American Institute of Mathematical Sciences

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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

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 &amp; Applied Analysis, 2013, 12 (5) : 2297-2318. doi: 10.3934/cpaa.2013.12.2297
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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.  Google Scholar

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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.  Google Scholar

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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.  Google Scholar

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R. L. Wheeden and A. Zygmund, "Measure and Integral: An Introduction to Real Analysis," Marcel Dekker, New York, 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), 197-208. 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), 381-388. 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), 322-341. 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, 106, Pitman, Boston, MA, 1985.  Google Scholar

[5]

J. I. Díaz, Qualitative study of nonlinear parabolic equations: an introduction, Extracta Math., 16 (2001), 303-341.  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 Sér. I Math., 329 (1999), 587-592. 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), 449-474. 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), 1-13.  Google Scholar

[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.  Google Scholar

[10]

H. L. Royden, "Real Analysis," Macmillan, New York, 1988.  Google Scholar

[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.  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), 128-153. 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), 52-64. 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, New York, 1977.  Google Scholar

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