\documentclass[twoside]{article} \usepackage{amsfonts, amsmath} % used for R in Real numbers \pagestyle{myheadings} \markboth{\hfil Exact multiplicity of a superlinear problem \hfil EJDE/Conf/10} {EJDE/Conf/10 \hfil Junping Shi \hfil} \begin{document} \setcounter{page}{257} \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)} \vspace{\bigskipamount} \\ % 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} \begin{thebibliography}{00} \frenchspacing \bibitem{BN} Brezis, Haim; Nirenberg, Louis, \textit{Positive solutions of nonlinear elliptic equations involving critical Sobolev exponents}. Comm. Pure Appl. 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