Classification of bifurcation diagrams of a P-Laplacian nonpositone problem

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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

Abstract
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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 and 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), 197-208. doi: 10.1016/S0022-247X(02)00539-5.
[2]	J. G. Cheng, Uniqueness results for the one-dimensional $p$-Laplacian, J. Math. Anal. Appl., 311 (2005), 381-388. doi: 10.1016/j.jmaa.2005.02.057.
[3]	J. G. Cheng, Exact number of positive solutions for semipositone problems, J. Math. Anal. Appl., 313 (2006), 322-341. doi: 10.1016/j.jmaa.2005.09.043.
[4]	J. I. Díaz, "Nonlinear Partial Differential Equations and Free Boundaries, Vol. I. Elliptic Equations," Research Notes in Mathematics, 106, Pitman, Boston, MA, 1985.
[5]	J. I. Díaz, Qualitative study of nonlinear parabolic equations: an introduction, Extracta Math., 16 (2001), 303-341.
[6]	J. I. Díaz and J. Hernández, Global bifurcation and continua of nonnegative solutions for a quasilinear elliptic problem, C. R. Acad. Sci. Paris Sér. I Math., 329 (1999), 587-592. doi: 10.1016/S0764-4442(00)80006-3.
[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), 449-474. doi: 10.1016/j.jmaa.2008.07.073.
[8]	T. Laetsch, The number of solutions of a nonlinear two point boundary value problem, Indiana Univ. Math. J., 20 (1970), 1-13.
[9]	A. Lakmeche and A. Hammoudi, Multiple positive solutions of the one-dimensional $p$-Laplacian, J. Math. Anal. Appl., 317 (2006), 43-49. doi: 10.1016/j.jmaa.2005.10.040.
[10]	H. L. Royden, "Real Analysis," Macmillan, New York, 1988.
[11]	J. Smoller and A. Wasserman, Global bifurcation of steady-state solutions, J. Differential Equations, 39 (1981), 269-290. doi: 10.1016/0022-0396(81)90077-2.
[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), 128-153. doi: 10.1016/j.jmaa.2003.10.021.
[13]	Z. L. Wei and C. C. Pang, Exact structure of positive solutions for some $p$-Laplacian equations, J. Math. Anal. Appl., 301 (2005), 52-64. doi: 10.1016/j.jmaa.2004.06.058.
[14]	R. L. Wheeden and A. Zygmund, "Measure and Integral: An Introduction to Real Analysis," Marcel Dekker, New York, 1977.

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), 197-208. doi: 10.1016/S0022-247X(02)00539-5.
[2]	J. G. Cheng, Uniqueness results for the one-dimensional $p$-Laplacian, J. Math. Anal. Appl., 311 (2005), 381-388. doi: 10.1016/j.jmaa.2005.02.057.
[3]	J. G. Cheng, Exact number of positive solutions for semipositone problems, J. Math. Anal. Appl., 313 (2006), 322-341. doi: 10.1016/j.jmaa.2005.09.043.
[4]	J. I. Díaz, "Nonlinear Partial Differential Equations and Free Boundaries, Vol. I. Elliptic Equations," Research Notes in Mathematics, 106, Pitman, Boston, MA, 1985.
[5]	J. I. Díaz, Qualitative study of nonlinear parabolic equations: an introduction, Extracta Math., 16 (2001), 303-341.
[6]	J. I. Díaz and J. Hernández, Global bifurcation and continua of nonnegative solutions for a quasilinear elliptic problem, C. R. Acad. Sci. Paris Sér. I Math., 329 (1999), 587-592. doi: 10.1016/S0764-4442(00)80006-3.
[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), 449-474. doi: 10.1016/j.jmaa.2008.07.073.
[8]	T. Laetsch, The number of solutions of a nonlinear two point boundary value problem, Indiana Univ. Math. J., 20 (1970), 1-13.
[9]	A. Lakmeche and A. Hammoudi, Multiple positive solutions of the one-dimensional $p$-Laplacian, J. Math. Anal. Appl., 317 (2006), 43-49. doi: 10.1016/j.jmaa.2005.10.040.
[10]	H. L. Royden, "Real Analysis," Macmillan, New York, 1988.
[11]	J. Smoller and A. Wasserman, Global bifurcation of steady-state solutions, J. Differential Equations, 39 (1981), 269-290. doi: 10.1016/0022-0396(81)90077-2.
[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), 128-153. doi: 10.1016/j.jmaa.2003.10.021.
[13]	Z. L. Wei and C. C. Pang, Exact structure of positive solutions for some $p$-Laplacian equations, J. Math. Anal. Appl., 301 (2005), 52-64. doi: 10.1016/j.jmaa.2004.06.058.
[14]	R. L. Wheeden and A. Zygmund, "Measure and Integral: An Introduction to Real Analysis," Marcel Dekker, New York, 1977.

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