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In this work, we are mainly concerned with the positive solutions for the 2nth-order p-Laplacian boundary-value problem $$displaylines{ -(((-1)^{n-1}x^{(2n-1)})^{p-1})' =f(t,x,x',ldots,(-1)^{n-1}x^{(2n-2)},(-1)^{n-1}x^{(2n-1)}),cr x^{(2i)}(0)=x^{(2i+1)}(1)=0,quad (i=0,1,ldots,n-1), }$$ where $nge 1$ and $fin C([0,1]imes mathbb{R}_+^{2n}, mathbb{R}_+)(mathbb{R}_+:=[0,infty))$. To overcome the difficulty resulting from all derivatives, we first convert the above problem into a boundary value problem for an associated second order integro-ordinary differential equation with p-Laplacian operator. Then, by virtue of the classic fixed point index theory, combined with a priori estimates of positive solutions, we establish some results on the existence and multiplicity of positive solutions for the above problem. Furthermore, our nonlinear term f is allowed to grow superlinearly and sublinearly.