Remote Access Proceedings of the American Mathematical Society
Green Open Access

Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

 
 

 

Non-singular solutions of two-point problems, with multiple changes of sign in the nonlinearity


Author: Philip Korman
Journal: Proc. Amer. Math. Soc. 144 (2016), 2539-2546
MSC (2010): Primary 34B15
DOI: https://doi.org/10.1090/proc/12905
Published electronically: October 22, 2015
MathSciNet review: 3477070
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: We prove that positive solutions of the two-point boundary value problem

$\displaystyle u''(x)+\lambda f(u(x))=0,\;\;$$\displaystyle \mbox {for $-1<x<1$},\;\; u(-1)=u(1)=0,$

satisfying $ \max u=u(0)>\gamma $, are non-singular, provided that $ f(u)$ is predominantly negative for $ u \in (0,\gamma ]$, and superlinear for $ u>\gamma $. This result adds a solution curve without turns to whatever is known about the solution set for $ u(0) \in (0,\gamma )$. In particular, we combine it with the well-known cases of parabola-like, or $ S$-shaped solution curves.

References [Enhancements On Off] (What's this?)

  • [1] B. Gidas, Wei Ming Ni, and L. Nirenberg, Symmetry and related properties via the maximum principle, Comm. Math. Phys. 68 (1979), no. 3, 209-243. MR 544879 (80h:35043)
  • [2] Kuo-Chih Hung, Exact multiplicity of positive solutions of a semipositone problem with concave-convex nonlinearity, J. Differential Equations 255 (2013), no. 11, 3811-3831. MR 3097237, https://doi.org/10.1016/j.jde.2013.07.033
  • [3] Philip Korman, Global solution curves for semilinear elliptic equations, World Scientific Publishing Co. Pte. Ltd., Hackensack, NJ, 2012. MR 2954053
  • [4] Philip Korman, Global solution branches and exact multiplicity of solutions for two point boundary value problems, Handbook of differential equations: ordinary differential equations. Vol. III, Handb. Differ. Equ., Elsevier/North-Holland, Amsterdam, 2006, pp. 547-606. MR 2457637 (2010e:34089), https://doi.org/10.1016/S1874-5725(06)80010-6
  • [5] Philip Korman and Yi Li, On the exactness of an S-shaped bifurcation curve, Proc. Amer. Math. Soc. 127 (1999), no. 4, 1011-1020. MR 1610804 (99f:34027), https://doi.org/10.1090/S0002-9939-99-04928-X
  • [6] Philip Korman and Yi Li, Exact multiplicity of positive solutions for concave-convex and convex-concave nonlinearities, J. Differential Equations 257 (2014), no. 10, 3730-3737. MR 3260239, https://doi.org/10.1016/j.jde.2014.07.007
  • [7] Philip Korman, Yi Li, and Tiancheng Ouyang, Exact multiplicity results for boundary value problems with nonlinearities generalising cubic, Proc. Roy. Soc. Edinburgh Sect. A 126 (1996), no. 3, 599-616. MR 1396280 (97c:34038), https://doi.org/10.1017/S0308210500022927
  • [8] Renate Schaaf, Global solution branches of two-point boundary value problems, Lecture Notes in Mathematics, vol. 1458, Springer-Verlag, Berlin, 1990. MR 1090827 (92a:34003)
  • [9] Junping Shi and Ratnasingham Shivaji, Exact multiplicity of solutions for classes of semipositone problems with concave-convex nonlinearity, Discrete Contin. Dynam. Systems 7 (2001), no. 3, 559-571. MR 1815768 (2001m:34047), https://doi.org/10.3934/dcds.2001.7.559
  • [10] Shin-Hwa Wang, A correction for a paper by J. Smoller and A. G. Wasserman: ``Global bifurcation of steady-state solutions'' [J. Differential Equations 39 (1981), no. 2, 269-290; MR0607786 (82d:58056)], J. Differential Equations 77 (1989), no. 1, 199-202. MR 980548 (90f:58136), https://doi.org/10.1016/0022-0396(89)90162-9
  • [11] Shin Hwa Wang, On $ {\mathsf {S}}$-shaped bifurcation curves, Nonlinear Anal. 22 (1994), no. 12, 1475-1485. MR 1285087 (96d:34021), https://doi.org/10.1016/0362-546X(94)90183-X

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC (2010): 34B15

Retrieve articles in all journals with MSC (2010): 34B15


Additional Information

Philip Korman
Affiliation: Department of Mathematical Sciences, University of Cincinnati, Cincinnati, Ohio 45221-0025
Email: kormanp@ucmail.uc.edu

DOI: https://doi.org/10.1090/proc/12905
Keywords: Global solution curves, non-singular positive solutions
Received by editor(s): October 27, 2014
Received by editor(s) in revised form: July 14, 2015
Published electronically: October 22, 2015
Communicated by: Joachim Krieger
Article copyright: © Copyright 2015 American Mathematical Society

American Mathematical Society