Abstract
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 $(-\Delta)^{m}$ , $m\geq2$ .
Highlights
Let be a smooth domain of Rn (n ≥ ) and f : [, ∞) → (, ∞) be a nondecreasing continuous function
We are interested in the existence of a positive solution of the following integral equation: u = λV af (u), ( . )
Where λ is a positive parameter, a is a nontrivial nonnegative measurable function satisfying an appropriate condition and V belongs to a class of positive kernels that contains in particular the potential kernel of the classical Laplacian V =
Summary
Let be a smooth domain of Rn (n ≥ ) and f : [ , ∞) → ( , ∞) be a nondecreasing continuous function. We are interested in the existence of a positive solution of the following integral equation: u = λV af (u) , ) has a positive solution for < λ < λ under the following hypotheses on the functions a and f : (H ) f : [ , ∞) → ( , ∞) is a nondecreasing continuous function.
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