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In this paper, we study the existence and nonexistence of positive bounded solutions of the integral equation \(u=\lambda V(af(u))\), where λ is a positive parameter, a is a nontrivial nonnegative measurable function with bounded potential and V belongs to a class of positive kernels that contains in particular the potential kernel of the classical Laplacian \(V=(-\Delta)^{-1}\) or \(V=(\frac{\partial}{\partial t }-\Delta)^{-1}\) or the inverse of the polyharmonic Laplacian \((-\Delta)^{m}\), \(m\geq2\).

Existence and global behavior of positive solutions for some eigenvalue problems