| Summary: | In this paper, we investigate the existence, multiplicity and uniqueness of positive solutions for the following system of $n$th-order nonlinear boundary value problems
\[\begin{cases}
u^{(n)}(t)+f(t,u(t),v(t))=0,0<t<1,\\v^{(n)}(t)+g(t,u(t),v(t))=0, 0<t<1,\\
u(0)=u'(0)=\ldots=u^{(n-2)}(0)=u(1)=0,\\
v(0)= v'(0)=\ldots=v^{(n-2)}(0)=v(1)=0.
\end{cases}\]
Based on a priori estimates achieved by using Jensen's integral inequality, we use fixed point index theory to establish our main results. Our assumptions on the nonlinearities are mostly formulated in terms of spectral radii of associated linear integral operators. In addition, concave and convex functions are utilized to characterize coupling behaviors of $f$ and $g$, so that we can treat the three cases: the first with both superlinear, the second with both sublinear, and the last with one superlinear and the other sublinear.
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