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\markboth{\hfil Exact multiplicity of a superlinear problem \hfil EJDE/Conf/10}
{EJDE/Conf/10 \hfil Junping Shi \hfil}
\begin{document}
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\title{\vspace{-1in}\parbox{\linewidth}{\footnotesize\noindent
Fifth Mississippi State Conference on Differential Equations and
Computational Simulations, \newline
Electronic Journal of Differential Equations,
Conference 10, 2003, pp 257--265. \newline
http://ejde.math.swt.edu or http://ejde.math.unt.edu
\newline ftp ejde.math.swt.edu (login: ftp)}
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Exact multiplicity of positive solutions to a superlinear problem
%
\thanks{ {\em Mathematics Subject Classifications:} 35J65, 35J60, 35B32.
\hfil\break\indent
{\em Key words:} Semilinear elliptic equations, exact multiplicity of solutions.
\hfil\break\indent
\copyright 2003 Southwest Texas State University. \hfil\break\indent
Published February 28, 2003. } }
\date{}
\author{Junping Shi}
\maketitle
\begin{abstract}
We generalize previous uniqueness results on a semilinear elliptic
equation with zero Dirichlet boundary condition and superlinear,
subcritical nonlinearity. Our proof is based on a bifurcation
approach and a Pohozaev type integral identity, which greatly
simplifies the previous arguments.
\end{abstract}
\newtheorem{theorem}{Theorem}[section]
\newtheorem{lemma}[theorem]{Lemma}
\newtheorem{corollary}[theorem]{Corollary}
\numberwithin{equation}{section}
\section{Introduction}
We consider the exact multiplicity of the solutions to the
semilinear elliptic equation
\begin{equation}\label{e:1.1}
\begin{gathered}
\Delta u+ \lambda f(u) = 0 \quad \mbox{in } B^n ,\\
u>0 \quad \mbox{in } B^n,\\
u=0 \quad \mbox{on } \partial B^n,
\end{gathered}
\end{equation}
where $B^n$ is the unit ball in $\mathbb{R}^n$ with $n\ge 3$, and
$\lambda$ is a positive parameter. The uniqueness and exact
multiplicity of the positive solutions to \eqref{e:1.1} have been
extensively studied in the past two decades, and in particular a
systematic approach has been developed in \cite{OS1} and
\cite{OS2}. (More references can be found therein.)
In this paper we assume that $f$ satisfies
\begin{enumerate}
\item[(D1)] $f \in C^1(\overline{\mathbb{R}^+})$, $f(0)=0$, $f(u)>0$, $f'(u)>0$ for
$u>0$;
\item[(D2)] There exists $p , q>0$ such that for all $u>0$,
\begin{equation}
\label{e:14.8} 1\le q\le K_f(u) \le p < \frac{n+2}{n-2}, \;\;
\text{ where } K_f(u)=\frac{uf'(u)}{f(u)};
\end{equation}
\item[(D3)] Let
\begin{equation}
\label{e:14.10} A_f(u)=(p-1)\big[ nF(u)-\frac{n-2}{2}
f(u)u\big]+[f'(u)u-pf(u)]u,
\end{equation}
where $F(u)=\int_0^u f(t)dt$. Then $A_f(u) \ge 0$ for $u\ge 0$.
\end{enumerate}
From (D2), $uf'(u)\ge f(u)$ for all $u>0$, thus the function
$f(u)/u$ is increasing for $u>0$. We define
\begin{equation}\label{e:a1}
\lambda_0=\frac{\lambda_1}{f'(0)}, \;\; \text{ and }
\lambda_{\infty}=\frac{\lambda_1}{f'(\infty)},
\end{equation}
where $f'(\infty)=\lim_{u\to \infty}f(u)/u$ and $\lambda_1$ is the
principal eigenvalue of $-\Delta$ in $H^1_0(B^n)$. When $f'(0)=0$,
we understand that $\lambda_0=\infty$ and when $f'(\infty)=\infty$,
$\lambda_{\infty}=0$.
Then our main result is as follows.
\begin{theorem} \label{thm:1}
Suppose that $f$ satisfies (D1), (D2), and (D3).
Then \eqref{e:1.1} has no solution for $0< \lambda \le
\lambda_{\infty}$ and $\lambda\ge \lambda_0$,
and has exactly one solution for $ \lambda_{\infty}<\lambda<\lambda_0$.
Moreover all solutions lie on a single smooth solution curve in
$(\lambda,u)$ space, which starts from $(\lambda_0,0)$ and continues to
the left up to $(\lambda_{\infty}, \infty)$, and there is no any
turning point on the curve. (see Figures. 1 and 2.)
\end{theorem}
In particular, for the special nonlinearity $f(u)=u^p+u^q$,
Theorem \ref{thm:1} implies that
\begin{corollary}\label{cor:1}
Let $f(u)=u^p+u^q$, and $p>q$.
\begin{enumerate}
\item If $q=1$ and $p < \frac{n+2}{n-2}$, then $\lambda_0=\lambda_1$ and
\eqref{e:1.1} has
no solution for $0< \lambda \le \lambda_{\infty}$ and $\lambda\ge
\lambda_1$,
and has exactly one solution for $ \lambda_*<\lambda<\lambda_1$; (see Figure 1)
\item If $q>1$, $p < \frac{n+2}{n-2} $ and
\begin{equation}\label{a3}
\frac{n(p-1)}{2(q+1)}\le 1,
\end{equation}
then \eqref{e:1.1} has exactly one
solution for $0<\lambda<\infty$. (see Figure 2)
\end{enumerate}
\end{corollary}
Our result is a generalization of previous results by Kwong and Li
\cite{KwL}, Srikanth \cite{Sr}, Yadava \cite{Y}, Zhang \cite{Z1}
where (1) of Corollary \ref{cor:1} was proved by different
methods, and Yadava \cite{Y}, Zhang \cite{Z2} where (2) of
Corollary \ref{cor:1} was proved. All these previous proofs seem
to be complicated and lengthy, and our proof is much simpler than
all of them. On the other hand, Erbe and Tang \cite{ET} prove the
results in Corollary \ref{cor:1} even without \eqref{a3}, but
their result can not imply Theorem \ref{thm:1}, and the methods
are quite different.
Our method also works for the case of $f(u)=u^q+u^p$ with
$p=(n+2)/(n-2)$, see Section 3 for details. In this case, Brezis
and Nirenberg \cite{BN} first showed the existence of a solution.
We use a bifurcation approach similar to that in \cite{OS1} and
\cite{OS2}, and some techniques in \cite{OS2} are also used here.
But the difference is that instead of showing that the degenerate
solution is neutrally stable (Morse index is $0$), we show that
the Morse index of the degenerate solution is very high ($\ge 2$),
thus turning points can not occur in a branch of solutions (which
have Morse index $1$) obtained from the Mountain Pass Lemma. Here
the function $A_f(u)$ introduced in (D3) provides a Pohozaev type
identity, which is the key of the proof. We introduce some
preliminaries in Section 2, and the main results are proved in
Section 3.
\begin{figure}
\begin{center}
\setlength{\unitlength}{1mm}
\begin{picture}(107,45)(-3,-11)
\put(-3,0){\line(1,0){45}}
\put(41.5,-.9){$\to$}
\put(42,-3.5){$\lambda$}
\put(0,-3){\line(0,1){35}}
\put(-.85,32.5){$\uparrow$}
\put(-3.5,33){$u$}
\qbezier(2,35)(2,18)(30,6)
\qbezier(30,6)(38,3)(38,0)
\put(-1,-7){Fig. 1: Bifurcation diagram}
\put(12,-11){for $f'(0)>0$}
%
\put(57,0){\line(1,0){45}}
\put(101.5,-.9){$\to$}
\put(102,-3.5){$\lambda$}
\put(60,-3){\line(0,1){35}}
\put(59.15,32.5){$\uparrow$}
\put(56.5,33){$u$}
\qbezier(62,35)(66,4)(104,2)
\put(59,-7){Fig. 2: Bifurcation diagram}
\put(72,-11){for $f'(0)=0$}
\end{picture}
\end{center}
\end{figure}
\section{Preliminaries}
A framework of using the bifurcation method to prove the exact
multiplicity of solutions of \eqref{e:1.1} was established in
Ouyang and Shi \cite{OS1}, \cite{OS2}. (see also \cite{KLO1},
\cite{KLO2}, \cite{K}.) Here we briefly recall the approach in
\cite{OS2} without the proof since all proofs can be found in
\cite{OS2}. One remarkable result regarding \eqref{e:1.1} was
proved by Gidas, Ni and Nirenberg \cite{GNN} in 1979. They showed
that if $f$ is locally Lipschitz continuous in $[0,\infty)$, then
all positive solutions of \eqref{e:1.1} are radially symmetric.
This result sets the foundation of our analysis of positive
solutions to \eqref{e:1.1}. We summarize some basic facts on
\eqref{e:1.1}.
\begin{lemma} \label{lem:5.1}
\begin{enumerate}
\item If $f$ is locally Lipschitz continuous in $[0,\infty)$,
then all positive solutions of \eqref{e:1.1} are radially
symmetric, and satisfy
\begin{equation}\label{e:6.1}
\begin{gathered}
(r^{n-1} u')' + \lambda r^{n-1}f(u) = 0, \quad r \in (0, 1),\\
u'(0) = u(1) = 0;
\end{gathered}
\end{equation}
\item If $u$ is a positive solution to \eqref{e:1.1}, and $w$ is a solution of
the linearized problem (if it exists):
\begin{equation}\label{e:5.3}
\begin{gathered}
\Delta w + \lambda f'(u)w = 0 \quad \text{in } B^n,\\
w = 0 \quad \text{on } \partial B^n.
\end{gathered}
\end{equation}
then $w$ is also radially symmetric and satisfies
\begin{equation} \label{e:6.2}
\begin{gathered}
(r^{n-1} w')' + \lambda r^{n-1} f'(u)w = 0, \quad r \in (0, 1),\\
w'(0) = w(1) = 0;
\end{gathered}
\end{equation}
\item For any $d>0$, there is at most one $\lambda_d >0$ such that
\eqref{e:1.1} has a
positive solution $u(\cdot)$ with $\lambda =\lambda_d$ and $u(0)=d$.
Let
$T=\{d>0 : \mbox{\eqref{e:1.1} has a positive solution with } u(0)=d \}$,
then $T$ is
open; $\lambda(d) =\lambda_d$ is a well-defined continuous
function from $T$ to $\mathbb{R}^+$.
\end{enumerate}
\end{lemma}
Because
of (3), we call $\mathbb{R}^+ \times \mathbb{R}^+ =\{ (\lambda, d) |
\lambda >0, d>0 \}$
the {\it phase space}, and $\Sigma=\{(\lambda(d),d):d\in T\}$ the
{\it bifurcation diagram}. A solution $(\lambda,u)$ of \eqref{e:1.1}
or \eqref{e:6.1} is a {\it degenerate} solution if \eqref{e:5.3}
or \eqref{e:6.2} has a non-trivial solution. At a degenerate
solution $(\lambda(d),u(d))$, $\lambda'(d)=0$, and it is referred as a
{\it turning point} of $\Sigma$ if $\lambda''(0)\ne 0$. We define the
\textit{Morse index} $M(u)$ of a solution $(\lambda,u)$ to be the
number of negative eigenvalues of the following eigenvalue
problem
\begin{equation} \label{e:41.2}
\begin{gathered}
(r^{n-1} \phi')'+ \lambda f'( u)\phi=-\mu \phi, \quad r\in (0,1) ,\\
\phi'(0)=\phi(1)=0.
\end{gathered}
\end{equation}
It is well-known that the eigenvalues $\mu_1, \mu_2, \dots$ of
\eqref{e:41.2} are all simple, and the eigenfunction $\phi_i$
corresponding to $\mu_i$ has exactly $i-1$ simple zeros in $(0,1)$
for $i \in {\mathbf N}$. We also call a solution $(\lambda,u)$ {\it
stable} if $\mu_1(u)>0$, otherwise it is {\it unstable}.
One of our main tools is the Sturm comparison lemma, which we
include for the sake of completeness.
\begin{lemma} \label{lem:6.1}
Let $Lu(t)=[(p(t)u^{\prime}(t)]^{\prime}+q(t)u(t)$, where $p(t)$ and $q(t)$ are
continuous in $[a,b]$ and $p(t)\ge 0$, $t \in [a,b]$. Suppose $Lw(t)=0$, $w
\not\equiv 0$.
\begin{enumerate}
\item If there exists $v \in C^2[a,b]$ such
that $Lv(t)\cdot v(t)\le (\not \equiv) 0$, then $w$ has at most one zero in
$[a,b]$. If in addition, $w^{\prime}(a)=0$ or $p(a)=0$, then $w$ does not have
any zero in $[a,b]$.
\item If there exists $v \in C^2[a,b]$ such
that $Lv(t)\cdot v(t)\ge (\not \equiv) 0$, and $v(a)=v(b)=0$, then $w$ has at
least one zero in $(a,b)$. If $w^{\prime}(a)=0$ or $p(a)=0$, then $w$ has at
least one zero in $[a,b]$ even if $v(a)\ne 0$.
\end{enumerate}
\end{lemma}
The proof is standard, and we refer to \cite{OS1}. In the
following, we will always use the notation $Lw(r)=(r^{n-1}w')'+\lambda
r^{n-1}f'(u)w$, where $u$ is a solution to \eqref{e:6.1}. We will
say that we apply the \textit{integral procedure} to
two equations: $Lu=g_1(r)$ and $Lv=g_2(r)$, which means we
multiply the first equation by $v$ and multiply the second
equation by $u$, integrate both over $[0,1]$ and subtract, so we
obtain $\int_0^1(vLu-uLv)dr+\int_0^1(vg_1-ug_2)dr=0$. The first
term can be simplified via the integration by parts and boundary
conditions of $u$ and $v$. The following are some calculation
which will be used in the proofs.
\begin{lemma} \label{lem:14.1}
Let $u$ and $w$ be the solutions of
\eqref{e:6.1} and \eqref{e:6.2} respectively, and let
$F(u)=\int_0^u f(t)dt$. Then
\begin{gather} \label{e:14.1}
Lu= \lambda r^{n-1}[f'(u)u-f(u)],\\
\label{e:14.2}
Lw=0,\\
\label{e:14.3}
L(ru_r)= -2\lambda r^{n-1} f(u),\\
\label{e:14.4}
\int_0^1 r^{n-1} f(u) w dr =\int_0^1 r^{n-1} f'(u)u w dr={1 \over 2 \lambda} u_r(1)
w_r(1),\\
\label{e:14.5} \int_0^1 r^{n-1} \big[ nF(u)-\frac{n-2}{2}
f(u)u\big] dr={1 \over 2 \lambda} u_r^2(1),\\
\label{e:14.6}
\int_0^1 r^{n-1} [ 2nF(u)-n f(u)u] dr -\int_0^1
r^{n-1} [f_u(u)u-f(u)] ru_r(r) dr=0.
\end{gather}
\end{lemma}
\paragraph{Proof} \eqref{e:14.1}-\eqref{e:14.3} are by direct calculations.
The first part of \eqref{e:14.4} is obtained by applying integral procedure
to \eqref{e:14.1} and \eqref{e:14.2}, and the second equality in \eqref{e:14.4}
is obtained by
applying the integral procedure
to \eqref{e:14.2} and \eqref{e:14.3}. (see also \cite{OS1}
for a more general identity.) \eqref{e:14.5} is
the well-known Pohozaev's identity, and it is obtained by
integrating $ru_rLu$. Finally, \eqref{e:14.6} is obtained by
applying the integral procedure
to \eqref{e:14.1} and \eqref{e:14.3}, and combining with
\eqref{e:14.5}. \hfill$\square$
\section{Proof of Main Results}
Note that (D1) and (D2) imply that for $u\ge 0$,
\begin{equation}
\label{e:14.9}
f'(u)u-p f(u) \le 0 \quad \text{and}\quad f'(u)u-q f(u) \ge 0.
\end{equation}
\begin{lemma}\label{lem:14.15}
Suppose that $f$ satisfies (D1) and (D2), and $u$ is a
degenerate solution of \eqref{e:6.1}. Let $w$ be a solution of
\eqref{e:6.2}. Then $w$ must change sign in $(0,1)$.
\end{lemma}
\paragraph{Proof}
By \eqref{e:14.4}, we have $\int_0^1 r^{n-1}[f'(u)u-f(u)]wdr=0$.
Since $q\ge 1$ and \eqref{e:14.9}, then $w$ must change sign in
$(0,1)$. \hfill$\square$
The following lemma is the key to our method.
\begin{lemma}\label{lem:14.2}
Suppose that $f$ satisfies (D1), (D2) and (D3), and $u$ is a
degenerate solution of \eqref{e:6.1}. Let $w$ be a solution of
\eqref{e:6.2}. Then $w$ has at least two zeros in $(0,1)$.
\end{lemma}
\paragraph{Proof}
We use a test function $v(r)=w(r)-u(r)$, where $w$ is a solution
of \eqref{e:6.2}. It is easy to see that $Lv=-Lu=-\lambda r^{n-1}
[f'(u)u-f(u)]\le 0$. Note that the solutions of \eqref{e:6.2} is a
one parameter family which can be parameterized by $w_r(1)$, and
we will specify $w_r(1)$ later. By \eqref{e:14.9} and $u>0$, we
have $\int_0^1 r^{n-1} [f'(u)u-p f(u)] u dr <0$. On the other
hand, by \eqref{e:14.4}, $\int_0^1 r^{n-1} [f'(u)u-p f(u)] w dr
=(2\lambda)^{-1}(1-p) u_r(1) w_r(1)$. Since $f(u)>0$, then $u_r(1)<0$
and $w_r(1)\ne 0$. therefore we can choose $w_r(1)$ such that
\begin{equation}\label{e:14.11}
\int_0^1 r^{n-1}
[f'(u)u-p f(u)] u dr =\int_0^1 r^{n-1} [f'(u)u-p f(u)] w dr.
\end{equation}
And by this choice, $w_r(1)<0$. Therefore, using \eqref{e:14.5},
we obtain
\begin{align*} \label{e:14.12}
&{1-p \over 2 \lambda} u_r(1) v_r(1)\\
&= {1-p \over 2 \lambda} u_r(1) w_r(1)-{1-p \over 2 \lambda} u_r^2(1) \\
&=\int_0^1
r^{n-1} [f'(u)u-p f(u)] w dr+(p-1)\int_0^1 r^{n-1}\big[ nF(u)-\frac{n-2}{2}
f(u)u\big]dr\\
&=\int_0^1 r^{n-1}A_f(u)dr >0.
\end{align*}
Thus $v_r(1)>0$. By \eqref{e:14.11}, $\int_0^1 r^{n-1} [f'(u)u-p f(u)] v dr
=0$, and $f'(u)u-p f(u)\le 0$ for $u \ge 0$. Hence $v$ must change sign in
$(0,1)$.
Let $r_1$ be the first zero of $v$ left of $1$. Then $v_r(1)>0$
implies $v(r)<0$ in $(r_1,1)$. Since $Lv \le 0$ in $(0,1)$, then
by Lemma \ref{lem:6.1}, $w$ has at least one zero in $(r_1,1)$.
Let $r_2(>r_1)$ be the first zero of $w$ left of $1$. Then
$w_r(1)<0$ implies $w(r)>0$ in $(r_2,1)$, and $w(r)<0$ in
$(r_2-\delta,r_2)$ for a small $\delta>0$. But
$w(r_1)=v(r_1)+u(r_1)=u(r_1)>0$, so $w$ has another zero in
$(r_1,r_2)$. Therefore $w$ has at least two zeros in $(0,1)$.
\hfill$\square$
\begin{corollary} \label{cor:14.4}
Suppose that $f$ satisfies (D1), (D2) and (D3), and $u$ is a
degenerate solution of \eqref{e:6.1}. Then the Morse index
$M(u)\ge 2$, and $0=\mu_i(u)$ for some $i\ge 3$.
\end{corollary}
\paragraph{Proof}
Since $w$ has at least two zeros in $(0,1)$, then $0=\mu_i(u)$ for
some $i\ge 3$. \hfill$\square$
Note that in the proof of Lemma \ref{lem:14.2}, the condition
$p<(n+2)/(n-2)$ is {\it not} needed. This fact is useful when
discussing the case of critical exponent.
\paragraph{Proof of Theorem \ref{thm:1}}
We first prove the case when $f'(0)>0$. In this case,
$\lambda_0=\lambda_1/f'(0)$ is a bifurcation point where a
bifurcation from the trivial solutions occurs. From a theorem of
Crandall and Rabinowitz \cite{CR1} (or see Theorem 3.1 (2) in
\cite{OS2}), the local structure of the solution set of
\eqref{e:1.1} near $(\lambda,u)=(\lambda_0,0)$ consists of two
parts: $\Sigma_0=\{(\lambda,0):\lambda>0\}$ and
$\Sigma_1=\{(\lambda(s),u(s)):|s|\le \delta\}$, where
$\lambda(0)=\lambda_0$, $u(s)=s\phi_1+o(|s|)$, and $\phi_1$ is the
positive eigenfunction corresponding to $\lambda_1$. Moreover,
from Proposition 3.4 (1) in \cite{OS2}, the bifurcation is
subcritical, so $\lambda'(s)\le 0$ for $s\in [0,\delta]$. On the
other hand, by Theorem 1.16 in \cite{CR2}, $\mu_1(s)\le 0$ where
$\mu_1(s)$ is the principal eigenvalue of \eqref{e:41.2} with
$u=u(s)$. If $\mu_1(s)=0$ for some $s\in (0,\delta)$, then $u(s)$
is a degenerate solution of \eqref{e:1.1}, that contradicts with
Corollary \ref{cor:14.4}. Thus $\mu_1(s)<0$ and $\mu_2(s)>0$ for
$s\in (0,\delta)$ with some small $\delta>0$ by the continuity of
the eigenvalues with respect to $s$. Thus $u(s)$ is a
non-degenerate solution with Morse index $1$, and in that case we
can apply the implicit function theorem to extend $\Sigma_1$
further. Suppose $s_0=sup \{s>0: \mu_1(s)<0$ and $\mu_2(s)>0\}$.
If $s_0<\infty$, then at $s=s_0$, $u(s)$ is still well-defined,
which is the solution of initial value problem
$(r^{n-1}u')'+\lambda(s_0)r^{n-1}f(u)=0$, $u'(0)=0$ and
$u(0)=s_0$. So either $\mu_1(s_0)=0$ or $\mu_2(s_0)=0$ by the
continuity, and the Morse index of $u(s_0)$ is either $0$ or $1$,
which again reaches a contradiction with Corollary \ref{cor:14.4}.
Therefore $s_0=\infty$, and $\lambda'(s)<0$ for all $s>0$. When
$f'(\infty)<\infty$, then $\lim_{s\to
\infty}\lambda(s)=\lambda_{\infty}$. When $f'(\infty)=\infty$,
then $\lim_{s\to \infty}\lambda(s)=0$. (see \cite{S2} for the
proofs).
Next we prove the case of $f'(0)=0$. In this case, the proof is
similar as long as we can show that for some $(\lambda,s)$ there
exists a solution \eqref{e:1.1} such that $u(s, 0)=s$,
$\mu_1(s)<0$ and $\mu_2(s)>0$. This can be obtained by the
well-known Mountain Pass Lemma. We verify that Theorem 2.15 in
Rabinowitz \cite{R} can be applied here. (For the convenience of
the readers, we include the statement of the theorem after the
proof.) Let $p(x,\xi)=f(\xi)$, and we would relate conditions
(D1-D3) to (p1-p4) in Theorem \ref{thm:14.5}. Obviously, (D1)
implies (p1) and we can assume (p3) since we only consider the
case of $f'(0)=0$. Also if $p<(n+2)/(n-2)$ in (D2), then (p2) is
true, since
\begin{equation}
\label{e:14.27} \Big[\frac{f(u)}{u^p}\Big]'=\frac{f'(u)u-p
f(u)}{u^{p+1}}\le 0,
\end{equation}
for all $u\ge 0$. Finally, we notice that in (D2), if $q>1$, then
(p4) is also satisfied. So if $q>1$, from the result of Rabinowitz
(see Theorem \ref{thm:14.5} below), for each $\lambda>0$,
\eqref{e:1.1} has a positive solution $u$. If $q=1$, we notice
that in the proof of the result of Rabinowitz, (p4) is only used
in proving that there is a function $u$ such that $I(u)=\int_{B^n}
[(1/2)|\nabla u|^2-\lambda P(u)]dx \le 0$, but that can also be
achieved if we let $\lambda$ be sufficiently large when $q=1$. So in
the case of $q=1$, \eqref{e:1.1} has a positive solution $u$ for
sufficiently large $\lambda$. (Indeed \eqref{e:1.1} may not have a
solution if $f'(\infty)<\infty$).
Thus in any case of $f'(0)=0$, we obtain a solution $(\lambda,u)$ of
\eqref{e:1.1} by the Mountain Pass Lemma. On the other hand, from
Theorem 1.6 and Corollary 3.1 in Chapter II of Chang \cite{Ch},
the Morse index of $(\lambda,u)$ is $1$ if it is non-degenerate, and
is $0$ if it is degenerate. But from Corollary \ref{cor:14.4}, the
latter case can not happen, so $(\lambda,u)$ must satisfy $\mu_1(s)<0$
and $\mu_2(s)>0$. Thus the continuation arguments in the proof of
the case $f'(0)>0$ can also be carried over to here. Finally, from
Proposition 6.6 in \cite{OS2}, since $p<(n+2)/(n-2)$, the domain
of the function $\lambda(s)$ should be all $(0,\infty)$, and
$\lim_{s\to 0^+} \lambda(s)=\infty$. Similar to the case of $f'(0)>0$,
when $f'(\infty)<\infty$, then $\lim_{s\to
\infty}\lambda(s)=\lambda_{\infty}$. When $f'(\infty)=\infty$, then
$\lim_{s\to \infty}\lambda(s)=0$. (see \cite{S2} for the proofs).
\hfill$\square$
The following is Theorem 2.15 and Corollary 2.23 in Rabinowitz
\cite{R}.
Let $\Omega$ be a bounded smooth domain in $\mathbb{R}^n$.
Consider the equation
\begin{equation} \label{e:14.26}
\begin{gathered}
\Delta u + \lambda p(x, u) = 0 \quad \mbox{in } \Omega ,\\
u=0 \quad \mbox{on } \partial\Omega.
\end{gathered}
\end{equation}
Assume that
\begin{enumerate}
\item[(p1)] $p(x,\xi) $ is locally Lipschitz continuous in $\overline\Omega \times
\mathbb{R}$,
\item[(p2)] there exists $a_1, a_2\ge 0$, such that $|p(x,\xi)|\le
a_1+a_2|\xi|^s$, where $0\le s <(n+2)/(n-2)$ if $n>2$,
\item[(p3)] $p(x,\xi)=o(|\xi|)$ as $\xi \to 0$, and
\item[(p4)] there exists constants $\mu>2$ and $r\ge 0$ such that for
$|\xi|\ge r$, $0<\mu P(x,\xi)\le \xi p(x,\xi)$.
\end{enumerate}
\begin{theorem} \label{thm:14.5} Under assumptions (p1)--(p4),
equation \eqref{e:14.26} possesses a positive classical solution.
\end{theorem}
Finally we discuss the critical exponent case. In fact, Lemma
\ref{lem:14.2} is even true when $p>(n+2)/(n-2)$, but in that case
the existence of the solution is not clear in general. When
$p=(n+2)/(n-2)$ in (D2), and $f(u)=u^p+u^q$, (D3) is also
satisfied if \eqref{a3} is also satisfied. So again if we can show
the existence of a solution with Morse index $1$, then the
uniqueness part is implied by Lemma \ref{lem:14.2} and the
continuity argument in the proof of Theorem \ref{thm:1}. In the
case of $q=1$, this can be done by the bifurcation result which we
used in the proof of Theorem \ref{thm:1}, but $\lim_{s\to \infty}
\lambda(s)$ may not be $0$ as shown in \cite{BN}. In the case of
$q>1$, \eqref{a3} can only be satisfied for $n\ge 4$, and in that
case, it is proved by Brezis and Nirenberg that \eqref{e:1.1}
always has a positive solution via a modified Mountain Pass Lemma,
so we can still prove that the Morse index of the solution is $1$
in that case. So summarizing these discussion, we have
\begin{theorem} Consider
\begin{equation} \label{e:11.1}
\begin{gathered}
\Delta u+ \lambda (u^p+u^q) = 0 \quad \mbox{in } B^n ,\\
u>0 \quad \mbox{in } B^n,\\
u=0 \quad \mbox{on } \partial B^n,
\end{gathered}
\end{equation}
where $p=(n+2)/(n-2)$. Then
\begin{enumerate}
\item If $q=1$, then \eqref{e:11.1} has no solution for $0< \lambda \le
\lambda_{*}$ and $\lambda\ge \lambda_0$,
and has exactly one solution for $ \lambda_{*}<\lambda<\lambda_0$, where
$\lambda_*=0$ when $n\ge 4$ and $\lambda_*=\lambda_1/4$ when $n=3$;
\item If $q>1$, $q$ satisfies \eqref{a3} and $n\ge 4$, then \eqref{e:11.1} has no
solution for $\lambda\ge \lambda_0$,
and has exactly one solution for $ 0<\lambda<\lambda_0$.
\end{enumerate}
\end{theorem}
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\noindent\textsc{Junping Shi}\\
Department of Mathematics, College of William and Mary, \\
Williamsburg, VA 23187, USA, and \\
Department of Mathematics, Harbin Normal University, \\
Harbin, Heilongjiang, China \\
e-mail: shij@math.wm.edu
\end{document}