\documentclass[twoside]{article} \usepackage{amsfonts, amsmath, epsf} \pagestyle{myheadings} \markboth{ Construction of solutions for a nonlinear Dirichlet problem} { Horacio Arango \& Jorge Cossio } \begin{document} \title{\vspace{-1in}\parbox{\linewidth}{\footnotesize\noindent Nonlinear Differential Equations, \newline Electron. J. Diff. Eqns., Conf. 05, 2000, pp. 1--12\newline http://ejde.math.swt.edu or http://ejde.math.unt.edu \newline ftp ejde.math.swt.edu or ejde.math.unt.edu (login: ftp)} \vspace{\bigskipamount} \\ % Explicit construction, uniqueness, and bifurcation curves of solutions for a \\ nonlinear Dirichlet problem in a ball % \thanks{ {\em Mathematics Subject Classifications:} 35B32, 35J60, 65D07, 65N99. \hfil\break\indent {\em Key words:} Nonlinear Dirichlet problem, radially symmetric solutions, bifurcation, \hfil\break\indent explicit solutions, spline. \hfil\break\indent \copyright 2000 Southwest Texas State University. \hfil\break\indent Published Ocotber 24, 2000. \hfil\break\indent Partially supported by Colciencias-BID grant 381-97.\hfil\break\indent Part of this research was done while the second author was visiting The University \hfil\break\indent of Texas at San Antonio } } \date{} \author{ Horacio Arango \& Jorge Cossio \\[12pt] {\em Dedicated to Alan Lazer} \\ {\em on his 60th birthday }} \maketitle \begin{abstract} This paper presents a method for the explicit construction of radially symmetric solutions to the semilinear elliptic problem $$\displaylines{ \Delta v + f(v) = 0 \quad \text{in }B\cr v = 0 \quad \text{on }\partial B\,, }$$ where $B$ is a ball in ${\mathbb R}^N$ and $f$ is a continuous piecewise linear function. Our construction method is inspired on a result by E. Deumens and H. Warchall [8], and uses spline of Bessel's functions. We prove uniqueness of solutions for this problem, with a given number of nodal regions and different sign at the origin. In addition, we give a bifurcation diagram when $f$ is multiplied by a parameter. \end{abstract} \newtheorem{theorem}{Theorem}[section] \newtheorem{lemma}[theorem]{Lemma} \section{Introduction} The purpose of this paper is to explicitly construct radially symmetric solutions $v: B \to {\mathbb R}$ to the nonlinear Dirichlet problem $$ \gathered \Delta v + f(v) = 0 \quad \text{in }B\\ v = 0 \quad \text{on }\partial B, \endgathered \eqno{(1.1)} $$ where $B$ is the ball in ${\mathbb R}^N$ centered at the origin with radius $\pi$, $\Delta$ is the Laplacian operator, and $f: {\mathbb R} \to {\mathbb R}$ is a continous piecewise-linear function such that $f(0)=0$, $f$ has a positive zero, and $f'(0) = f'(\infty)$. We construct solutions to (1.1) with a given number of zeros in their radial profiles. Our method provides an explicit calculation rather than the existence result presented in \cite{c1,c2,c3,c4,e1,g1,k1,l1}. Our constructions further develops the authors' work in \cite{a2} and the paper by E. Deumens and H. Warchall \cite{d1}. Let $\lambda_1<\lambda_2<\cdots<\lambda_k<\cdots$ be the eigenvalues of $-\Delta$ acting on radial functions of $H^1_0(B)$ (see \cite{a1}) and $\{\varphi_1,\varphi_2,\cdots,\varphi_k,\cdots\}$ be the corresponding complete set of eigenfunctions. Let $\lambda_{j+1} > \alpha^2 > \lambda_j$, $\beta >0$, and $$f(t)=\begin{cases} \alpha^2\, t &\text{if $t\le\frac{\beta}2$},\\ -\alpha^2\,t + \alpha^2\,\beta &\text{if $\frac{\beta}2 \le t \le \beta$},\\ \alpha^2\,t - \alpha^2\,\beta &\text{if $t \ge \beta$}. \end{cases} \eqno{(1.2)} $$ In Section 2, we shall construct radially symmetric solutions to (1.1) with the above nonlinear function $f$. We recall that the radial solutions to (1.1) are the solutions to the ordinary differential equation $$\gathered v'' + \frac {N-1}r v' + f(v) = 0 \quad (0 \beta >0$ and $0 < u_i(0) < \beta$. \end{theorem} In Section 4, we obtain a description of the graph of the set of radial solutions to $$\gathered \Delta v + \lambda \, f(v) = 0 \quad \text{in }B\\ v = 0 \quad \text{on }\partial B, \endgathered \eqno{(1.4)} $$ where $\lambda \in {\mathbb R}$ is a parameter (see Figures 4 and 5). Figures 6, 7, and 8 were generated with software, written by the authors, following the method of construction given in Section 2. \section{Explicit construction of radially symmetric solutions} In each $r$-interval where $v(r)$ lies between $-\infty$ and $\frac{\beta}2$, or between $\frac{\beta}2$ and $\beta$, or between $\beta$ and $+\infty$, the equation (1.3) has the form $$ v'' + \frac {N-1}r \, v' + K_1\,v + K_2 = 0, \eqno{(2.1)} $$ with $K_1$ and $K_2$ constants depending only on $f'(0)= \alpha^2$ and $\beta$. The solution to this equation is $$N\ge2:\quad \quad v(r)= A r^{-\nu} J_{\nu}(kr) + B r^{-\nu} N_{\nu}(kr) - \frac {K_2}{K_1}, $$ where $k^2 = K_1$, $\nu = \frac{N-2}{2}$, and $J_{\nu}$ and $N_{\nu}$ are the Bessel and Neumann functions (see \cite{a2}). To build solutions to (1.3) we put together several of the above pieces, subject to continuity conditions for $v$ and its first two derivatives, and subject to the boundary conditions $v'(0) = 0$ and $v(\pi) = 0$. For the sake of clarity and easy of manipulations, we henceforth deal with the three-dimensional case. We discuss the construction of a solution $v$ to problem (1.3) under assumption (1.2) with $i$ nodes in $(0,\pi)$ $(0\le i \le {j-1})$ and $v(0)=d > \beta$. The construction of a solution $u$ with $i$ nodes in $(0,\pi)$ $(0\le i \le {j-1})$ and $0< u(0)=d < \beta$ follows a similar pattern. \begin{figure} \begin{center} \epsffile{fig1.eps} \caption{Radial profile of a solution of (1.3) with 2 nodes} \end{center} \end{figure} For $0\le r \le p$ we take $v(r) \ge \beta$. Thus $f(v) = \alpha^2\,v - \alpha^2\,\beta$, and the solution to (1.3) is $$v_1(r) = \beta + \frac{p_1}{r}\, \sin{\alpha\,(r-P_1)}.$$ For $p\le r \le q$, $\frac{\beta}2 \le v(r) \le \beta$. Thus $f(v) = -\alpha^2\,v + \alpha^2 \,\beta$, and the solution is $$v_2(r) = \beta + \frac{p_2}{r}\, \sinh{\alpha\,(r-P_2)}.$$ For $q\le r \le \pi$, \, $0\le v(r) \le \frac{\beta}2$. Thus $f(v)= \alpha^2\,v$, and the solution is $$v_3(r) = \frac{p_3}{r}\, \sin{\alpha\,(r-P_3)}.$$ This ansatz specifies the solution in terms of 3 coefficients $p_1, p_2, p_3$ and 2 welding points $p$ and $q$, and 3 unknowns $P_1, P_2, P_3$. These 8 unknowns are to be found from the equations stating that $v, v'$, and $v''$ are continuous at the 2 welding points and the boundary conditions. The weld point $p$ and the 3 unknowns $P_1, P_2,$ and $P_3$ are determined by the conditions $v_1'(0)=0, v_1(p) = v_2(p) = \beta$, and $v_3(\pi) = 0$, and we find $$ p = \frac{\pi}{\alpha}, \quad P_1=0, \quad P_2= \frac{\pi}{\alpha}, \quad P_3 = \frac{\pi}{\alpha}\,(\alpha - k), $$ where $k\in \mathbb Z -\{0\}$. \smallskip \noindent{\bf Remark 1:} Note that all solutions of (1.3) with $v(0)>\beta$ satisfy $v(\frac{\pi}{\alpha}) =\beta$. \smallskip Since $v_3(r)$ has $(k-1)$ nodes in $(\frac{\pi}{\alpha}(\alpha -k), \pi)$, in order to construct a solution with $i$ nodes in $(0,\pi)$ to problem (1.3) we take $k= i+1$. Let $z= \alpha\,q$. Since $v_2'(q) = v_3'(q)$ it follows that $z$ must be a solution of the equation $$ g(z):=\frac{z}{\tan (z-\pi\,\alpha)} + \frac{z}{\tanh(z-\pi)} - 2 =0. \eqno{(2.2)} $$ Equation (2.2) has a unique solution $z$ over the interval $(\pi\,(\alpha-k), \break \pi\,(\alpha-k+1))$ (see Figure 2), which can be found by using Newton's method with initial condition $z_0 \in (\pi\,(\alpha-k), \pi\,(\alpha-k+1))$ and $z_0 \simeq \pi\,(\alpha-k+1)$. Using the solution $z$ we get the weld point $q=\frac{z}{\alpha}$. \begin{figure} \begin{center} \epsffile{fig2.eps} \caption{Solutions to (2.2) with $\alpha=4.9$} \end{center} \end{figure} The remaining continuity conditions yield $$ p_1=-p_2 = \frac{\beta\, z}{2\alpha\,\sinh (z-\pi)}$$ and $$ p_3 = \frac{\beta\, z}{2\alpha\,\sin (z-\pi(\alpha -i-1))}.$$ Since $\lim_{r\to 0^+} v_1(r) =d$, it follows that $$ d= \beta + \frac{\beta\,z}{2\sinh(z-\pi)} \quad (z>\pi). \eqno{(2.3)} $$ Thus, we have constructed a solution with $i$ nodes in $(0,\pi)$ and initial condition $d= v(0)>\beta$. \smallskip \noindent {\bf Remark 2:} For each positive integer $m$ with $1\le m\le j$, let $\alpha_m = \alpha-j+m$. Since $j<\alpha<{j+1}$, it follows that $$ m<\alpha_m<{m+1}.$$ Therefore, using our method of construction we can obtain solutions with $i$ nodes in $(0,\pi)$ $(0\le i \le {m-1})$ to (1.3) with nonlinearity $f$ given by (1.2) with $\alpha = \alpha_m$. Let us call $d_{mi}$ the initial data corresponding to this solution, which can be found by using (2.3). \smallskip Let $l$ be a positive integer less than or equal to $i$. Since $$ (\pi\,(\alpha_m - (i+1)), \pi\,(\alpha_m -i)) = (\pi\,(\alpha_m - l- (i-l+1)), \pi\,(\alpha_m -l - (i-l))), $$ we see that finding a solution of (2.3) on $ (\pi\,(\alpha_m - (i+1)), \pi\,(\alpha_m -i))$ it is equivalent to find a solution of (2.3) over the interval $ (\pi\,(\alpha_m - l- (i-l+1)),\break \pi\,(\alpha_m -l - (i-l)))$. Therefore, $$ d_{mi} = d_{(m-l)(i-l)}, \quad (1\le m\le j,\,\, 0\le i\le {m-1},\,\,0\le l\le i).$$ We summarize the above discussion in Table \ref{tbl1} which will be useful for constructing bifurcation diagrams in Section 4. \begin{table}[ht]\label{tbl1}\begin{center} \begin{tabular}{|c|c|c|c|c|c|} \hline $\alpha\;\backslash$ nodes & 0 & 1 & 2 & \dots & $m$-1 \\ \hline $1<\alpha_1<2$ & $d_{10}$ & & & & \\ \hline $2<\alpha_2<3$ & $d_{20}$ & $d_{21}=d_{10}$ & & & \\ \hline $3<\alpha_3<4$ & $d_{30}$ & $d_{31}=d_{20}$ & $d_{32}=d_{10}$ & & \\ \hline \vdots & \vdots & \vdots & \vdots & & \\ \hline %$m<\alpha_m$ & & & & & \\ %$ \beta$. \begin{figure} \begin{center} \epsffile{fig3.eps} \caption{Radial profile of a solution $v(r)$ to problem (1.3) } \end{center} \end{figure} As we mentioned in Remark 1, solutions to (1.3) satisfy the equation \break $v(p)= v(\frac{\pi}{\alpha}) = \beta$. Next we derive a basic lemma about the solutions of (1.3). \begin{lemma} \label{lemma3.1} Let $v_1$ and $v_2$ be two solutions of (1.3) such that $v_1(q) = v_2(q)$. Then $$ v_1 = v_2 \quad \text{on } [p, q]\,. $$ \end{lemma} \noindent{\bf Proof.} Let $$ w(r) = v_1(r) - v_2(r), \quad r\in [p, q]. $$ Because $v_1$ and $v_2$ are solutions of (1.3), $w$ satisfies $$\gathered w'' + \frac {2}r w' + f(v_1)-f(v_2) = 0 \quad p\le r \le q\\ w(p) = w(q) = 0\,. \endgathered $$ Using the Mean Value Theorem, we see that there exists $\xi$ such that $$ w'' + \frac {2}r \, w' + f'(\xi)\,w(r) = 0 \quad r\in [p, q]. \eqno{(3.1)} $$ We multiply (3.1) by $r^2$. This yields $$ (r^2\,w')' + r^2\,f'(\xi)\,w = 0, \quad r\in [p, q]. $$ Now we multiply by $w$ and integrate by parts over $[p, q]$, we obtain $$ -\int_{p}^q r^2\,(w')^2 + \int_{p}^q r^2\,f'(\xi)\,w^2 = 0\,. \eqno{(3.2)} $$ To prove the lemma we proceed by contradiction. Suppose $w \ne 0$ on $ [p, q]$. Since $r\in (p,q)$ we know that $v,\xi\in(\frac{\beta}2,\beta)$ so that $f'(\xi) < 0$ on $ [p, q]$, we see that $$ -\int_{p}^q r^2\,(w')^2 +\int_{p}^q r^2\,f'(\xi)\,w^2 < 0\,. \eqno{(3.3)} $$ This contradicts (3.2). The contradiction shows that $w \equiv 0$ on $[p, q].$ The proof of the lemma follows. \hfill \vskip 18 true pt \noindent{\bf Proof of Theorem \ref{thmA}.} Let $v_1$ and $v_2$ be solutions to (1.3), with $v_1(0)=d_1$ and $v_2(0)=d_2$. Since $v_1(p) = v_2(p) = \beta$, by uniqueness of the initial value problem for ordinary differential equations applied to (1.3) on $[0, p]$, we see that $$ d_1 \ne d_2 \Longrightarrow v_1'(p) \ne v_2'(p). $$ Using Lemma \ref{lemma3.1} we obtain $$ v_1'(p) \ne v_2'(p) \Longrightarrow v_1(q) \ne v_2(q). $$ Finally, using again the uniqueness of the initial value problem for ordinary differential equations, we obtain $$ v_1(q) \ne v_2(q) \Longrightarrow v_1(\pi) \ne v_2(\pi). $$ Therefore, if $d_1 \ne d_2$ we infer that $$ v_1(\pi) \ne v_2(\pi), $$ which is a contradiction because $v_1(\pi)= 0 = v_2(\pi)$. Hence $d_1 = d_2$. This proves uniqueness of solutions to (1.3). Thus, we have proved Theorem \ref{thmA}. \hfill \section{Construction of bifurcation curves and graphs of solutions} In this section we give a description of the graph of the set of radial solutions to $$\gathered \Delta v + \lambda \, f(v) = 0 \quad \text{in }B\\ v = 0 \quad \text{on }\partial B, \endgathered \eqno{(4.1)} $$ where $\lambda \in {\mathbb R}^+$ is a parameter. Let $\lambda \in {\mathbb R}^+$, $m\in {\mathbb N}$ be such that $m <\lambda\,\alpha < {m+1}$, and $i=0,1,\cdots,m-1$. Now, as we have seen in Section 2, we can find a unique solution $z= z(\lambda)$ to the equation $$ \frac{z}{\tan (z-\pi\,(\lambda\,\alpha))} + \frac{z}{\tanh(z-\pi)} - 2 =0, \quad\text{on }(\pi\,(\lambda\,\alpha - (i+1)), \pi\,(\lambda\,\alpha -i)).$$ With this solution and (2.3) we find the initial data $d_{mi}>\beta$ corresponding to the solution with $i$ nodes in $(0,\pi)$. Since $$ d= \beta + \frac{\beta\,z}{2\sinh(z-\pi)} \quad (z>\pi), $$ we see that $$\gathered d'(z) < 0 \quad (z>\pi),\\ \lim_{z\to \infty} d(z) =\beta, \quad \text{and}\\ d'(\lambda) <0. \endgathered $$ The sequence $\{d_{mi}\}_{i=m-1}^0 =\{d_{j0}\}_{j=1}^m$ is decreasing. Thus, using Table 1 and the previous information, we obtain the following bifurcation diagram \begin{figure} \begin{center} \epsffile{fig4.eps} \caption{Bifurcation diagram for (4.1) with initial data $d_{mi} >\beta$ } \end{center} \end{figure} Similarly, we can construct the bifurcation diagram for solutions with initial data $0< d_{mi}<\beta$ (see Figure 5). In this case, since $$ d= \beta - \frac{\beta\,z}{2\sinh(z)} \quad (z>0), $$ we see that $$\gathered d'(z) > 0 \quad (z>0),\\ \lim_{z\to \infty} d(z) =\beta, \quad \text{and}\\ d'(\lambda) >0\,. \endgathered $$ The sequence $\{d_{mi}\}_{i=m-1}^0 =\{d_{j0}\}_{j=1}^m$ is increasing. \begin{figure} \begin{center} \epsffile{fig5.eps} \caption{Bifurcation diagram for (4.1) with initial data $0< d_{mi}<\beta$ } \end{center} \end{figure} Figures 6-8 of radially symmetric solutions to problem (1.1) were generated with software, written by the authors, following the method of construction given in Section 2. \begin{figure} \begin{center} \epsffile{fig6.eps} \caption{Radial solution in three dimensions with $\alpha=5.1$, $\beta=2.0$, and $i=4$} \end{center} \end{figure} \begin{figure} \begin{center} \epsffile{fig7.eps} \caption{Radial profile of the solution with $\alpha=8.9$, $\beta =3.0 $, and $i=7$} \end{center} \end{figure} \begin{figure} \begin{center} \epsffile{fig8.eps} \caption{Radial profile of the solution with $\alpha=40.3$, $\beta =3.0 $, and $i=25$} \end{center} \end{figure} \paragraph{\bf Acknowledgment.} The authors want to express their gratitude to Professor Alfonso Castro for his comments about Theorem \ref{thmA}. \begin{thebibliography}{00} {\frenchspacing \bibitem{a1} R. 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Derrick, {\em Positive and oscilatory radial solutions of semilinear elliptic equations}, to appear in J. Applied Math. Stochastic Analysis. \bibitem{d1} E. Deumens and H. Warchall, {\em Explicit construction of all spherically symmetric solitary waves for a nonlinear wave equation in multiple dimensions}, Nonlinear Analysis, Theory, Methods and Applications, Vol. 12 (1988), No. 4, pp. 419--447. \bibitem{e1} M. Esteban, {\em Multiple solutions of semilinear elliptic problems in a ball}, J. Differential Equations, Vol. 57 (1985), pp. 112--137. \bibitem{g1} M. Grillakis {\em Existence of nodal solutions of semilinear equations in ${\mathbb R}^N$}, J. Differential Equations, Vol. 85 (1990), pp. 367--400. \bibitem{k1} S. Kichenassamy and J. Smoller, {\em On the existence of radial solutions of quasi-li\-near elliptic equations}, Nonlinearity, Vol. 3 (1990), pp. 677--694. \bibitem{l1} P. L. Lions, {\em On the existence of positive solutions in semilinear elliptic equations}, SIAM Review, Vol. 24 (1982), pp. 441-467. }\end{thebibliography} \medskip \noindent{\sc Horacio Arango} (e-mail: harango@perseus.unalmed.edu.co)\\ {\sc Jorge Cossio } (e-mail: jcossio@perseus.unalmed.edu.co)\\ Departamento de Matem\'aticas \\ Universidad Nacional de Colombia \\ Apartado A\'ereo 3840 \\ Medell\'\i n, Colombia \enddocument