## Abstract and Applied Analysis

### Positive Solutions of the One-Dimensional $p$-Laplacian with Nonlinearity Defined on a Finite Interval

#### Abstract

We use the quadrature method to show the existence and multiplicity of positive solutions of the boundary value problems involving one-dimensional $p$-Laplacian ${\left({u}^{\prime }\left(t\right){|}^{p-2}{u}^{\prime }\left(t\right)\right)}^{\prime }+\lambda f\left(u\left(t\right)\right)=0$, $t\in \left(0,1\right)$, $u\left(0\right)=u\left(1\right)=0$, where $p\in \left(1,2\right]$, $\lambda \in \left(0,\mathrm{\infty }\right)$ is a parameter, $f\in {C}^{1}\left(\left[0,r\right),\left[0,\mathrm{\infty }\right)\right)$ for some constant $r>0$, $f\left(s\right)>0$ in $\left(0,r\right)$, and ${\mathrm{lim}}_{s\to {r}^{-}}\left(r-s{\right)}^{p-\mathrm{1}}f\left(s\right)=+\infty$.

#### Article information

Source
Abstr. Appl. Anal., Volume 2013 (2013), Article ID 492026, 6 pages.

Dates
First available in Project Euclid: 27 February 2014

https://projecteuclid.org/euclid.aaa/1393511829

Digital Object Identifier
doi:10.1155/2013/492026

Mathematical Reviews number (MathSciNet)
MR3039178

Zentralblatt MATH identifier
1276.34019

#### Citation

Ma, Ruyun; Xie, Chunjie; Ahmed, Abubaker. Positive Solutions of the One-Dimensional $p$ -Laplacian with Nonlinearity Defined on a Finite Interval. Abstr. Appl. Anal. 2013 (2013), Article ID 492026, 6 pages. doi:10.1155/2013/492026. https://projecteuclid.org/euclid.aaa/1393511829

#### References

• H. Wang, “On the existence of positive solutions for semilinear elliptic equations in the annulus,” Journal of Differential Equations, vol. 109, no. 1, pp. 1–7, 1994.
• W. Ge, Boundary Value Problems of Nonlinear Ordinary Differential Equations, Science Press, Beijing, China, 2007.
• J. Wang, “The existence of positive solutions for the one-dimensional $p$-Laplacian,” Proceedings of the American Mathematical Society, vol. 125, no. 8, pp. 2275–2283, 1997.
• J. Henderson and H. Wang, “Positive solutions for nonlinear eigenvalue problems,” Journal of Mathematical Analysis and Applications, vol. 208, no. 1, pp. 252–259, 1997.
• R. P. Agarwal, F. Wang, and W. Lian, “Positive solutions for nonlinear singular boundary value problems,” Computers & Mathematics with Applications, vol. 35, pp. 81–87, 1998.
• L. H. Erbe and H. Wang, “On the existence of positive solutions of ordinary differential equations,” Proceedings of the American Mathematical Society, vol. 120, no. 3, pp. 743–748, 1994.
• K. Lan and J. R. L. Webb, “Positive solutions of semilinear differential equations with singularities,” Journal of Differential Equations, vol. 148, no. 2, pp. 407–421, 1998.
• L. H. Erbe, S. C. Hu, and H. Wang, “Multiple positive solutions of some boundary value problems,” Journal of Mathematical Analysis and Applications, vol. 184, no. 3, pp. 640–648, 1994.
• W. C. Lian, F. H. Wong, and C. C. Yeh, “On the existence of positive solutions of nonlinear second order differential equations,” Proceedings of the American Mathematical Society, vol. 124, no. 4, pp. 1117–1126, 1996.
• Z. Liu and F. Li, “Multiple positive solutions of nonlinear two-point boundary value problems,” Journal of Mathematical Analysis and Applications, vol. 203, no. 3, pp. 610–625, 1996.
• R. Ma, “Positive solutions of a nonlinear three-point boundary-value problem,” Electronic Journal of Differential Equations, vol. 1999, article 34, 8 pages, 1999.
• A. M. Fink, J. A. Gatica, and G. E. Hernández, “Eigenvalues of generalized Gelfand models,” Nonlinear Analysis: Theory, Methods & Applications, vol. 20, no. 12, pp. 1453–1468, 1993.
• K. J. Brown and H. Budin, “On the existence of positive solutions for a class of semilinear elliptic boundary value problems,” SIAM Journal on Mathematical Analysis, vol. 10, no. 5, pp. 875–883, 1979.
• I. Addou and S.-H. Wang, “Exact multiplicity results for a $p$-Laplacian problem with concave-convex-concave nonlinearities,” Nonlinear Analysis: Theory, Methods & Applications, vol. 53, no. 1, pp. 111–137, 2003.
• J. Cheng and Y. Shao, “The positive solutions of boundary value problems for a class of one-dimensional $p$-Laplacians,” Nonlinear Analysis: Theory, Methods & Applications, vol. 68, no. 4, pp. 883–891, 2008.
• J. Karátson and P. L. Simon, “Exact multiplicitly for degenerate two-point boundary value problems with $p$-convex nonlinearity,” Nonlinear Analysis: Theory, Methods & Applications, vol. 52, no. 6, pp. 1569–1590, 2003.
• T. Laetsch, “The number of solutions of a nonlinear two point boundary value problem,” Indiana University Mathematics Journal, vol. 20, pp. 1–13, 1970-1971.
• C.-C. Tzeng, K.-C. Hung, and S.-H. Wang, “Global bifurcation and exact multiplicity of positive solutions for a positone problem with cubic nonlinearity,” Journal of Differential Equations, vol. 252, no. 12, pp. 6250–6274, 2012.
• K.-C. Hung and S.-H. Wang, “Classification and evolution of bifurcation curves for a multiparameter $p$-Laplacian Dirichlet problem,” Nonlinear Analysis: Theory, Methods & Applications, vol. 74, no. 11, pp. 3589–3598, 2011.
• W. Reichel and W. Walter, “Radial solutions of equations and inequalities involving the $p$-Laplacian,” Journal of Inequalities and Applications, vol. 1, no. 1, pp. 47–71, 1997.