\documentclass[reqno]{amsart}
\usepackage{graphicx, amssymb}
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\AtBeginDocument{{\noindent\small
\emph{Electronic Journal of Differential Equations},
Vol. 2008(2008), No. 101, pp. 1--7.\newline
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu
\newline ftp ejde.math.txstate.edu  (login: ftp)}
\thanks{\copyright 2008 Texas State University - San Marcos.}
\vspace{9mm}}

\begin{document}
\title[\hfilneg EJDE-2008/101\hfil Infinitely many solutions]
{Infinitely many solutions for the $p$-Laplace
equations with nonsymmetric perturbations}

\author[D. Liu, D. Geng \hfil EJDE-2008/101\hfilneg]
{Disheng Liu, Di Geng} 

\address{School of Mathematical Sciences, South China Normal University,
Guangzhou 510631, China}
\email[Disheng Liu]{dison\_lau@yahoo.cn}
\email[Di Geng (Corresponding author)]{gengdi@scnu.edu.cn}

\thanks{Submitted February 28, 2008. Published July 30, 2008.}
\thanks{Supported by grant 7005795 from  Guangdong Provincial Natural Science
 Foundation of China}
\subjclass[2000]{35J70, 35D50}
\keywords{$p$-Laplacian; large Morse index; nonsymmetric
 perturbation; \hfill\break\indent infinitely many solutions}

\begin{abstract}
 In this article, we study Dirichlet problems involving the $p$-Lapla\-cian
 with a nonsymmetric term. By using the large Morse index of the
 corresponding Laplace equation, we establish an estimate on the
 growth of the min-max values for a functional associated with
 the problem. The estimate is better than the given result in some range.
 We show that the problem possesses infinitely many weak solutions.
\end{abstract}

\maketitle
\numberwithin{equation}{section}
\newtheorem{theorem}{Theorem}[section]
\newtheorem{lemma}[theorem]{Lemma}

\section{Introduction and statement of main results}

In this paper, we investigate the existence of infinitely many
weak solutions for the Dirichlet problem, involving $p$-Laplacian,
\begin{equation} \label{e1}
\begin{gathered}
-\Delta_pu = |u|^{q-2}u+f(x),\quad \text{in }\Omega\\
u=0,\quad \text{on }\partial\Omega,
\end{gathered}
\end{equation}
where $\Omega$ is a bounded domain in $\mathbb{R}^N$ with smooth
boundary and $N>p>1$;
$\Delta_pu=\mathop{\rm div}(|\nabla u|^{p-2}\nabla u)$ is so-called
$p$-Laplace operator and $p<q<p^*=Np/(N-p)$.

 Rabinowitz \cite{R},  Bahri \& Berestycki \cite{BB} and
Struwe \cite{Struwe} discussed this type problem in the special
case of $p=2$ using perturbation method for even  symmetric
functional.  Bahri \&  Lions \cite{BL} and  Tanaka \cite{T}
employed the Morse index theory and obtained the best result up to
now.  Garcia Azorero \&  Peral Alonso \cite{AA} generalized the
multiple solutions results as in \cite{BB, R} into $p$-Laplacian
equation. In the case of $p$-Laplacian with $p\not=2$, not as in
Hilbert space $H_0^1(\Omega)$, because of the lack of Hilbert
space framework, in the Banach space $W_0^{1,p}(\Omega)$, Morse
index theory cannot directly be applied into this class of
equation, yet corresponding multiple solutions problem had not
reached the optimal result as in the situation of Laplace
equation. In this note, we compare the variational functional of
problem \eqref{e1} involving $p$-Lapacian ($p>2$) with that
involving Laplacian, establish some estimate on growth of critical
values of $p$-Laplace equation with the large Morse index of
Laplace equation and show existence of infinitely many solutions
of $p$-Laplace problem with nonsymmetric perturbation \eqref{e1}.
On the some range, the result we obtain improves the known
conclusion of  Garcia Azorero and  Peral Alonso \cite{AA}.

The main result in this paper is the following theorem.

\begin{theorem} \label{thm1}
Suppose $f(x)\in C(\bar\Omega)$ and
$$
2<p<q<\max\big\{\bar q,\,\min\{\bar{ \bar q},
{Np-2N+4\over Np-2N+2p}{Np\over N-2}\}\big\},
$$
where $\bar q$ and $\,\bar{ \bar q}$ are the largest roots of the
equations
\begin{equation} \label{e2} % \label{e2}
{q\over q-1}={Np-q(N-p)\over(q-p)N}\quad \text{and}\quad
{q\over q-1}={p\over N}{q(Np-2N+2p)-pN(p-2)\over(q-p)(Np-2N+2p)},
\end{equation}
respectively. Then the problem \eqref{e1} possesses infinitely
many (weak) solutions $\{u_k\}\subset W_0^{1,p}(\Omega)$, and the
corresponding critical values of variational functional tend to
the positive infinity.
\end{theorem}

The weak solutions of \eqref{e1} are the critical points of the
$C^1$ functional for $\theta=1$ as follows:
\begin{equation} \label{e3}
J_\theta(u)={1\over p}\int_\Omega|\nabla u|^pdx-{1\over q}
\int_\Omega|u|^q\,dx -\theta\int_\Omega f(x)udx,\quad
(u,\theta)\in W^{1,p}_0(\Omega)\times[0,1].
\end{equation}
The equivalent norm in $W_0^{1,p}(\Omega)$ is $\|\nabla u\|_p$. If
$\theta=0$, $J_0(u)$ is an even functional on $W_0^{1,p}(\Omega)$.
It is clear that, if $p>2$, ${Np-2N+4\over Np-2N+2p}{Np\over
N-2}<p^*$. Therefore, under the assumptions of Theorem \ref{thm1}, the
functional $J_\theta(u)$ satisfies Palais-Smale condition in
$W_0^{1,p}(\Omega)$ (see, for instance, \cite{Struwe}).


\section{Preliminaries}

To seek a series of critical values of $J_1(u)$, we
introduce the following facts: In the Sobolev space
$W_0^{1,p}(\Omega)$, we know that (see, for example, Triebel
\cite{Triebel}) there is Schauder basis.
And in the Sobolev space
$W_0^{1,p}(\Omega)$, the Sobolev inequality is valid.

\begin{lemma}[Triebel \cite{Triebel}] \label{triebel}
Let $\Omega\subset\mathbb{R}^N$ be a bounded domain of cone-type.
There exists a Schauder basis $\{w_k\}_{k=1}^\infty$ in
$W^{1,p}(\Omega)$, such that for $q\in[p,p^*)$, there is some
positive constant $C_1>0$, it holds for
$u\in\overline{\mathop{\rm span}}\{w_k,w_{k+1},\dots\}$
\begin{equation} \label{e4}
C_1k^{{1\over N}+{1\over q}-{1\over
p}}\|u\|_{L_q(\Omega)}\leq\|u\|_{W^{1,p}(\Omega)}.
\end{equation}
\end{lemma}

Denote the Schauder basis in $W_0^{1,p}(\Omega)$ satisfied the
above lemma by $\{w_k\}$. The basis $\{w_k\}$ is also Schauder
basis in $H_0^1(\Omega)$.
Denote by
$$
E_k={\rm span}\{w_1,w_2,\dots,w_k\},\quad
E_k^\bot=\overline{\rm span}\{w_k,w_{k+1},\dots\}.
$$
Define family of maps  and series of min-max values of $J_0(u)$ as
follows:
\begin{gather*}
\Gamma_k(p)=\{\gamma\in C(E_k\cap\bar B_{R_k}(0),W_0^{1,p}(\Omega));
\gamma\text{ is odd, and }\gamma|_{E_k\cap\partial B_{R_k}(0)}
=\mathop{\rm id}\}; \\
c_k=\inf_{\gamma\in\Gamma_k(p)}\max_{u\in E_k\cap B_{R_k}(0)}J_0(\gamma(u)),
\end{gather*}
where $R_k$ is a series of positive constants tend to positive infinite,
such that $J_0(u)<0$ for all $u\in E_k$ and $\|u\|\geq R_k$.

\begin{lemma} \label{lem2}
For every $k$ large enough, there exists $\rho_k>0$ such that
$$
J_\theta(u)\geq C_2k^{pN-q(N-p)\over N(q-p)},\quad\text{when }
u\in E_k^\bot\cap\partial B_{\rho_k}(0),
$$
where $C_2$ is independent of $k$.
\end{lemma}

\begin{proof}
 For $u\in E_k^\bot$, applying Lemma \ref{triebel},
we estimate the functional $J_\theta(u)$:
\begin{align*}
J_\theta(u)
&= {1\over p}\|\nabla u\|_p^p-{1\over q}\|u\|_q^q-\theta\int_\Omega f(x)udx\\
&\geq {1\over p}\|\nabla u\|_p^p-Ck^{{q\over p^*}-1}\|\nabla u\|_p^q-C',
\end{align*}
where $C'$ is a constant depending only on $f$.

By letting $\rho_k=(k^{1-{q\over p}+{q\over N}}/(2qC))^{1/(q-p)}$,
one have, for all $k$ large enough and
$u\in E_k^\bot\cap\partial B_{\rho_k}(0)$,
$$
J_\theta(u)\geq\big[{1\over2p}(2qC)^{-{p\over q-p}}-C(2qC)^{{-q\over
q-p}}\big] k^{({q\over p^*}-1){p\over q-p}}-C'\geq
C_2k^{{qp\over N(q-p)}-1},
$$
where $C_2$ is a positive constant independent of $k$.
\end{proof}

 From the above lemma, using the Borsuk theorem, we obtain
\begin{equation} \label{e5}   %\label{b_k-}
c_k\geq Ck^{pN-q(N-p)\over N(q-p)}.
\end{equation}
This estimate on $c_k$ in the case $p=2$ was obtained by
Rabinowitz with the growth of eigenvalue of Laplacian due to
Hilbert-Courant;  Garcia Azorero and  Peral Alonso
\cite{AA} got the similar result in the special case
$\Omega=[0,1]^N$ for $p\not=2$.

\section{Proof of the main theorem}

As a matter of convenience, we write the functional $J_0(u)$ as
$I_{p,q}(u)$; that is,
$$
I_{p,q}(u)={1\over p}\int_\Omega|\nabla u|^pdx-{C\over q}\int_\Omega|u|^qdx,
$$
where $C$ is a positive constant depends only on $p$, $q$ and
$|\Omega|$.

\begin{lemma}  \label{lem3}
Suppose $q<p^*=pN/(N-p)$. There exits a positive constant $C_3$
dependent only on $p$ and $q$, such that
\begin{equation} \label{e6} % \label{2,r}
I_{p,q}(u)\geq C_3(I_{2,r}(u))^{p/2},\quad u\in W^{1,p}(\Omega),
\end{equation}
where $r=(q(Np-2N+2p)-pN(p-2))/p^2$; moreover if
\begin{equation} \label{e7}  % \label{r<2^*}
 q<{Np-2N+4\over Np-2N+2p}{Np\over N-2},
\end{equation}
 then $r<2^*$.
\end{lemma}

\begin{proof} By the assumption $q<p^*=pN/(N-p)$ and the
interpolation inequality and Sobolev inequality, one can estimate
the second term in $I_{p,q}(u)$ as follows:
\begin{equation} \label{e8}
\begin{aligned}
\int_\Omega|u|^qdx
&\leq \Big(\int_\Omega|u|^{p^*}dx\Big)^{q{(1-\alpha)\over p^*}}
\Big(\int_\Omega|u|^rdx\Big)^{q{\alpha\over r}} \\
&\leq \Big(S\int_\Omega|\nabla u|^pdx\Big)^{q{1-\alpha\over p}}
\Big(\int_\Omega|u|^rdx\Big)^{q{\alpha\over r}}\\
&\leq {q\over2p}\int_\Omega|\nabla u|^pdx
+C\Big(\int_\Omega|u|^rdx\Big)^{q{\alpha\over r}{p\over p-(1-\alpha)q}},
\end{aligned}
\end{equation}
where the parameters $r$ and $\alpha$ satisfy: $q>r>1$,
$0<\alpha<1$, $p>q(1-\alpha)$ and
\begin{equation} \label{e9} % \label{zb}
q{1-\alpha\over p^*}+q{\alpha\over r}=1.
\end{equation}
When $u\in W_0^{1,p}(\Omega)$ and $J_0(u)=I_{p,q}(u)\gg0$, we also
have
\begin{equation} \label{e10}
\begin{aligned}
I_{p,q}(u)&\geq {1\over p}\int_\Omega|\nabla u|^pdx-{1\over q}
\Big[{q\over2p}\int_\Omega|\nabla u|^pdx
+C\Big(\int_\Omega|u|^rdx\Big)^{q{\alpha\over r}{p\over p-(1-\alpha)q}}
\Big]\\
&\geq {1\over2p}\int_\Omega|\nabla u|^pdx-{C\over q}
\Big(\int_\Omega|u|^rdx\Big)^{q{\alpha\over r}{p\over p-(1-\alpha)q}}\\
&\geq \Big[\Big({1\over2p}\int_\Omega|\nabla u|^pdx\Big)^{2/p}
-\Big({C\over q}\int_\Omega|u|^rdx\Big)
^{{q{\alpha\over r}{p\over p-(1-\alpha)q}}{2\over p}}\Big]^{p/2}\\
&\geq \Big[C(|\Omega|)\int_\Omega|\nabla u|^2dx- \Big({C\over
q}\int_\Omega|u|^rdx\Big)^{{q{\alpha\over r}{2\over p-(1-\alpha)q}}}
\Big]^{p/2}.
\end{aligned}
\end{equation}
In addition to \eqref{e9}, we  require that $r$ and $\alpha$
satisfy
\begin{equation} \label{e11} % \label{ze}
q{\alpha\over r}{2\over p-(1-\alpha)q}=1.
\end{equation}
Solving the simultaneous equations \eqref{e9} and \eqref{e11}, we obtain
$$
\alpha=1-{1\over q}{{p\over2}-1\over{1\over2}-{1\over p^*}},\quad
r={q({1\over2}-{1\over p^*})-({p\over2}-1)\over{p\over2}\Big({1\over p}-{1\over p^*}\Big)}={q(Np-2N+2p)-pN(p-2)\over p^2}.
$$
Therefore, the inequality \eqref{e10} becomes
\begin{equation} \label{e12}
\begin{aligned}
I_{p,q}(u)
&\geq \Big[C(|\Omega|)\int_\Omega|\nabla u|^2dx-C\int_\Omega|u|^rdx\Big]^{p/2}\\
&\geq C_3\Big[{1\over2}\int_\Omega|\nabla u|^2dx-{C\over
r}\int_\Omega|u|^rdx\Big]^{p/2}=C_3I_{2,r}(u)^{p/2},
\end{aligned}
\end{equation}
where
$$
I_{2,r}(u)={1\over2}\int_\Omega|\nabla u|^2dx-{C\over r}\int_\Omega|u|^rdx,
$$
and $C_3$ and $C$ are positive constants, without loss of generality,
 we suppose that $C\geq1$ and $C_3\leq1$.

As regards \eqref{e7}, which is a simple fact, we skip over the
detail. \end{proof}

Next, we present an estimate on $c_k$ under the
condition $p\geq2$ which is superior to \eqref{e5} in a certain
extent.

\begin{lemma} \label{lem4}
Suppose that $2<p<q<{Np-2N+4\over Np-2N+2p}{Np\over N-2}$.
There exists positive constant $C_4$ independent of $k$, such that
for all $k$ large enough,
$$
c_k\geq C_4k^{{p\over N}\,{q(Np-2N+2p)-pN(p-2)\over(q-p)(Np-2N+2p)}}.
$$
\end{lemma}

\begin{proof}
We define a series of min-max values of the
functional $I_{2,r}(u)$ in the previous lemma as follows:
$$
\tilde c_k=\inf_{\gamma\in\Gamma_k(p)}
\max_{u\in E_k\cap B_{R_k}(0)}I_{2,r}(\gamma(u)).
$$
Obviously,  \eqref{e12} indicates that if $\tilde c_k$
is sufficiently large that for every $\gamma\in\Gamma_k(p)$, and
$w_k$ satisfying
$$
I_{2,r}(w_k)=\max_{u\in E_k\cap B_{R_k}(0)}I_{2,r}(\gamma(u)),
$$
there holds $I_{2,r}(w_k)\gg0$. Therefore, the inequality
\eqref{e12} leads to
\begin{equation} \label{e13}  %\label{v}
c_k\geq C_3\tilde c_k^{p/2}.
\end{equation}
On the other hand, for every $\gamma$ in $\Gamma_k(p)$,
$\gamma(E_k\cap\bar B_{R_k}(0))\subset W^{1,p}_0(\Omega)\subset
H_0^1(\Omega)$, which implies $\Gamma_k(p)\subset\Gamma_k(2)$,
hence we have
\begin{equation} \label{e14} %\label{zg}
\tilde c_k\geq b_k=\inf_{\gamma\in\Gamma_k(2)}
\max_{u\in E_k\cap B_{R_k}(0)}I_{2,r}(\gamma(u)).
\end{equation}
Moreover, from \eqref{e7} it follows that
$q<{Np-2N+4\over Np-2N+2p}{Np\over N-2}$ implies $r<2^*$.
Therefore, the functional $I_{2,r}(u)$ fulfill the Palais-Smale
condition in $H_0^1(\Omega)$. By the results due to  Tanaka \cite{T}
or Bahri-Lions \cite{BL}, we know that $b_k$ are the critical
values of $I_{2,r}(u)$. And the large Morse index of $I_{2,r}(u)$
at the critical point $u$ corresponding to $b_k$ implies that
\begin{equation} \label{e15}
b_k\geq Ck^{{2\over N}{r\over r-2}}=Ck^{{2\over
N}{q(Np-2N+2p)-pN(p-2)\over q(Np-2N+2p)-pN(p-2)-2p^2}},
\end{equation}
where $C$ is a positive constant independent of $k$.
Thus $\tilde c_k\to+\infty$, that is, \eqref{e13} holds for all $k$ large
enough. Again by \eqref{e12},
\begin{equation} \label{e16} %\label{zh}
c_k\geq C_3b_k^{p/2}\geq C_4k^{{2\over N}
{q(Np-2N+2p)-pN(p-2)\over q(Np-2N+2p)-pN(p-2)-2p^2}{p\over2}},
\end{equation}
the conclusion of the lemma follows.                                                                                      \end{proof}

By H\"older inequality, if $u\in W^{1,p}_0(\Omega)$ is a critical
point of $J_\theta(u)$, we  obtain
\begin{equation}  \label{4a}
\big|{\partial\over\partial\theta}J_\theta(u)\big|
=\big|\int_\Omega f(x)udx\big| \leq C_5(|J_\theta(u)|+1)^{1/q},
\end{equation}
where $C_5$ is a positive constant depending only on
$\|f\|_{C(\bar\Omega)}$.

The method in Hilbert spaces  developed by Bolle \cite{B} can be
generalized into Banach spaces (just replace the gradient vector
fields with the pseudo-gradient vector fields), which can be used
to deduce the following fact, if the functional $J_1(u)$ has at
most finite critical values, by using \eqref{4a} and the proof of
\cite[Theorem 2.2]{BGT}, we get
\begin{equation} \label{4b}
c_{k+1}-c_k\leq C(c_{k+1}^{1/q}+1).
\end{equation}
With the facts above, we can establish the following estimates on
$c_k$.

\begin{lemma} \label{lem5}
If there are at most finite critical values for the functional
$J_1(u)$, there exits a positive constant $C_6$ independent of
$k$, such that
$$
c_k\leq C_6k^{q\over q-1}.
$$
\end{lemma}

\begin{proof}
Without loss of generality, one can suppose $c_k\geq1$,
then from \eqref{4b} it follows that
\begin{gather*}
c_{k+1}-c_k \leq 2C\cdot c_{k+1}^{1/q};\\
c_k-c_{k-1} \leq 2C\cdot c_{k}^{1/q}\leq2C\cdot c_{k+1}^{1/q};\\
\dots\\
c_2-c_1\leq 2C\cdot c_{2}^{1/q}\leq2C\cdot c_{k+1}^{1/q}.
\end{gather*}
Adding the two sides, respectively, we have
 $c_{k+1}-c_1\leq2k\cdot C\cdot c_{k+1}^{1/q}$,
which implies
$$
c_{k+1}\leq2k\cdot C\cdot
c_{k+1}^{1/q}+c_1<2(k+1)\cdot C\cdot c_{k+1}^{1/q},
$$
where the constant $C$ may be changed. And then the conclusion follows.
\end{proof}

\begin{proof}[Proof of the main theorem]
Combining the above lemmas and
the method developed by  Bolle-Ghoussoub-Tehrani
\cite{BGT}, we can deduce the existence of infinitely many
solutions of \eqref{e1}. In fact, if $J_1(u)$ has at most finite
critical values, the conclusions of Lemma \ref{lem2}, Lemma
\ref{lem3} and Lemma \ref{lem4} imply that, for all sufficiently
large $k$,
$$
\max\{C_4k^{{p\over N} {q(Np-2N+2p)-pN(p-2)\over(q-p)(Np-2N+2p)}},
C_3k^{{p\over N}{2q-N(p-2)\over2(q-p)}}\}
\leq c_k\leq C_6k^{q\over q-1}.
$$
However, according to the assumptions of Theorem \ref{thm1}, we have
$$
{q\over q-1}<\max\Big\{{p\over N}{2q-N(p-2)\over2(q-p)}, \,{p\over
N}\,{q(Np-2N+2p)-pN(p-2)\over(q-p)(Np-2N+2p)}\Big\},
$$
which yields contradiction.
\end{proof}

\subsection*{Remark}
The conclusion of infinitely many solutions based
on \eqref{e6} is better than that of Garcia Azorero and Peral
Alonso \cite{AA} in some range. In fact, according to the
conclusions in \cite{AA}, we know that, when $q\in(p,\bar q)$, the
problem \eqref{e1} has infinitely many solutions, where
$\bar q=\bar q(p)$ is the largest root of the first equation in
\eqref{e2} as follows:
\begin{equation} \label{e19} % \label{2---}
{q\over q-1}={pN-q(N-p)\over N(q-p)},
\end{equation}
the expression on the right hand of the equation \eqref{e19} is
also the exponent  in Lemma \ref{triebel}. Notice that if we
denote the largest root of the second equation in \eqref{e2} by
$\bar{\bar q}=\bar{\bar q}(p)$, it is clear that
$$
\bar{ \bar q}(2)={2(N-1)\over N-2}>\bar q(2),
$$
so there exits some $p_0\in(2,+\infty]$, such that
$\bar{ \bar q}(p)>\bar q(p)$ for all $p\in[2,p_0)$. Setting
$\bar q(p)=\bar{\bar q}(p)$, we can find out $p_0$. Since
$\bar q(p)$ and $\bar{\bar q}(p)$ satisfy the equations in \eqref{e2},
we have
$$
{p\over N}{q(Np-2N+2p)-pN(p-2)\over(q-p)(Np-2N+2p)}={pN-q(N-p)\over N(q-p)},
$$
that is, $q=(Np-2N+p^2)p/(Np-2N+2p)$. From the equation
$\bar q(p)$ or $\bar{ \bar q}(p)$ satisfies, it follows that $p_0$ meets
a quartic equation as follows:
\begin{equation} \label{e20}
p^4-(2+N+N^2)p^2+(2N+4N^2)p-4N^2=0.
\end{equation}
 Analysis on this quartic
equation with Mathematica yields some interesting facts: when
$N=3,4,5,6$, the equation has no real root greater than 2,
that is, in those cases, our result under the hypothesis in
Theorem \ref{thm1} is better than that in \cite{AA}; when $N\geq7$, the
equation \eqref{e20} has two real roots greater than 2, the first one
is $p_0$, which is in the interval $(2,3)$. Therefore, we can
conclude that, under the conditions of Theorem \ref{thm1}, if $p\in(2,p_0)$
and $q\in(p,\bar{\bar q}(p))$, the problem \eqref{e1} possesses
infinitely many solutions. The conclusion is better than that in
\cite{AA}, since
$(p,\bar q(p))\varsubsetneqq (p,\bar{ \bar q}(p))$.


\begin{figure}[ht]
\begin{center}
\includegraphics{fig1} \quad\includegraphics{fig2} % N=6.eps N=8.eps
\end{center}
\caption{Graphs of $\bar q$, $\bar{ \bar q}$, $p^*$ for $N=6$ and $N=8$}
\end{figure}

Figure 1 illustrates the relationship among $\bar q$, $\bar{ \bar q}$, $p^*$
and $\bar r(p)={Np-2N+4\over Np-2N+2p}{Np\over N-2}$ in the cases $N=6$
and $N=8$. In each figure, two dashed curves are $q=p^*(p)$ and $q=\bar r(p)$;
the two solid curves represent $q=\bar q(p)$ and
$q=\bar{ \bar q}(p)$. Notice that the curve
$q=\bar{\bar q}(p)$ is always over $q=\bar q(p)$ for all $p\geq2$ when
$N=6$, and the curve $q=\bar{ \bar q}(p)$ is over $q=\bar q(p)$
near $p=2$ when $N=8$. The two figures were produced with Mathematica.

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