\documentclass[twoside]{article} \pagestyle{myheadings} \markboth{ Infinitely many solutions at a resonance } { Philip Korman \& Yi Li } \begin{document} \setcounter{page}{105} \title{\vspace{-1in}\parbox{\linewidth}{\footnotesize\noindent Nonlinear Differential Equations, \newline Electron. J. Diff. Eqns., Conf. 05, 2000, pp. 105--111\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} \\ % Infinitely many solutions at a resonance % \thanks{ {\em Mathematics Subject Classifications:} 34B15. \hfil\break\indent {\em Key words:} Bifurcation of solutions, the global solution curve. \hfil\break\indent \copyright 2000 Southwest Texas State University. \hfil\break\indent Published October 25, 2000. \hfil\break\indent Partially supported by a Taft Faculty Grant at the University of Cincinnati } } \date{} \author{ Philip Korman \& Yi Li \$12pt] {\em Dedicated to Alan Lazer} \\ {\em on his 60th birthday }} \maketitle \begin{abstract} We use bifurcation theory to show the existence of infinitely many solutions at the first eigenvalue for a class of Dirichlet problems in one dimension. \end{abstract} \newtheorem{thm}{Theorem} \newtheorem{lma}{Lemma} It has been observed that complexity of the solution curve for the boundary value problem \label{1} u'' +\lambda f(u)=0 \quad \mbox{for 00 small, and for \lambda large. We did not attempt to get the optimal bounds. \begin{lma} If the problem (\ref{2}) has a positive solution, then $$\label{3} \frac{\lambda _1}{2} <\lambda < \frac{\pi}{\pi-1}\lambda _1.$$ \end{lma} \paragraph{Proof:} Set f(u)=u+\sin u. Observe that 0< f(u) < 2u for all u > 0. Multiplying the equation (\ref{2}) by u, integrating by parts, and using the Poincare inequality \[ 2\lambda \int_0^L u^2 \, dx > \lambda \int_0^L f(u)u \, dx = \int_0^L {u'}^2 \, dx \geq \lambda _1 \int_0^L u^2 \, dx,$ from which the left inequality in (\ref{3}) follows. \smallskip On the other hand, multiplying the equation (\ref{3}) by $\phi _1 =\sin{\frac{\pi}{L}x}$, and integrating over $(0,L)$, we obtain (with $\lambda _1 = \frac{\pi^2}{L^2}$) $$\label{4} \int_0^L \left(-\lambda _1u+\lambda u+\lambda \sin u \right) \phi _1 \, dx=0.$$ Since $\phi _1 >0$, we obtain a contradiction in (\ref{4}), provided that $g(u) \equiv \frac{(\lambda -\lambda _1)}{\lambda } u+ \sin u >0$ for all $u>0$. This will certainly happen if $\lambda >\lambda _1$, and $g(\pi) \geq 1$, which leads to the right inequality in (\ref{3}). \paragraph{Proof of the Theorem 1.} It is well-known that there is a curve of positive solutions of (\ref{2}) bifurcating off the trivial one (to the right) at $\frac{\lambda _1}{2}$. Similarly there a curve of positive solutions bifurcating from infinity at $\lambda =\lambda _1$, see e.g. p. 673 in E. Zeidler \cite{Z}. We claim that both branches link up, giving a unique solution curve. Indeed, let us start with the branch bifurcating from infinity. This branch extends globally, since at each of its points either the implicit function theorem or a bifurcation theorem of Crandall-Rabinowitz applies, see \cite{KLO} for details. By Lemma 1 this branch is constrained to a strip $\frac{\lambda _1}{2}<\lambda <\frac{\pi}{\pi-1}\lambda _1$, and so it has to go to zero at $\lambda _1$. By uniqueness of the bifurcating solution, this branch has to link up with the lower one. \smallskip We show next that the solution curve changes its direction infinitely many times. Define $F(u)=\int_0^u f(t) \, dt$, $G(u)=uf(u)-2F(u)$. Notice that $G'(u)=uf'(u)-f(u)$. Differentiating the problem (\ref{1}) with respect to $\lambda$, we get \label{5} u''_{\lambda} +\lambda f'(u)u_{\lambda }+f(u)=0 \quad \mbox{for $00, & \mbox{if$u(\frac{L}{2})=n\pi$, and$n$is odd} \$2pt] <0 & \mbox{if u(\frac{L}{2})=n\pi, and n is even.} \end{array} \right.$ Hence$J(s)$changes sign infinitely many times. It follows that$J(s)$has infinitely many points of negative minimum, where by (\ref{8})$I'(s)$is positive, and infinitely many points of positive maximum where$I'(s)$is negative. It follows that$I'(s)$changes sign infinitely many times, and so$I(s)$is not monotone, and hence the solution curve changes its direction infinitely many times. \smallskip It remains to show that the solution curve intersects the line$\lambda =\lambda _1$infinitely many times. From (\ref{4}) we conclude $$\label{9} (\lambda _1 - \lambda ) \int_0^L u\phi _1 \, dx= \lambda \int_0^L \sin u \; \phi _1 \, dx = \lambda \int_0^L \sin s\phi \; \phi _1 \, dx.$$ It follows from a standard analysis of bifurcation from infinity that$\phi(x)=\phi _1 (x) +o(1)$for$s$large. We then have $$\label{10} \int_0^L \sin s\phi \; \phi _1 \, dx=J(s)+o(1)\int_0^L \sin (s\phi) \, dx.$$ We now recall an asymptotic formula from sec. 40 in Y.V. Sidorov et al \cite{S} (see also Corollary 1 in sec. 39). \begin{lma} Assume that the functions$f(x)$and$g(x)$are infinitely differentiable on the interval$[0,1]$, and assume that $g(x)0 for all u>0. Then $$\label{13} \int _0^{x_n} f(u)g(u) \, du >0, \quad \int _0^{y_n} f(u)g(u) \, du <0.$$ \end{lma} \paragraph{Proof:} Using (\ref{11}) and (\ref{12}) we have \begin{eqnarray*} & \int _0^{x_n} f(u)g(u) \, du =\int _0^{x_n} g(u) d \left[F(u)-F(x_n) \right] \\ & = g(0)F(x_n)- \int _0^{x_n} g'(u) \left[F(u)-F(x_n) \right] \, du >0, \end{eqnarray*} and the second inequality in (\ref{13}) is proved similarly. \hfill\diamondsuit \begin{thm} Consider the problem \label{14} u'' +\lambda \left(u+f(u)\right)=0 \quad \mbox{for 0-1, $$\label{15a} u+f(u)>0 \quad \mbox{for all u>0},$$ $$\label{16} \frac{f(u)}{u} \to 0 \quad \mbox{as u \to \infty}.$$ Assume finally that either f(u) or uf(u) satisfies Schaaf-Schmitt condition. Then the problem (\ref{14}) has a C^1 curve of positive solutions bifurcating off the trivial solution at \lambda =\frac{\lambda _1}{1+f'(0)}. This curve makes infinitely many turns, and tends to infinity at \lambda =\lambda _1. This curve exhausts the set of positive solutions of (\ref{14}). \end{thm} \paragraph{Proof:} Similar to Theorem 1. Examining our derivation of (\ref{*}), one sees that this time \[ J(s)=\frac{2}{s} \int _0^{u(\frac{L}{2})} f(u) \frac{dx}{du}u \, du= \frac{2}{s} \int _0^s f(u) \frac{dx}{du}u \, du.$ Since by (\ref{15a}) the function$\frac{dx}{du}$is increasing, applying Lemma 3 we see that$J(s)$changes sign infinitely many times, and hence the solution curve changes direction infinitely many times. Clearly, an analog of Lemma 1 holds, so that the curves bifurcating from zero and infinity have to link up. \hfill$\diamondsuit$\paragraph{Remarks.} \begin{enumerate} \item One can have several variations of the Theorem 2. For example, we could drop the condition (\ref{15}), and have a curve bifurcating from infinity, and making infinitely many turns. Alternatively, we could drop the condition (\ref{16}) and obtain a curve bifurcating from zero, and making infinitely many turns. \item It follows from \cite{SS} that the problem (\ref{14}) has infinitely many solutions at$\lambda =\lambda _1$if$f(u)$satisfies Schaaf-Schmitt condition. Clearly, it may happen that$uf(u)$satisfies Schaaf-Schmitt condition, while$f(u)$does not. \item Another novelty of our approach is that we can estimate the number of turns the solution curve makes until it reaches any level$u(0)= c$. \item Arguing as in Lemma 1, we see that either the solution curve crosses the line$\lambda =\lambda _1$infinitely many times, or else it makes infinitely many turns to the left of this line. (Assuming that$\lambda >\lambda _1$for sufficiently large$s$, we multiply the equation (\ref{14}) by$u\$, integrate by parts, and obtain a contradiction.) \end{enumerate} \paragraph{Acknowledgment.} We wish to thank Y. Cheng and S.-H. Wang for sending us preprints of their work, and K.R. Meyer, T. Ouyang and J. Shi for useful comments. \begin{thebibliography}{99} \bibitem{C} Y. Cheng, On an open problem of Ambrosetti, Brezis and Cerami, {\em Differential Integral Equations}, to appear. \bibitem{SS1} D. Costa, H. Jeggle, R. Schaaf and K. Schmitt, Oscillatory perturbations of linear problems at resonance, {\em Results in Mathematics} {\bf 14}, 275-287 (1988). \bibitem{GNN} B. Gidas, W.-M. Ni and L. Nirenberg, Symmetry and related properties via the maximum principle, {\em Commun. Math. Phys.} {\bf 68}, 209-243 (1979). \bibitem{KM} H. Kielhofer and S. Maier, Infinitely many positive solutions of semilinear elliptic problems via sub- and supersolutions, {\em Comm. Partial Differential Equations} {\bf 18} , 1219-1229 (1993). \bibitem{KL} P. Korman and Y. Li, On the exactness of an S-shaped bifurcation curve, {\em Proc. Amer. Math. Soc.} {\bf 127(4)}, 1011--1020 (1999). \bibitem{KLO} P. Korman, Y. Li and T. Ouyang, Exact multiplicity results for boundary-value problems with nonlinearities generalising cubic, {\em Proc. Royal Soc. Edinburgh, Ser. A}. 126A, 599-616 (1996). \bibitem{LL} E.M. Landesman and A.C. Lazer, Nonlinear perturbations of linear elliptic boundary value problems at resonance, {\em J. Math. Mech.} {\bf 19}, 609-623 (1970). \bibitem{SS} R. Schaaf and K. Schmitt, A class of nonlinear Sturm-Liouville problems with infinitely many solution, {\em Trans. Amer. Math. Soc.} {\bf 306(2)}, 853-859 (1988). \bibitem{S} Y.V. Sidorov, M.V. Fedoryuk and M.I. Shabunin, Lectures on the Theory of Functions of Complex Variable, {\em Mir Publishers}, Moscow (1985). \bibitem{W} S.-H. Wang, On the bifurcation curve of positive solutions of a boundary value problem, Preprint. \bibitem{Z} E. Zeidler, Nonlinear Functional Analysis and its Applications, vol. I, {\em Springer} (1986). \end{thebibliography} \medskip \noindent{\sc Philip Korman} \\ Institute for Dynamics and \\ Department of Mathematical Sciences \\ University of Cincinnati \\ Cincinnati Ohio 45221-0025 \\ e-mail: kormanp@math.uc.edu \medskip \noindent{\sc Yi Li } \\ Department of Mathematics \\ University of Iowa \\ Iowa City Iowa 52242 \\ e-mail: yi-li@uiowa.edu \end{document}