Abstract
We study the bifurcation problem of positive solutions for the Moore-Nehari differential equation, \(u''+h(x,\lambda )u^p=0\), \(u>0\) in \((-1,1)\) with \(u(-1)=u(1)=0\), where \(p>1\), \(h(x,\lambda )=0\) for \(|x|<\lambda \) and \(h(x,\lambda )=1\) for \(\lambda \le |x| \le 1\) and \(\lambda \in (0,1)\) is a bifurcation parameter. We shall show that the problem has a unique even positive solution \(U(x,\lambda )\) for each \(\lambda \in (0,1)\). We shall prove that there exists a unique \(\lambda _*\in (0,1)\) such that a non-even positive solution bifurcates at \(\lambda _*\) from the curve \((\lambda , U(x,\lambda ))\), where \(\lambda _*\) is explicitly represented as a function of p.
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Amadori, A.L., Gladiali, F.: Bifurcation and symmetry breaking for the Henon equation. Adv. Differ. Equ. 19, 755–782 (2014)
Brezis, H.: Functional Analysis, Sobolev Spaces and Partial Differential Equations (Universitext). Springer, New York (2011)
Courant, R., Hilbert, D.: Methods of Mathematical Physics, vol. 1. Interscience, New York (1989)
Drábek, P., Manásevich, R.: On the closed solution to some nonhomogeneous eigenvalue problems with \(p\)-Laplacian. Differ. Integral Equ. 12, 773–788 (1999)
Gritsans, A., Sadyrbaev, F.: Extension of the example by Moore–Nehari. Tatra Mt. Math. Publ. 63, 115–127 (2015)
Kajikiya, R.: Non-even least energy solutions of the Emden–Fowler equation. Proc. Am. Math. Soc. 140, 1353–1362 (2012)
Kajikiya, R.: Non-radial least energy solutions of the generalized Hénon equation. J. Differ. Equ. 252, 1987–2003 (2012)
Kajikiya, R.: Nonradial positive solutions of the \(p\)-Laplace Emden–Fowler equation with sign-changing weight. Math. Nachr. 289, 290–299 (2016)
Le, V.K., Schmitt, K.: Global Bifurcation in Variational Inequalities: Applications to obstacle and unilateral problems (Applied Mathematical Sciences). Springer, Berlin (1997)
López-Gómez, J., Rabinowitz, P.H.: Nodal solutions for a class of degenerate boundary value problems. Adv. Nonlinear Stud. 15, 253–288 (2015)
López-Gómez, J., Rabinowitz, P.H.: The effects of spatial heterogeneities on some multiplicity results. Discrete Contin. Dyn. Syst. 36, 941–952 (2016)
López-Gómez, J., Rabinowitz, P.H.: Nodal solutions for a class of degenerate one dimensional BVP’s. Topol. Methods Nonlinear Anal. 49, 359–376 (2017)
López-Gómez, J., Molina-Meyer, M., Rabinowitz, P.H.: Global bifurcation diagrams of one node solutions in a class of degenerate boundary value problems. Discrete Contin. Dyn. Syst. Ser. B 22, 923–946 (2017)
Moore, R.A., Nehari, Z.: Nonoscillation theorems for a class of nonlinear differential equations. Trans. Am. Math. Soc. 93, 30–52 (1959)
Rabinowitz, P.: Some aspects of nonlinear eigenvalue problems. Rocky Mt. J. Math. 3, 161–202 (1973)
Schmitt, K., Thompson, R.: Nonlinear Analysis and Differential Equations: An introduction. University of Utah Lecture Note, Salt Lake City (2004)
Sim, I., Tanaka, S.: Symmetry-breaking bifurcation for the one-dimensional Hénon equation. Commun. Contemp. Math. (to appear)
Smets, D., Willem, M., Su, J.: Non-radial ground states for the Hénon equation. Commun. Contemp. Math. 4, 467–480 (2002)
Takeuchi, S.: Generalized Jacobian elliptic functions and their application to bifurcation problems associated with \(p\)-Laplacian. J. Math. Anal. Appl. 385, 24–35 (2012)
Takeuchi, S.: The basis property of generalized Jacobian elliptic functions. Commun. Pure Appl. Anal. 13, 2675–2692 (2014)
Takeuchi, S.: Multiple-angle formulas of generalized trigonometric functions with two parameters. J. Math. Anal. Appl. 444, 1000–1014 (2016)
Tanaka, S.: Morse index and symmetry-breaking for positive solutions of one-dimensional Hénon type equations. J. Differ. Equ. 255, 1709–1733 (2013)
Whyburn, G.T.: Topological Analysis, revised edn. Princeton University Press, Princeton (1964)
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Ryuji Kajikiya was supported by JSPS KAKENHI Grant Number 16K05236. Inbo Sim was supported by NRF Grant No. 2015R1D1A3A01019789. Satoshi Tanaka was supported by JSPS KAKENHI Grant Number 26400182.
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Kajikiya, R., Sim, I. & Tanaka, S. Symmetry-breaking bifurcation for the Moore–Nehari differential equation. Nonlinear Differ. Equ. Appl. 25, 54 (2018). https://doi.org/10.1007/s00030-018-0545-3
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DOI: https://doi.org/10.1007/s00030-018-0545-3