p^*$, respectively, where $$ p^* := \begin{cases} \frac{\mu+1+2m}{\mu+1-2m} & \hbox{if }\mu+1>2m,\\ \infty & \hbox{if }\mu+1 \le 2m. \end{cases} $$ If $\mu=N-1$, then $p^*$ is the Sobolev critical exponent associated with Equation~\eqref{eU}, that is, the critical exponent for the embedding of $H^m(B)$ into $L^{p+1}(B)$. In this terminology, the preceding observation regarding entire solutions of \eqref{eVW} simply says that no such solutions exist if the system is subcritical. This is, of course, a well-established fact, at least in the case of integer $\mu$ (see \cite{Mitidieri, Soranzo, Wei-Xu}). The result is optimal in the sense that a family of entire solutions of \eqref{eVW} is known explicitly in the critical case. In fact, if $\mu=2q+2m-1$, then the function $r\mapsto(-1)^m\,2^qQ/(1+r^2)^q$ is the first component of such a solution, and additional ones are obtained by rescaling. If $\mu=N-1$, these solutions correspond to ``minimum-energy solutions'' of Equation~\eqref{eU} on $\mathbb{R}^N$ and determine the norm of the embedding of the associated ``finite-energy space'' into $L^{p^*+1}(\mathbb{R}^N)$ (see \cite{Swanson92}). \end{remark} \begin{remark} \rm \label{dirichlet} We call a nontrivial even solution $(v,w)$ of \eqref{eVW} a {\it solution of the Dirichlet problem\/} if there exists a point in the solution's interval of existence where the first $m$ components of $(v,w)=(v_1,w_1,\dots,v_m,w_m)$ vanish simultaneously. More precisely, if $R$ is a common zero of the first~$m$ components, we say that $(v,w)$ solves the Dirichlet problem for \eqref{eVW} on the interval $[0,R]$. Clearly, if $\mu=N-1$, any such solution corresponds to a nontrivial radial solution of Equation~\eqref{eU} satisfying Dirichlet conditions on the boundary of the ball $B_R(0)$. Let $R\in(0,\infty)$ and suppose that $(v,w)$ solves the Dirichlet problem for \eqref{eVW} on $[0,R]$. If $m=2k-1$ for some $k\in\mathbb{N}$, then $v_1$ is negative and increasing on $[0,R]$, and while $v_1,w_1,\dots,v_{k-1},w_{k-1},v_k$ vanish at~$R$, the next component, $w_k$, does not; similarly, if $m=2k$ for some $k\in\mathbb{N}$, then $v_1$ is positive and decreasing on $[0,R]$, and while $v_1,w_1,\dots,v_k,w_k$ vanish at~$R$, $v_{k+1}$ does not (see \cite[Theorem~3.3]{LS}). Applying the Rellich-type identity from Remark~\ref{rellich} to the solution $(v,w)$, we see that all but one of the terms on the right-hand side vanish; in fact, we have \begin{align*} &\frac{m}{q+m}\,(\mu+1-2q-2m)\int_0^R\!r^{\mu}v_{1}(r)|v_1(r)|^p\,dr\\ &=\begin{cases} \displaystyle \phantom{-}R^{\mu+1}w_k^2(R)>0&\hbox{if }m=2k-1,\ k\in\mathbb{N},\\[5pt] \displaystyle -R^{\mu+1}v_{k+1}^2(R)<0&\hbox{if }m=2k,\ k\in\mathbb{N}. \end{cases} \end{align*} In either case, it follows that $\mu+1-2q-2m<0$; that is, solutions of the Dirichlet problem do not exist unless \eqref{eVW} is subcritical. Like the complementary result for entire solutions in Remark~\ref{rellich}, this is well known, at least for integer $\mu$ (see \cite{Pucci-Serrin} for the supercritical case and \cite{Soranzo} for the critical case). \end{remark} \begin{remark} \rm \label{supercritical} According to Remark~\ref{rellich}, the system \eqref{eVW} has no entire solutions if it is subcritical and at least one scaling-equivalence class of entire solutions if it is critical. Moreover, the explicitly known solutions in the critical case can be shown to be the only entire solutions with {\it finite energy\/}. For integer~$\mu$, this was done by Swanson \cite{Swanson92, Swanson} and again by Wei and Xu~\cite{Wei-Xu}, who claim implicitly that {\it every\/} entire solution has finite energy. This is correct for $m\le2$ (see~\cite{Lin, Xu} or the discussion below) and probably for any~$m$; but the argument in~\cite{Wei-Xu} (specifically the proof of Lemma~4.3 ibidem) appears to be inconclusive, and we have not been able to close the gap (see, however, the ``note added in proof'' at the end of the~paper). We conjecture, in fact, that \eqref{eVW} has exactly one scaling-equivalence class of entire solutions not only in the critical, but also in the supercritical case. This result is extractable from the literature (see below) if $m\le2$, but appears to be a wide open problem if $m\ge3$. We include short proofs in the cases $m=1$ and $m=2$ for the sake of completeness. First suppose that $m=1$. Then uniqueness is trivial, since any entire solution of \eqref{eVW} must be scaling-equivalent to the unique even solution starting at $-1$. Furthermore, the first component of this solution is either negative throughout, in which case the solution is global, or it has a zero, in which case the solution solves the Dirichlet problem. By Remark~\ref{dirichlet}, the latter is impossible if \eqref{eVW} is critical or supercritical, so that in this case, the even solution starting at $-1$ is an entire solution. Now suppose that $m=2$. Then any entire solution of \eqref{eVW} is scaling-equivalent to an even solution $(v,w)$ with $v_1(0)=1$ and $v_1$ positive. Since $v_2(0)$ is the only free parameter, any two solutions satisfying these conditions are ordered, and Proposition~\ref{rho}(b) implies that not both can be global unless they are equal. This proves uniqueness. Now, given $\alpha\in\mathbb{R}$, let $(v^\alpha,w^\alpha)$ denote the even solution of \eqref{eVW} starting at $(1,\alpha)$; further, define $I:=\{\alpha\in\mathbb{R} \,|\, v^\alpha_1\ge0\}$. By the comparison principle, $I$ is an interval containing $\mathbb{R}_+$. Since the solution $(v^\alpha,w^\alpha)$ depends continuously on~$\alpha$, $I$ is closed. For the same reason, $0$ is not a lower bound for~$I$ (note that $v^0_1$ is increasing and thus satisfies $v^0_1\ge1$). However, $I$ is bounded from below, else Proposition~\ref{rho}(b) would imply that the exit point of $(v^\alpha,w^\alpha)$ increases as $\alpha\to-\infty$, contradicting Proposition~\ref{rho}(c). In conclusion,~$I$ has a negative minimum $\alpha^*$. Denote by $(v^*,w^*)$ the even solution starting at $(1,\alpha^*)$. Its first component, $v^*_1$, is either monotonically decreasing throughout, or it decreases to a global minimum, necessarily with value $0$, and increases to infinity thereafter. In the first case, $(v^*,w^*)$ is an entire solution; in the second case, it solves the Dirichlet problem. If \eqref{eVW} is critical or supercritical, the latter is impossible, by Remark~\ref{dirichlet}, and $(v^*,w^*)$ is an entire solution. (For the supercritical case, a similar proof was given in~\cite{Gazzola-Grunau}; more general existence and nonexistence results can be found in \cite{Dalmasso, Serrin-Zou}.) In conclusion, \eqref{eVW} has exactly one scaling-equivalence class of entire solutions if the system is critical or supercritical and if $m\le2$. If $m\ge3$, the above arguments fail, with regard to both existence and uniqueness. As for uniqueness, note that Proposition~\ref{rho}(b) implies that \eqref{eVW} cannot have two distinct global even solutions that are initially ordered and have nonnegative first components. Since entire solutions have {\it positive\/} first components if $m$ is {\it even\/}, but {\it negative\/} first components if $m$ is {\it odd\/}, this observation is relevant if $m$ is even, but does not imply uniqueness unless~$m=2$. \end{remark} The following theorem and corollary gather the main results of this section. \begin{theorem}\label{entire-vw} Let $\mu^*:=2q+2m-1$. The system \eqref{eVW}, for arbitrary $m\in\mathbb{N}$, has no entire solutions if $\mu<\mu^*$ and at least one scaling-equivalence class of entire solutions if $\mu=\mu^*$. For $m\in\{1,2\}$, the system has exactly one scaling-equivalence class of entire solutions if $\mu\ge\mu^*$. \end{theorem} \begin{corollary}\label{entire-u} Equation \eqref{eU}, for arbitrary $m\in\mathbb{N}$, has no entire radial solutions in the subcritical and at least one scaling-equivalence class of entire radial solutions in the critical case. For $m=1$ or $m=2$, there is exactly one scaling-equivalence class of entire radial solutions if the equation is critical or supercritical. \end{corollary} \section{Large Radial Solutions}\label{large} Recall from Section~\ref{ode} that we denote the even solution of \eqref{eVW} starting at $x\in\mathbb{R}^m$ by $(v^x,w^x)$, its exit point by $\rho(x)$. The rescalings of a point $x\in\mathbb{R}^m$ are defined by $\sigma_{x}(\lambda):=\lambda^q\Lambda x:=\lambda^q(x_1,\lambda^2x_2,\dots,\lambda^{2(m-1)}x_m)$ for $\lambda\in(0,\infty)$. If $x \ne 0$, the curve~$\Sigma_x$ parametrized by $\sigma_x$ is called the scaling-parabola through~$x$; it intersects the unit sphere $S^{m-1}$ in exactly one point, denoted by $\pi(x)$. We call the mapping $\pi \,{:}\ \mathbb{R}^m\setminus\{0\}\to S^{m-1}$ the scaling-projection. Along each scaling-parabola, the exit point $\rho$ is either identically equal to~$\infty$ or strictly decreasing from $\infty$ to $0$; in fact, $\rho(\sigma_x(\lambda))=\rho(x)/\lambda$ for all $x\in\mathbb{R}^m$ and $\lambda\in(0,\infty)$. It follows that if $\rho(x)<\infty$, then the curve $\Sigma_x$ intersects the set $\mathcal{H}:=\{\xi\in\mathbb{R}^m \,|\, \rho(\xi)=1\}$ exactly once, at the point $\sigma_x(\rho(x))$. On the other hand, if $\rho(x)=\infty$, then $\Sigma_x$ does not intersect $\mathcal{H}$. We conclude that $$ \mathcal{H}=\{\sigma_\xi(\rho(\xi)) \,|\, \xi\in S^{m-1}\!,\,\rho(\xi)<\infty\}. $$ Define $\mathcal{S}:=\{\xi\in S^{m-1} \,|\, \rho(\xi)<\infty\}$. By Proposition~\ref{rho}(a), the set $\mathcal{S}$ is relatively open in $S^{m-1}$, and the mapping $\xi\mapsto\sigma_\xi(\rho(\xi))$ is a homeomorphism from $\mathcal{S}$ onto $\mathcal{H}$; its inverse is the scaling-projection $\pi$, restricted to $\mathcal{H}$. Not only is $\mathcal{H}$ homeomorphic to an open subset of $S^{m-1}$ under the scaling-projection; it is in fact a ``separating hypersurface'' in $\mathbb{R}^m$, namely, the common boundary of the connected open sets $\mathcal{A}$ and $\mathcal{B}$, defined by \begin{gather*} \mathcal{A}:=\{\xi\in\mathbb{R}^m \,|\, \rho(\xi)>1\} =\{0\}\cup\{\sigma_\xi(\lambda) \,|\, \xi\in S^{m-1}\!,\,\lambda\in(0,\rho(\xi))\}, \\ \mathcal{B}:=\{\xi\in\mathbb{R}^m \,|\, \rho(\xi)<1\}= \{\sigma_\xi(\lambda) \,|\, \xi\in S^{m-1}\!,\,\lambda\in(\rho(\xi),\infty)\}, \end{gather*} the ``inside'' and ``outside'' of $\mathcal{H}$, relative to the scaling-projection. Note, however, that $\mathcal{H}$ is a closed subset of $\mathbb{R}^m$ and thus, cannot be bounded unless $\mathcal{S}$ is compact; if $m\ge2$, this requires that $\mathcal{S}=S^{m-1}$. In any case (including $m=1$), the set $\mathcal{A}$ is bounded if and only if $\mathcal{S}=S^{m-1}$. Recall from Proposition~\ref{rho}(a) that $S^{m-1}\setminus\mathcal{S}$ is contained in the open orthant $$ {\mathcal O}:=\{\xi\in\mathbb{R}^m \,|\, (-1)^{m-i}\xi_i<0\;\forall\;i\in\{1,\dots,m\}\} $$ (the negative half-axis if $m=1$). This implies that $\mathcal{H}\cap(\mathbb{R}^m\setminus{\mathcal O})$ is compact and, in particular, bounded; but if $m\ge2$, then $\mathcal{H}\cap{\mathcal O}$ is bounded if and only if $\mathcal{S}=S^{m-1}$. (If $m=1$, then $\mathcal{H}\cap{\mathcal O}$ is empty unless $\mathcal{S}=S^0=\{\pm1\}$.) In particular, $\mathcal{S}$ contains $S^{m-1}_+:=S^{m-1}\cap\mathbb{R}^m_+$, and $\mathcal{H}_+:=\mathcal{H}\cap\mathbb{R}^m_+$ is compact and homeomorphic to $S^{m-1}_+$ under the scaling-projection. Moreover, it follows from Proposition~\ref{rho}(b) that every half-line in $\mathbb{R}^m_+$, emanating from the origin, intersects $\mathcal{H}_+$ exactly once. Thus, $\mathcal{H}_+$ is homeomorphic to $S^{m-1}_+$ also under the {\it radial\/} projection. As another consequence of Proposition~\ref{rho}(b), the set $\mathcal{H}_+$ is the boundary of an order decomposition of $\mathbb{R}^m_+$. By this we mean a pair $(A,B)$ of nonempty closed sets $A,\,B\subset\mathbb{R}^m_+$ with $A\cup B=\mathbb{R}^m_+$ and $\hbox{int}(A\cap B)=\emptyset$ such that $A$ is lower-closed (that is, $\{\xi\in\mathbb{R}^m_+ \,|\, \xi\le x\}\subset A$ for every $x\in A$) and $B$ is upper-closed (that is, $\{\xi\in\mathbb{R}^m_+ \,|\, \xi\ge x\}\subset B$ for every $x\in B$). The set $A\cap B$, the common boundary of~$A$ and $B$ relative to~$\mathbb{R}^m_+$, is called the boundary of the order decomposition $(A,B)$. (These notions, due to Hirsch \cite{Hirsch3}, are useful in the theory of monotone dynamical systems.) The set $\mathcal{H}_+$ is the boundary of the order decomposition $(\bar{\mathcal{A}}_+,\bar{\mathcal{B}}_+)$ of $\mathbb{R}^m_+$, given by $\bar{\mathcal{A}}_+:=\bar{\mathcal{A}}\cap\mathbb{R}^m_+=\{\xi\in\mathbb{R}^m_+ \,|\, \rho(\xi)\ge1\}$ (lower-closed) and $\bar{\mathcal{B}}_+:=\bar{\mathcal{B}}\cap\mathbb{R}^m_+=\{\xi\in\mathbb{R}^m_+ \,|\, \rho(\xi)\le1\}$ (upper-closed). Moreover, $\mathcal{H}_+=\bar{\mathcal{A}}_+\cap\bar{\mathcal{B}}_+$ is {\it unordered\/}, that is, it does not contain any two distinct points that are ordered. We collect the basic properties of $\mathcal{H}$ in the following proposition. \begin{proposition}\label{H}\hfill \noindent{\rm (a)}\ The set $\mathcal{H}:=\{\xi\in\mathbb{R}^m \,|\, \rho(\xi)=1\}$ is a closed subset of $\mathbb{R}^m$; it is homeomorphic, under the scaling-projection, to the relatively open subset $\mathcal{S}$ of $S^{m-1}$, defined by $\mathcal{S}:=\{\xi\in S^{m-1} \,|\, \rho(\xi)<\infty\}$; and it is the common boundary of the connected open sets $\mathcal{A}:=\{\xi\in\mathbb{R}^m \,|\, \rho(\xi)>1\}$ and $\mathcal{B}:=\{\xi\in\mathbb{R}^m \,|\, \rho(\xi)<1\}$ {\rm(}the ``inside'' and ``outside'' of $\mathcal{H}$, relative to the scaling-projection{\rm)}. \noindent{\rm (b)}\ The set $S^{m-1}\setminus\mathcal{S}$ is contained in ${\mathcal O}:= \{\xi\in\mathbb{R}^m | (-1)^{m-i} \xi_i<0\,\forall\,i\in\{1,\dots,m\}\}$; $\mathcal{H}\cap(\mathbb{R}^m\setminus{\mathcal O})$ is compact; and if $m\ge2$, then $\mathcal{H}\cap{\mathcal O}$ is bounded if and only if $\mathcal{S}=S^{m-1}$. \noindent{\rm (c)}\ The set $\mathcal{H}_+:=\mathcal{H}\cap\mathbb{R}^m_+$ is compact, homeomorphic to $S^{m-1}_+:=S^{m-1}\cap\mathbb{R}^m_+$ under the radial projection, unordered, and the boundary of the order decomposition $(\bar{\mathcal{A}}_+,\bar{\mathcal{B}}_+)$ of $\mathbb{R}^m_+$ with $\bar{\mathcal{A}}_+:=\bar{\mathcal{A}}\cap\mathbb{R}^m_+$ and $\bar{\mathcal{B}}_+:=\bar{\mathcal{B}}\cap\mathbb{R}^m_+$. \end{proposition} A complete characterization of the set $\mathcal{H}$, beyond the general properties gathered above, requires the determination of the set $\mathcal{S}$. Note that the points of $S^{m-1}\setminus\mathcal{S}$ are in one-to-one correspondence with the scaling-equivalence classes of entire radial solutions of \eqref{eVW}. Recalling Theorem~\ref{entire-vw}, we conclude that $\mathcal{S}=S^{m-1}$ whenever \eqref{eVW} is subcritical. For $m\le2$, we infer that $\mathcal{S}=S^{m-1}\setminus\{\xi_0\}$, for some point $\xi_0\in S^{m-1}\cap{\mathcal O}$, whenever \eqref{eVW} is critical or supercritical. In view of our conjecture regarding entire solutions of \eqref{eVW} (see Remark~\ref{supercritical}), we expect the latter to hold for $m\ge3$ as well. This leads to the following theorem and conjecture. \begin{theorem}\label{H1}\hfill \noindent{\rm (a)}\ Suppose that $m=1$. Then the set $\mathcal{H}$ consists of exactly two real numbers of opposite sign if \eqref{eVW} is subcritical, of exactly one positive number if \eqref{eVW} is critical or supercritical. \noindent{\rm (b)}\ Suppose that $m=2$. Then the set $\mathcal{H}$ is a closed simple curve in $\mathbb{R}^2$ if \eqref{eVW} is subcritical, an unbounded simple curve, asymptotic to a unique scaling-parabola in the fourth quadrant of $\mathbb{R}^2$, if \eqref{eVW} is critical or supercritical; in either case, the origin belongs to the interior of the curve. \noindent{\rm (c)}\ Suppose that $m\ge3$ and that \eqref{eVW} is subcritical. Then the set $\mathcal{H}$ is a closed hypersurface in $\mathbb{R}^m$, homeomorphic to $S^{m-1}$ under the scaling-projection. \end{theorem} \begin{conjecture}\label{H2} Suppose that $m\ge3$ and that \eqref{eVW} is critical or supercritical. Then the set $\mathcal{H}$ is an unbounded hypersurface in $\mathbb{R}^m$, and there exists a point \hbox{$\xi_0\in S^{m-1}\cap{\mathcal O}$} such that $\mathcal{H}$ is homeomorphic to $S^{m-1}\setminus\{\xi_0\}$ under the scaling-projection. \end{conjecture} \begin{remark} \rm While the conjecture is completely open in the supercritical case, Theorem~\ref{entire-vw} and Proposition~\ref{H} yield at least a partial result if \eqref{eVW} is critical: the set $\mathcal{H}$ is then an unbounded hypersurface in $\mathbb{R}^m$, and there exists a nonempty closed set $X_0\subset S^{m-1}\cap{\mathcal O}$ such that $\mathcal{H}$ is homeomorphic to $S^{m-1}\setminus X_0$ under the scaling-projection. \end{remark} \begin{remark} \rm As discussed in Remark~\ref{correspondence}, the set $\mathcal{H}$ (for $\mu=N-1$) is homeomorphic to the set $\mathcal{L}$ of all large radial solutions of Equation~\eqref{eU} on the unit ball $B$, endowed with the natural topology of the space $C^{2m}(B)$. Thus, by Theorem~\ref{H1}, $\mathcal{L}$ is a full $(m{-}1)$-sphere whenever \eqref{eU} is subcritical; if \eqref{eU} is critical or supercritical, and if $m\le2$, then $\mathcal{L}$ is a punctured $(m{-}1)$-sphere. If proved, Conjecture~\ref{H2} would imply the latter to hold without the restriction $m\le2$. \end{remark} \begin{remark} \rm \label{monotonicity1} In Remark~\ref{monotonicity} we conjectured that on every line parallel to one of the coordinate axes in $\mathbb{R}^m$, the function $\rho$ attains its global maximum value at exactly one point and is strictly monotonic on either side of that point. This is equivalent to saying that any such line intersects the set $\mathcal{H}$ at most twice, which would shed further light on the geometric structure of $\mathcal{H}$, with interesting implications regarding the exact multiplicity of large radial solutions of Equation~\eqref{eU}. In fact, given any $m{-}1$ of the values $u(0),\Delta u(0),\dots,\Delta\!^{m-1}u(0)$, there would be at most two large radial solutions of \eqref{eU} on $B$ with these prescribed center values. \end{remark} In the remainder of this section, we illustrate our results and derive further information in the second and fourth-order cases. We note that the accompanying graphs are not schematic drawings; they are based on high-accuracy numerical computations (the method will be described in \cite{DLS2}). The depictions of typical large radial solutions illustrate qualitative features that are consequences of the present analysis, as discussed below. These graphs do not and cannot resolve fine points of the solutions' blow-up behavior; those will be addressed in~\cite{DLS2}. First consider the familiar second-order case, $m=1$. If the equation~\eqref{eU} is subcritical, there are exactly two large radial solutions on the unit ball, one with a positive, the other with a negative center value. If \eqref{eU} is critical or supercritical, there is exactly one such solution, with a positive center value. In either case, the solutions are strictly increasing with respect to the radial variable. Figure~1 shows the large radial solutions of \eqref{eU} on $B$ in space dimension $N=3$ for $p=3$, $p=4$, and $p=4.5$ (subcritical, two solutions) as well as for $p=5$ (critical, one solution). \begin{figure}[ht] \begin{center} \includegraphics[width=\textwidth,trim=115 490 100 0]{figures/fig1} \end{center} \caption{Large radial solutions of \eqref{eU} on $B$ for $m=1$ and $N=3$, in the subcritical cases $p=3$ (top-left), $p=4$ (top-right), $p=4.5$ (bottom-left), and in the critical case $p=5$ (bottom-right).} \end{figure} Now consider the fourth-order case, $m=2$. If Equation~\eqref{eU} is subcritical, the set $\mathcal{H}$ is a closed simple curve in $\mathbb{R}^2$, containing the origin in its interior. Hence there exist numbers ${\underline\alpha},\,{\overline\alpha}\in\mathbb{R}$ with ${\underline\alpha}<0<{\overline\alpha}$ such that, given $\alpha\in\mathbb{R}$, Equation~\eqref{eU} has no large radial solution on $B$ with center value $\alpha$ if $\alpha<{\underline\alpha}$ or $\alpha>{\overline\alpha}$; at least one such solution if $\alpha={\underline\alpha}$ or $\alpha={\overline\alpha}$; and at least two such solutions if ${\underline\alpha}<\alpha<{\overline\alpha}$. (As~noted in Remark~\ref{monotonicity1}, we could say ``exactly'' instead of ``at least'' if our conjecture regarding the monotonicity of $\rho$ on lines parallel to the coordinate axes were proved.) Figure~2 depicts the set $\mathcal{H}$ for $p=3$ and $N=3$, along with the scaling-parabolae passing through the four extremal points of $\mathcal{H}$. The close-up in the graph on the right reveals that there are indeed two extremal points in the fourth quadrant, albeit very close to each other. Figure~3 shows typical large radial solutions of \eqref{eU} on $B$, including the ones with the smallest and largest center values (top-left graph). \begin{figure}[p] \begin{center} \includegraphics[width=\textwidth,trim=115 676 99 0]{figures/fig2} \end{center} \caption{The set $\mathcal{H}$ for $m=2$, $p=3$, $N=3$ (subcritical, $\mathcal{H}$ compact).} \end{figure} \begin{figure}[p] \begin{center} \vskip15pt \includegraphics[width=\textwidth,trim=118 492 98 0]{figures/fig3} \end{center} \caption{Large radial solutions of \eqref{eU} on $B$ for $m=2$, $p=3$, $N=3$ (subcritical).} \end{figure} \begin{figure}[ht] \begin{center} \includegraphics[width=\textwidth,trim=117 676 99 0]{figures/fig4} \end{center} \caption{The set $\mathcal{H}$ for $m=2$, $p=3$, $N=13$ (supercritical, $\mathcal{H}$ unbounded).} \end{figure} If Equation~\eqref{eU} is critical or supercritical, the set $\mathcal{H}$ is an unbounded simple curve in $\mathbb{R}^2$, asymptotic to a unique scaling-parabola in the fourth quadrant and containing the origin in its ``interior.'' Hence there exists a number ${\underline\alpha}\in\mathbb{R}$ with ${\underline\alpha}<0$ such that, given $\alpha\in\mathbb{R}$, Equation~\eqref{eU} has no large radial solution on $B$ with center value $\alpha$ if $\alpha<{\underline\alpha}$; at least one such solution if $\alpha={\underline\alpha}$; and at least two such solutions if $\alpha>{\underline\alpha}$. (Again, we could say ``exactly'' instead of ``at least'' if the conjecture in Remark~\ref{monotonicity} were proved.) Figure~4 depicts the set $\mathcal{H}$ for $p=3$ and $N=13$, along with the scaling-parabolae passing through the two extremal points of $\mathcal{H}$ and the unique scaling-parabola in the fourth quadrant that does not intersect $\mathcal{H}$. Figure~5 shows typical large radial solutions of \eqref{eU} on $B$, including the one with the smallest center value (on the left). Despite its appearance, the graph on the right contains six solutions --- two for each of the center values, one positive, the other sign-changing. \begin{figure}[ht] \begin{center} \includegraphics[width=\textwidth,trim=117 674 100 0]{figures/fig5} \end{center} \caption{Large radial solutions of \eqref{eU} on $B$ for $m=2$, $p=3$, $N=13$ (supercritical).} \end{figure} All large radial solutions of \eqref{eU} on $B$ that start in the upper half-plane or on the horizontal axis (that is, solutions $u$ with $\Delta u(0)\ge0$) are strictly increasing with respect to the radial variable; those starting in the lower half-plane (that is, solutions $u$ with $\Delta u(0)<0$) are strictly decreasing to a global minimum and strictly increasing thereafter. In the subcritical case, there exists a solution of the second kind whose minimum value is $0$ and which, therefore, solves the Dirichlet problem for Equation \eqref{eU} on a ball centered at the origin. This solution necessarily starts at a point $(\alpha^*,\beta^*)$ on the ``upper'' part of $\mathcal{H}$ in the fourth quadrant. The segment of $\mathcal{H}$ in the first quadrant and its continuation into the fourth quadrant, down to the point $(\alpha^*,\beta^*)$, is the locus of the starting-points of the nonnegative large radial solutions of \eqref{eU} on $B$. As a consequence of Proposition~\ref{rho}(b), this segment is unordered and thus the graph of a strictly decreasing continuous function $\phi \,{:}\ [0,\alpha^*]\to\mathbb{R}$ with $\phi(0)=:\beta_*>0$ and $\phi(\alpha^*)=\beta^*<0$. It follows that, given $\alpha\in\mathbb{R}$, Equation~\eqref{eU} has a unique nonnegative large radial solution $u$ on $B$ with center value $u(0)=\alpha$ if and only if $0\le\alpha\le\alpha^*$. As~$\alpha$ increases from $0$ to~$\alpha^*$, the second center value $\Delta u(0)$ decreases from $\beta_*>0$ to $\beta^*<0$. Further, the solution is strictly positive except if $\alpha=0$ or $\alpha=\alpha^*$. In the critical or supercritical case, radial solutions of \eqref{eU} starting on the unique scaling-parabola in the fourth quadrant that misses the set $\mathcal{H}$ are entire solutions and positive. Consequently, the segment of $\mathcal{H}$ ``above'' this parabola is comprised of starting-points of positive large radial solutions of \eqref{eU} on $B$. Proposition~\ref{rho}(b) implies that this segment, including its end-point above the origin, is unordered and thus the graph of a strictly decreasing continuous function $\phi \,{:}\ [0,\infty)\to\mathbb{R}$ with $\phi(0)=:\beta_*>0$ and $\phi(\alpha)\to-\infty$ as $\alpha\to\infty$. We conclude that Equation~\eqref{eU} has a unique nonnegative large radial solution $u$ on $B$ with $u(0)=\alpha$ for every center value $\alpha\in[0,\infty)$. As~$\alpha$ increases from $0$ to $\infty$, the second center value $\Delta u(0)$ decreases from $\beta_*>0$ to $-\infty$, and the solution is strictly positive except if $\alpha=0$. In the critical case, similar results were obtained in~\cite{Grunau-Reichel}. Figure~6 shows typical nonnegative large radial solutions of \eqref{eU} on $B$ in a subcritical and a supercritical case. Included are the extremal solutions with center value $0$ (in both cases) and the one with center value~$\alpha^*$ in the subcritical case. \begin{figure}[ht] \begin{center} \includegraphics[width=\textwidth,trim=117 679 99 7,clip]{figures/fig6} \end{center} \caption{Nonnegative large radial solutions of \eqref{eU} on $B$ for $m=2$ and $p=3$, in the cases $N=3$ (subcritical, left) and $N=13$ (supercritical, right).} \end{figure} \subsection*{Note Added in Proof} After this paper was accepted and edited for publication, we became aware of Reference~\cite{Chen-Li-Ou}, which appears to close the gap in \cite{Wei-Xu} discussed at the beginning of Remark~\ref{supercritical}. It would follow that, in the critical case with arbitrary $m\in{\mathbb N}$, Equation~\eqref{eU} has exactly one scaling-equivalence class of entire radial solutions, and then the set of all large radial solutions on the unit ball is homeomorphic to a punctured $(m{-}1)$-sphere. This lends further credence to Conjecture~\ref{H2}. We thank Tobias Weth of the University of Giessen for drawing our attention to~\cite{Chen-Li-Ou}. \subsection*{Acknowledgements} Ildefonso D\'{\i}az was supported by Research Project MCT-2004-05417 (Spain). Monica Lazzo was supported by MIUR, Project ``Metodi va\-ria\-zio\-na\-li e topologici ed equazioni differenziali non lineari.'' 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