Global and local structures of oscillatory bifurcation curves with application to inverse bifurcation problem

Tetsutaro Shibata

DOI: http://dx.doi.org/10.12775/TMNA.2017.032

Abstract


We consider the bifurcation problem $$ -u''(t) = \lambda (u(t) + g(u(t))), \quad u(t) > 0, \quad t \in I := (-1,1), \quad u(\pm 1) = 0, $$% where $g(u) = g_1(u) := \sin \sqrt{u}$ and $g_2(u) := \sin u^2 (= \sin (u^2))$, and $\lambda > 0$ is a bifurcation parameter. It is known that $\lambda$ is parameterized by the maximum norm $\alpha = \Vert u_\lambda\Vert_\infty$ of the solution $u_\lambda$ associated with $\lambda$ and is written as $\lambda = \lambda(g,\alpha)$. When $g(u) = g_1(u)$, this problem has been proposed in Cheng [4] as an example which has arbitrary many solutions near $\lambda = \pi^2/4$. We show that the bifurcation diagram of $\lambda(g_1,\alpha)$ intersects the line $\lambda = \pi^2/4$ infinitely many times by establishing the precise asymptotic formula for $\lambda(g_1,\alpha)$ as $\alpha \to \infty$. We also establish the precise asymptotic formulas for $\lambda(g_i,\alpha)$ ($i = 1,2$) as $\alpha \to \infty$ and $\alpha \to 0$. We apply these results to the new concept of inverse bifurcation problems.

Keywords


Oscillatory bifurcation; global and local structure of bifurcation curves

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References


A. Ambrosetti, H. Brezis and G. Cerami, Combined effects of concave and convex nonlinearities in some elliptic problems, J. Funct. Anal. 122 (1994), 519–543.

S. Cano-Casanova and J. López-Gómez, Existence, uniqueness and blow-up rate of large solutions for a canonical class of one-dimensional problems on the half-line, J. Differential Equations 244 (2008), 3180–3203.

S. Cano-Casanova and J. López-Gómez, Blow-up rates of radially symmetric large solutions, J. Math. Anal. Appl. 352 (2009), 166–174.

Y.J. Cheng, On an open problem of Ambrosetti, Brezis and Cerami, Differential Integral Equations 15 (2002), 1025–1044.

R. Chiappinelli, On spectral asymptotics and bifurcation for elliptic operators with odd superlinear term, Nonlinear Anal. 13 (1989), 871–878.

J.M. Fraile, J. López-Gómez and J. Sabina de Lis, On the global structure of the set of positive solutions of some semilinear elliptic boundary value problems, J. Differential Equations 123 (1995), 180–212.

A. Galstian, P. Korman and Y. Li, On the oscillations of the solution curve for a class of semilinear equations, J. Math. Anal. Appl. 321 (2006), 576–588.

I.S. Gradshteyn and I.M. Ryzhik, Table of Integrals, Series, and Products, Translated from the Russian. Translation edited and with a preface by Daniel Zwillinger and Victor Moll. Eighth edition. Elsevier/Academic Press, Amsterdam, 2015.

P. Korman and Y. Li, Infinitely many solutions at a resonance, Electron. J. Differ. Equ. Conf. 05, 105–111.

P. Korman, An oscillatory bifurcation from infinity, and from zero, NoDEA Nonlinear Differential Equations Appl. 15 (2008), 335–345.

P. Korman, Global solution curves for semilinear elliptic equations, World Scientific, Hackensack, 2012.

I. Krasikov, Approximations for the Bessel and Airy functions with an explicit error term, LMS J. Comput. Math. 17 (2014), 209–225.

T. Laetsch, The number of solutions of a nonlinear two point boundary value problem, Indiana Univ. Math. J. 20 (1970/1971), 1–13.

T. Shibata, Asymptotic behavior of bifurcation curve for sine-Gordon type differential equation, Abstract and Applied Analysis 2012 (2012), Article ID 753857, 16 pages.

T. Shibata, Asymptotic length of bifurcation curves related to inverse bifurcation problems, J. Math. Anal. Appl. 438 (2016), 629–642.

T. Shibata, Oscillatory bifurcation for semilinear ordinary differential equations, Electron. J. Qual. Theory Differ. Equ. 44 (2016), 1–13.


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