered}
\end{equation}
where $\Omega$ is defined as above and $a$, $f$ are two smooth $G$-invariant
functions.

For any small  $\varepsilon>0$ and some $m>0$ we consider the
family of expanding domains:
$$
\Omega_\varepsilon=\varepsilon^{-m}\Omega
=\{\varepsilon^{-m}x:x\in\Omega\}.
$$
Then, it is very simple to confirm that the $\Omega_\varepsilon$s inherit
the symmetry properties of $\Omega$ for any $\varepsilon$.
We consider,  also,  the transformation:
\begin{equation}\label{E1}
\phi:\Omega\to\Omega_\varepsilon:\;
X=\phi(x),\;x\in\Omega,\; X\in\Omega_\varepsilon,
\end{equation}
and for $l>0$, we set
\begin{equation}\label{E2}
u(x)= \varepsilon^{-l}u_\varepsilon(X).
\end{equation}
In particular, we obtain
\begin{gather}\label{E3}
|\nabla u|=\varepsilon^{-l-m}|\nabla u_\varepsilon|, \\
\label{E4}
\Delta  u= \varepsilon^{-l-2m}\; \Delta u_\varepsilon.
\end{gather}
Applying the  transformation \eqref{E1})in the equation of
problem \eqref{eP0}, because of \eqref{E2}, \eqref{E3} and
\eqref{E4}  we obtain the equation
$$
- \Delta  u_\varepsilon + \varepsilon^{2m}a(x)u_\varepsilon =
\varepsilon^{2m-l{\frac{4}{n-2}}}f(x)|u_\varepsilon|^{\frac{4}{n-2}}u_\varepsilon,
$$
where we denote again by $a$ and $f$ the functions $a\circ \phi^{-1}$
and $f\circ \phi^{-1}$,
respectively and the independent variable by $x$.
Since  $l$ is an arbitrary positive real, we can choose
 $l=2m\frac{n-2}{4}$ and thus we have
\begin{equation}\label{E5}
 -\Delta  u_\varepsilon + \varepsilon^{2m}a(x)u_\varepsilon =
 f(x)|u_\varepsilon|^{\frac{4}{n-2}}u_\varepsilon.
\end{equation}
From  \eqref{E5}, replacing the $\varepsilon^{2m}$ by
$\varepsilon$, we obtain
\begin{equation*}
- \Delta   u_\varepsilon + \varepsilon a(x)u_\varepsilon
  =  f(x)|u_\varepsilon|^{\frac{4}{n-2}}u_\varepsilon.
\end{equation*}
So, we have to solve the  critical problem
\begin{equation} \label{ePe2}
\begin{gathered}
-\Delta   u_\varepsilon + \varepsilon a(x)u_\varepsilon
=f(x)|u_\varepsilon|^{\frac{4}{n-2}}u_\varepsilon,\quad n\geq3, \\
u_\varepsilon \not  \equiv  0\quad\text{in }\Omega_\varepsilon  ,
\quad u_\varepsilon = 0\quad\text{on }\partial \Omega _\varepsilon .
\end{gathered}
\end{equation}
We consider the functional
 $$
 J(u_\varepsilon)=\int_{\Omega_\varepsilon} (|\nabla
 u_\varepsilon|^2+\varepsilon a(x)u_\varepsilon^2 )dx.
$$
Since the operator  $L (u)=-\Delta u+ a(x)u$ is considered to be
coercive in $\Omega$  the operator $
 L (u_\varepsilon)=-\Delta
u_\varepsilon+\varepsilon a(x)u_\varepsilon $ is coercive in
$\Omega_\varepsilon$.

Denote
$$
 \mathcal{H}_\varepsilon=\big\{u_\varepsilon\in{\mathaccent"7017 H}^2_{1,
 G}(\Omega_\varepsilon)  :
\int_{\Omega_\varepsilon} f(x)|u_\varepsilon|^{\frac{2n}{n-2}}
dx=1\big\},
$$
and suppose that an isometry $\sigma$ such as
$\sigma(\Omega_\varepsilon)=\Omega_\varepsilon$ exists.
Furthermore, suppose also that the functions $a(x)$ and $f(x)$ are
invariant under the action of  $\sigma$ and that
$$
\mathcal{H}_\varepsilon^\sigma= \mathcal{H}\cap
\big\{u_\varepsilon\in{\mathaccent"7017 H}^2_{1}(\Omega_\varepsilon)
: u_\varepsilon\circ \sigma =-u_\varepsilon \big\}\neq
\emptyset.
$$
By definition, a function  $u$ which satisfies
$u\circ \sigma =-u$ is called  \emph{antisymmetrical}.

Under the above considerations the following
theorem holds (see in \cite{Cot-Ili}).

\begin{theorem}\label{thm2.1}
Problem \eqref{ePe},  always,  has a non-radial
 nodal solution $u$. Moreover, if  $f(x)>0$ for all
$x\in \overline \Omega_\varepsilon $, \eqref{ePe} has an
infinity sequence $\{u_{\varepsilon_i}\}$ of non-radial nodal
solutions, such that
$$
\lim_{i\to \infty} \int_{\Omega_\varepsilon} (|\nabla
 u_{\varepsilon_i}|^2+ u_{\varepsilon_i}^2 )dx=+\infty.
$$
 In addition,  $u$ and
$\{u_{\varepsilon_i}\}_{i=1,2,\dots}$ are $G$-invariant and
$\sigma$-antisymmetrical.
\end{theorem}

\begin{remark} \label{rmk2.1}\rm
Theorem  \ref{thm2.1} holds for the supercritical case and even
to the critical of the supercritical case, (see in
\cite{Cot-Ili}), namely for every $p$ such that:
$$
\frac{2n}{n-2}<p\leq \frac{2(n-k)}{n-k-2}.
$$
\end{remark}

\section{Solution of problem \eqref{eP}}

Because of the double lack of compactness, direct variational
methods are not applicable to the limit problem
\begin{equation} \label{eP2}
-\Delta  u  = |u|^{\frac{4}{n-2}}u ,\quad u\in C^2(\mathbb{R}^n), \quad
   n\geq3.
\end{equation}
However, this method is successful in  approximating a solution to
the problem \eqref{eP}  by solutions  in the open
domains $\Omega_{\varepsilon_j}$. Thus, a solution to
\eqref{eP} may be then obtained by the limit procedure as
$\varepsilon_j\to 0$.

 Before we approximate the solutions in
$\mathbb{R}^n$ by solutions in bounded domains $\Omega_\varepsilon
\in\mathbb{R}^n$, we note that, in the generalized setting of the
problems in $\Omega_\varepsilon $s, the Dirichlet condition
$u_\varepsilon(x)= 0$ on $\partial \Omega_\varepsilon $ may actually be
included in the condition
$u_\varepsilon \in {\mathaccent"7017 H}^2_{1}(\Omega_\varepsilon)$.
 Moreover, since any function $u_\varepsilon
\in {\mathaccent"7017 H}^2_{1}(\Omega_\varepsilon )$ can be
extended onto $\mathbb{R}^n$ by
$$
\tilde u_\varepsilon ( x)
= \begin{cases}
 u_\varepsilon  ( x), & x \in \Omega_\varepsilon   \\
 0,& x \in \mathbb{R}^n \backslash \Omega_\varepsilon ,
\end{cases}
 $$
generalized solutions may be defined in $\Omega_\varepsilon$s
analogously to the case in  $\mathbb{R}^n$.
We need now the following two definitions:

\begin{definition} \label{def3.1} \rm
A function $u_\varepsilon \in{\mathaccent"7017 H}^2_{1}(\Omega_\varepsilon )$ is a
\emph{generalized solution} of \eqref{ePe} if
the function
$$
g(x,u_\varepsilon)=\varepsilon a(x)u_\varepsilon
 -f(x)|u_\varepsilon |^{\frac{4}{n-2}}u_\varepsilon
$$
is  locally integrable and for all
$\varphi\in C^\infty_0(  \Omega_\varepsilon )$, the following holds:
$$
\int_{\Omega_\varepsilon }(\nabla u_\varepsilon,
 \nabla\varphi)dx+\int_{ \Omega_\varepsilon }g(x,u_\varepsilon)\varphi dx=0.
$$
\end{definition}

\begin{definition} \label{def3.2} \rm
A function $u_\varepsilon \in C^2(\Omega_\varepsilon)\cap
C(\overline \Omega_\varepsilon)$ is a
\emph{classical solution} to \eqref{ePe} if
after substituting it into the equation of \eqref{ePe},
 this equation becomes an identity at each $x\in \Omega_\varepsilon$ and
$u_\varepsilon(x)=0$ provided $x\in \partial \Omega_\varepsilon$.
\end{definition}

 Consider now a sequence of real numbers
$\{\varepsilon_j\}_{ j=1,2,\dots }$ such that
$\varepsilon_j\to 0$ as $j\to \infty$ and the
sequence of problems:
\begin{equation} \label{ePej}
\begin{gathered}
-\Delta   u_{\varepsilon_j} +
\varepsilon_j a(x)u_{\varepsilon_j}
=  f(x)|u_{\varepsilon_j}|^{\frac{4}{n-2}}u_{\varepsilon_j},\quad
n\geq3 \\
u_{\varepsilon_j} \not  \equiv 0\quad\text{in } \Omega _{\varepsilon_j}
,\quad u_{\varepsilon_j} = 0\quad \text{on }\partial \Omega _{\varepsilon_j},
\end{gathered}
\end{equation}
where  $f(x)>0$ 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:
$$
1<p<\frac{2(n-k)}{n-k-2}
$$
this embedding is compact and then from
the Sobolev and Kondrashov theorems together and \eqref{E7} arises
that:
\begin{equation}\label{E10}
\upsilon_j\to \upsilon\quad\text{in }L^{p_0-1}(\Omega_{\varepsilon_0}),\;
2<p_0<\frac{2(n-k)}{n-k-2}+1,\quad \text{as }j\to+\infty.
\end{equation}
Furthermore, by definition of $a(x)$ and $f(x)$, there exists a
positive constant $C$ such that:
$$
|g(x,t)|\leq C(|t|+|t|^{p_0-1}),\quad
2<p_0<\frac{2(n-k)}{n-k-2}+1,
$$
for almost all $x\in \Omega _{\varepsilon_j}$, $j=1,2,\dots$ and for all
$t\in \mathbb{R}$. Thus, the Vainberg-Krasnoselskii Theorem (see \cite{Kra}
or  \cite{Vai}) gives:
\begin{equation}\label{E11}
\varphi g(\cdot,\upsilon_j(\cdot))\to \varphi
g(\cdot,\upsilon(\cdot))\quad \text{in } L^{\frac{n}{n-2}}
(\Omega_{\varepsilon_0})\quad \text{as } j\to+\infty.
\end{equation}
By the H\"older inequality from \eqref{E11} follows that
\begin{equation}\label{E12}
\varphi g(\cdot,\upsilon_j(\cdot))\to \varphi
g(\cdot,\upsilon(\cdot))\quad \text{in }
L^1(\Omega_{\varepsilon_0})\quad \text{as } j\to+\infty.
\end{equation}
By \eqref{E12} the limit relation  from the right hand-side of
\eqref{E8} yields:
\begin{equation}\label{E13}
\lim_{j \to \infty}\int_{\Omega_{\varepsilon_0}}
 g(x,\upsilon_j)\varphi\,dx
=\int_{\Omega_{\varepsilon_0}} g(x,\upsilon)\varphi\, dx.
\end{equation}
Finally, passing to the limit in \eqref{E8} because of \eqref{E7}
and \eqref{E13}, we obtain
\begin{equation*}
\int_{\mathbb{R}^n} \nabla\upsilon\nabla\varphi\,dx
=-\int_{\Omega_{\varepsilon_0}} g(x,\upsilon)\varphi\,dx
=-\int_{\mathbb{R}^n} g(x,\upsilon)\varphi dx,
\end{equation*}
which corresponds to the definition of a weak solution. It is
generalized by the force of \eqref{E7} and since  the function $f$
is regular enough  it is a classical solution, (see \cite[Secs. 1.2 and 3.1]{Kuz-Poh}).
 As convergence in $L^p$ spaces implies
a.e. convergence  by \eqref{E10} follows that the function
$\upsilon$ will be $G$-invariant.

It remains to prove that this solution is nontrivial. Suppose,  by
contradiction, that $\upsilon\equiv 0$. Then, for any $\varepsilon
>0$ 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}.


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\end{document}