Electron. J. Diff. Equ., Vol. 2012 (2012), No. 106, pp. 1-12.

Global continuum and multiple positive solutions to a p-Laplacian boundary-value problem.

Chan-Gyun Kim, Junping Shi

Abstract:
A p-Laplacian boundary-value problem with positive nonlinearity is considered. The existence of a continuum of positive solutions emanating from $(\lambda,u)=(0,0)$ is shown, and it can be extended to $\lambda=\infty$. Under an additional condition on the nonlinearity, it is shown that the positive solution is unique for any $\lambda>0$; thus the continuum $\mathcal{C}$ is indeed a continuous curve globally defined for all $\lambda>0$. In addition, by the upper and lower solutions method, existence of three positive solutions is established under some conditions on the nonlinearity.

Submitted May 25, 2012. Published June 25, 2012.
Math Subject Classifications: 34B18, 34C23, 35J25.
Key Words: Upper and lower solution; positive solution; p-Laplacian; uniqueness; multiplicity.

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Chan-Gyun Kim
Department of Mathematics, College of William and Mary
Williamsburg, Virginia 23187-8795, USA
email: cgkim75@gmail.com
Junping Shi
Department of Mathematics, College of William and Mary
Williamsburg, Virginia 23187-8795, USA.
School of Mathematical Sciences, Shanxi University
Taiyuan, Shanxi 030006, China
email: shij@math.wm.edu

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