American Institute of Mathematical Sciences

Advanced
September  2013, 12(5): 2297-2318. doi: 10.3934/cpaa.2013.12.2297

Classification of bifurcation diagrams of a $P$-Laplacian nonpositone problem

Po-Chun Huang 1, , Shin-Hwa Wang 2, and  Tzung-Shin Yeh 3,

1. 

Department of Mathematics, National Tsing Hua University, Hsinchu 300, Taiwan
2. 

Department of Mathematics, National Tsing Hua University, Hsinchu, Taiwan 300
3. 

Department of Applied Mathematics, National University of Tainan, Tainan 700, Taiwan

Received  February 2012 Revised  April 2012 Published  January 2013

We study the bifurcation diagrams of positive solutions of the $p$-Laplacian Dirichlet problem \begin{eqnarray*} (\varphi_p(u'(x)))'+f_\lambda(u(x))=0, -1 < x < 1, \\ u(-1)=u(1)=0, \end{eqnarray*} where $\varphi_p(y)=|y|^{p-2}y$, $(\varphi_p(u'))'$ is the one-dimensional $p$-Laplacian, $p>1$, the nonlinearity $f_\lambda(u)=\lambda g(u)-h(u),$ $g,h\in C[0,\infty)\cap C^2(0,\infty )$, and $\lambda >0$ is a bifurcation parameter. Under certain hypotheses on functions $g$ and $h$, we give a complete classification of bifurcation diagrams. We prove that, on the $(\lambda, |u|_\infty)$-plane, each bifurcation diagram consists of exactly one curve which has exactly one turning point where the curve turns to the right. Hence we are able to determine the exact multiplicity of positive solutions for each $\lambda >0.$ In addition, we show the evolution phenomena of bifurcation diagrams of polynomial nonlinearities with positive coefficients.
Keywords: Bifurcation diagram, nonpositone problem, positive solution, exact multiplicity., $p$-Laplacian.
Mathematics Subject Classification: 34B15, 34B1.
Citation: Po-Chun Huang, Shin-Hwa Wang, Tzung-Shin Yeh. Classification of bifurcation diagrams of a $P$-Laplacian nonpositone problem. Communications on Pure & Applied Analysis, 2013, 12 (5) : 2297-2318. doi: 10.3934/cpaa.2013.12.2297
References:
[1]

J. G. Cheng, Exact number of positive solutions for a class of semipositone problems,, J. Math. Anal. Appl., 280 (2003), 197.  doi: 10.1016/S0022-247X(02)00539-5.  Google Scholar

[2]

J. G. Cheng, Uniqueness results for the one-dimensional $p$-Laplacian,, J. Math. Anal. Appl., 311 (2005), 381.  doi: 10.1016/j.jmaa.2005.02.057.  Google Scholar

[3]

J. G. Cheng, Exact number of positive solutions for semipositone problems,, J. Math. Anal. Appl., 313 (2006), 322.  doi: 10.1016/j.jmaa.2005.09.043.  Google Scholar

[4]

J. I. Díaz, "Nonlinear Partial Differential Equations and Free Boundaries, Vol. I. Elliptic Equations,", Research Notes in Mathematics, (1985).   Google Scholar

[5]

J. I. Díaz, Qualitative study of nonlinear parabolic equations: an introduction,, Extracta Math., 16 (2001), 303.   Google Scholar

[6]

J. I. Díaz and J. Hernández, Global bifurcation and continua of nonnegative solutions for a quasilinear elliptic problem,, C. R. Acad. Sci. Paris S\'er. I Math., 329 (1999), 587.  doi: 10.1016/S0764-4442(00)80006-3.  Google Scholar

[7]

J. I. Díaz, J. Hernández and F. J. Mancebo, Branches of positive and free boundary solutions for some singular quasilinear elliptic problems,, J. Math. Anal. Appl., 352 (2009), 449.  doi: 10.1016/j.jmaa.2008.07.073.  Google Scholar

[8]

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

[9]

A. Lakmeche and A. Hammoudi, Multiple positive solutions of the one-dimensional $p$-Laplacian,, J. Math. Anal. Appl., 317 (2006), 43.  doi: 10.1016/j.jmaa.2005.10.040.  Google Scholar

[10]

H. L. Royden, "Real Analysis,", Macmillan, (1988).   Google Scholar

[11]

J. Smoller and A. Wasserman, Global bifurcation of steady-state solutions,, J. Differential Equations, 39 (1981), 269.  doi: 10.1016/0022-0396(81)90077-2.  Google Scholar

[12]

S.-H. Wang and T.-S. Yeh, A complete classification of bifurcation diagrams of a Dirichlet problem with concave-convex nonlinearities,, J. Math. Anal. Appl., 291 (2004), 128.  doi: 10.1016/j.jmaa.2003.10.021.  Google Scholar

[13]

Z. L. Wei and C. C. Pang, Exact structure of positive solutions for some $p$-Laplacian equations,, J. Math. Anal. Appl., 301 (2005), 52.  doi: 10.1016/j.jmaa.2004.06.058.  Google Scholar

[14]

R. L. Wheeden and A. Zygmund, "Measure and Integral: An Introduction to Real Analysis,", Marcel Dekker, (1977).   Google Scholar

show all references

References:
[1]

J. G. Cheng, Exact number of positive solutions for a class of semipositone problems,, J. Math. Anal. Appl., 280 (2003), 197.  doi: 10.1016/S0022-247X(02)00539-5.  Google Scholar

[2]

J. G. Cheng, Uniqueness results for the one-dimensional $p$-Laplacian,, J. Math. Anal. Appl., 311 (2005), 381.  doi: 10.1016/j.jmaa.2005.02.057.  Google Scholar

[3]

J. G. Cheng, Exact number of positive solutions for semipositone problems,, J. Math. Anal. Appl., 313 (2006), 322.  doi: 10.1016/j.jmaa.2005.09.043.  Google Scholar

[4]

J. I. Díaz, "Nonlinear Partial Differential Equations and Free Boundaries, Vol. I. Elliptic Equations,", Research Notes in Mathematics, (1985).   Google Scholar

[5]

J. I. Díaz, Qualitative study of nonlinear parabolic equations: an introduction,, Extracta Math., 16 (2001), 303.   Google Scholar

[6]

J. I. Díaz and J. Hernández, Global bifurcation and continua of nonnegative solutions for a quasilinear elliptic problem,, C. R. Acad. Sci. Paris S\'er. I Math., 329 (1999), 587.  doi: 10.1016/S0764-4442(00)80006-3.  Google Scholar

[7]

J. I. Díaz, J. Hernández and F. J. Mancebo, Branches of positive and free boundary solutions for some singular quasilinear elliptic problems,, J. Math. Anal. Appl., 352 (2009), 449.  doi: 10.1016/j.jmaa.2008.07.073.  Google Scholar

[8]

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

[9]

A. Lakmeche and A. Hammoudi, Multiple positive solutions of the one-dimensional $p$-Laplacian,, J. Math. Anal. Appl., 317 (2006), 43.  doi: 10.1016/j.jmaa.2005.10.040.  Google Scholar

[10]

H. L. Royden, "Real Analysis,", Macmillan, (1988).   Google Scholar

[11]

J. Smoller and A. Wasserman, Global bifurcation of steady-state solutions,, J. Differential Equations, 39 (1981), 269.  doi: 10.1016/0022-0396(81)90077-2.  Google Scholar

[12]

S.-H. Wang and T.-S. Yeh, A complete classification of bifurcation diagrams of a Dirichlet problem with concave-convex nonlinearities,, J. Math. Anal. Appl., 291 (2004), 128.  doi: 10.1016/j.jmaa.2003.10.021.  Google Scholar

[13]

Z. L. Wei and C. C. Pang, Exact structure of positive solutions for some $p$-Laplacian equations,, J. Math. Anal. Appl., 301 (2005), 52.  doi: 10.1016/j.jmaa.2004.06.058.  Google Scholar

[14]

R. L. Wheeden and A. Zygmund, "Measure and Integral: An Introduction to Real Analysis,", Marcel Dekker, (1977).   Google Scholar

[1]

Kuan-Ju Huang, Yi-Jung Lee, Tzung-Shin Yeh. Classification of bifurcation curves of positive solutions for a nonpositone problem with a quartic polynomial. Communications on Pure & Applied Analysis, 2016, 15 (4) : 1497-1514. doi: 10.3934/cpaa.2016.15.1497
[2]

Shao-Yuan Huang. Exact multiplicity and bifurcation curves of positive solutions of a one-dimensional Minkowski-curvature problem and its application. Communications on Pure & Applied Analysis, 2018, 17 (3) : 1271-1294. doi: 10.3934/cpaa.2018061
[3]

Shao-Yuan Huang. Global bifurcation and exact multiplicity of positive solutions for the one-dimensional Minkowski-curvature problem with sign-changing nonlinearity. Communications on Pure & Applied Analysis, 2019, 18 (6) : 3267-3284. doi: 10.3934/cpaa.2019147
[4]

Michael E. Filippakis, Nikolaos S. Papageorgiou. Existence and multiplicity of positive solutions for nonlinear boundary value problems driven by the scalar $p$-Laplacian. Communications on Pure & Applied Analysis, 2004, 3 (4) : 729-756. doi: 10.3934/cpaa.2004.3.729
[5]

Mohan Mallick, R. Shivaji, Byungjae Son, S. Sundar. Bifurcation and multiplicity results for a class of $n\times n$ $p$-Laplacian system. Communications on Pure & Applied Analysis, 2018, 17 (3) : 1295-1304. doi: 10.3934/cpaa.2018062
[6]

Salvatore A. Marano, Nikolaos S. Papageorgiou. Positive solutions to a Dirichlet problem with $p$-Laplacian and concave-convex nonlinearity depending on a parameter. Communications on Pure & Applied Analysis, 2013, 12 (2) : 815-829. doi: 10.3934/cpaa.2013.12.815
[7]

K. D. Chu, D. D. Hai. Positive solutions for the one-dimensional singular superlinear $ p $-Laplacian problem. Communications on Pure & Applied Analysis, 2020, 19 (1) : 241-252. doi: 10.3934/cpaa.2020013
[8]

Nikolaos S. Papageorgiou, George Smyrlis. Positive solutions for parametric $p$-Laplacian equations. Communications on Pure & Applied Analysis, 2016, 15 (5) : 1545-1570. doi: 10.3934/cpaa.2016002
[9]

Friedemann Brock, Leonelo Iturriaga, Justino Sánchez, Pedro Ubilla. Existence of positive solutions for $p$--Laplacian problems with weights. Communications on Pure & Applied Analysis, 2006, 5 (4) : 941-952. doi: 10.3934/cpaa.2006.5.941
[10]

Leonelo Iturriaga, Eugenio Massa. Existence, nonexistence and multiplicity of positive solutions for the poly-Laplacian and nonlinearities with zeros. Discrete & Continuous Dynamical Systems - A, 2018, 38 (8) : 3831-3850. doi: 10.3934/dcds.2018166
[11]

Michael Filippakis, Alexandru Kristály, Nikolaos S. Papageorgiou. Existence of five nonzero solutions with exact sign for a $p$-Laplacian equation. Discrete & Continuous Dynamical Systems - A, 2009, 24 (2) : 405-440. doi: 10.3934/dcds.2009.24.405
[12]

Shin-Yi Lee, Shin-Hwa Wang, Chiou-Ping Ye. Explicit necessary and sufficient conditions for the existence of a dead core solution of a p-laplacian steady-state reaction-diffusion problem. Conference Publications, 2005, 2005 (Special) : 587-596. doi: 10.3934/proc.2005.2005.587
[13]

Guowei Dai. Bifurcation and one-sign solutions of the $p$-Laplacian involving a nonlinearity with zeros. Discrete & Continuous Dynamical Systems - A, 2016, 36 (10) : 5323-5345. doi: 10.3934/dcds.2016034
[14]

Guowei Dai, Ruyun Ma, Haiyan Wang. Eigenvalues, bifurcation and one-sign solutions for the periodic $p$-Laplacian. Communications on Pure & Applied Analysis, 2013, 12 (6) : 2839-2872. doi: 10.3934/cpaa.2013.12.2839
[15]

Marta García-Huidobro, Raul Manásevich, J. R. Ward. Vector p-Laplacian like operators, pseudo-eigenvalues, and bifurcation. Discrete & Continuous Dynamical Systems - A, 2007, 19 (2) : 299-321. doi: 10.3934/dcds.2007.19.299
[16]

Yansheng Zhong, Yongqing Li. On a p-Laplacian eigenvalue problem with supercritical exponent. Communications on Pure & Applied Analysis, 2019, 18 (1) : 227-236. doi: 10.3934/cpaa.2019012
[17]

Francesca Colasuonno, Benedetta Noris. A p-Laplacian supercritical Neumann problem. Discrete & Continuous Dynamical Systems - A, 2017, 37 (6) : 3025-3057. doi: 10.3934/dcds.2017130
[18]

Giuseppina Barletta, Roberto Livrea, Nikolaos S. Papageorgiou. A nonlinear eigenvalue problem for the periodic scalar $p$-Laplacian. Communications on Pure & Applied Analysis, 2014, 13 (3) : 1075-1086. doi: 10.3934/cpaa.2014.13.1075
[19]

Francesca Colasuonno, Fausto Ferrari. The Soap Bubble Theorem and a $ p $-Laplacian overdetermined problem. Communications on Pure & Applied Analysis, 2020, 19 (2) : 983-1000. doi: 10.3934/cpaa.2020045
[20]

Tomás Caraballo, Marta Herrera-Cobos, Pedro Marín-Rubio. Global attractor for a nonlocal p-Laplacian equation without uniqueness of solution. Discrete & Continuous Dynamical Systems - B, 2017, 22 (5) : 1801-1816. doi: 10.3934/dcdsb.2017107

2018 Impact Factor: 0.925

Tools

Metrics

  • PDF downloads (11)
  • HTML views (0)
  • Cited by (0)

Other articles
by authors

[Back to Top]

Copyright © 2019 American Institute of Mathematical Sciences