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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.