Rocky Mountain Journal of Mathematics

Sign-changing solutions to a class of nonlinear equations involving the $p$-Laplacian

Wei-Chuan Wang and Yan-Hsiou Cheng

Full-text: Access denied (no subscription detected)

We're sorry, but we are unable to provide you with the full text of this article because we are not able to identify you as a subscriber. If you have a personal subscription to this journal, then please login. If you are already logged in, then you may need to update your profile to register your subscription. Read more about accessing full-text


This paper deals with a class of nonlinear problems $$ -(r^{n-1}|u'|^{p-2}u')'+r^{n-1}q(r)|u|^{p-2}u =r^{n-1}w(r)f(u) $$ in $(0,1)$, where $1\leq n\lt p\lt \infty $ and $'={d}/{dr}$. We study the existence of nodal solutions to this nonautonomous system. We give necessary and sufficient conditions for the existence of sign-changing solutions and also observe an application related to the case of multi-point boundary conditions. Methods used here are energy function control, shooting arguments and Pr\"{u}fer-type substitutions.

Article information

Rocky Mountain J. Math., Volume 47, Number 3 (2017), 971-996.

First available in Project Euclid: 24 June 2017

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 34A12: Initial value problems, existence, uniqueness, continuous dependence and continuation of solutions 34B15: Nonlinear boundary value problems

Sign-changing radial solution nonlinear $p$-Laplacian equation


Wang, Wei-Chuan; Cheng, Yan-Hsiou. Sign-changing solutions to a class of nonlinear equations involving the $p$-Laplacian. Rocky Mountain J. Math. 47 (2017), no. 3, 971--996. doi:10.1216/RMJ-2017-47-3-971.

Export citation


  • J. Benedikt, P. Drabek and P. Girg, The second eigenfunction of the $p$-Laplacian on the disk in not radial, Nonlin. Anal. 75 (2012), 4422–4435.
  • P. Binding and P. Drabek, Sturm-Liouville theory for the $p$-Laplacian, Stud. Sci. Math. Hungar. 40 (2003), 373–396.
  • G. Birkhoff and G. Rota, Ordinary differential equations, John Wiley and Sons, New York, 1989.
  • B.M. Brown and M.S.P. Eastham, Titchmarsh's asymptotic formula for periodic eigenvalues and an extension to the $p$-Laplacian, J. Math. Anal. Appl. 338 (2008), 1255–1266.
  • B.M. Brown and W. Reichel, Computing eigenvalues and Fucik-spectrum of the radially symmetric $p$-Laplacian, J. Comp. Appl. Math. 148 (2002), 183–211.
  • ––––, Eigenvalues of the radial symmetric $p$-Laplacian in $\mathbb{R}^n$, J. Lond. Math. Soc. 69 (2004), 657–675.
  • C. Cortazar, M. Garcia-Huidobro and C.S. Yarur, On the existence of sign changing bound state solutions of a quasilinear equation, J. Diff. Eq. 254 (2013), 2603–2625.
  • Manuel A. del Pino and Raúl F Manásevich, Global bifurcation from the eigenvalues of the $p$-Laplacian, J. Diff. Eq. 92 (1991), 226–251.
  • J.I. Diaz, Nonlinear partial differential equations and free boundaries, Volume 1, Elliptic equations, Res. Notes Math. 106, Pitman, Boston, 1985.
  • ––––, Qualitative study of nonlinear parabolic equations: An introduction, Extract. Math. 16 (2001), 303–341.
  • A. Elbert, A half-linear second order differential equation, Colloq. Math. Soc. J. Bolyai \bf30 (1979), 153–180.
  • H. Feng, W. Ge and M. Jiang, Multiple positive solutions for $m$-point boundary-value problems with a one-dimensional $p$-Laplacian, Nonlin. Anal. 68 (2008), 2269–2279.
  • M. Garcia-Huidobro, R. Manasevich and C.S. Yarur, On the structure of positive radial solutions to an equation containing a $p$-Laplacian with weight, J. Diff. Eq. 223 (2006), 51–95.
  • Chunhua Jin, Jingxue Yin and Zejia Wang, Positive radial solutions of $p$-Laplacian equation with sign changing nonlinear sources, Math. Meth. Appl. Sci. 30 (2007), 1–14.
  • L. Kong and Q. Kong, Existence of nodal solutions of multi-point boundary value problems, Discr. Cont. Dynam. Syst. (2009), 457–465.
  • Q. Kong, Existence and nonexistence of solutions of second-order nonlinear boundary value problems, Nonlin. Anal. 66 (2007), 2635–2651.
  • O.A. Ladyzhenskaya, The mathematical theory of viscous incompressible flow, Second edition, Gordon and Breach, New York, 1969.
  • C.K. Law, W.C. Lian and W.C. Wang, Inverse nodal problem and Ambarzumyan problem for the $p$-Laplacian, Proc. Royal Soc. Edinburgh 139 (2009), 1261–1273.
  • P. Lindqvist, Some remarkable sine and cosine functions, Ricer. Mat. 44 (1995), 269–290.
  • J.L. Lions, Quelques methodes de resolution des problemes aux limites non lineaires, Dunod, Paris, 1969.
  • B. Liu, Positive solutions of singular three-point boundary value problems for the one-dimensional $p$-Laplacian, Comp. Math. Appl. 48 (2004), 913–925.
  • R. Ma, Positive solutions for multipoint boundary value problem with a one-dimensional $p$-Laplacian, Comp. Math. Appl. 42 (2001), 755–765.
  • P.J. Mckenna, W. Reichel and W. Walter, Symmetry and multiplicity for nonlinear elliptic differential equations with boundary blow-up, Nonlin. Anal. Th. Meth. Appl. 28 (1997), 1213–1225.
  • M.C. Pelissier and M.L. Reynaud, Etude d'ún modele mathematique d'écoulement de glacier, C.R. Acad. Sci. Paris 279 (1974), 531–534.
  • M. Ramaswamy and R. Shivaji, Multiple positive solutions for classes of $p$-Laplacian equations, Diff. Int. Eq. 17 (2004), 1255–1261.
  • W. Reichel and W. Walter, Radial solutions of equations and inequalities involving the $p$-Laplacian, J. Inequal. Appl. 1 (1997), 47–71.
  • ––––, Sturm-Liouville type problems for the $p$-Laplacian under asymptotic non-resonance conditions, J. Diff. Eq. 156 (1999), 50–70.
  • R.E. Showalter and N.J. Walkington, Diffusion of fluid in a fissured medium with microstructure, SIAM J. Math. Anal. 22 (1991), 1702–1722.
  • S. Takeuchi, Generalized Jacobian elliptic functions and their application to bifurcation problems associated with $p$-Laplacian, J. Math. Anal. Appl. 385 (2012), 24–35.
  • W. Walter, Sturm-Liouville theory for the radial $\vartriangle_p$-operator, Math. Z. 227 (1998), 175–185.
  • W.C. Wang and Y.H. Cheng, On the existence of sign-changing radial solutions to nonlinear $p$-Laplacian equations in $\mathbb{R}^n$, Nonlin. Anal. 102 (2014), 14–22.
  • Y. Wang and C. Hou, Existence of multiple positive solutions for one-dimensional $p$-Laplacian, J. Math. Anal. Appl. 315 (2006), 144–153.