Progress in Fractional Differentiation & Applications

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In this paper, we consider the fractional Laplacian equation (−∆ )s u = λ f (u) on a smooth bounded domain Ω ⊂ RN with zero Dirichlet boundary condition, where λ > 0 is a parameter and s ∈ (0, 1). At first, under typical assumptions on f and whenever it is a convex function we provide some new regularity results for the extremal solution u∗. Unlike the known results, our contributions do not require the function f(t)f′′(t) tends to a limit at infinity. After that, we rule out the convexity assumption on f and prove some f′(t)2 new L∞ estimates for the extremal solution u∗ under some suitable conditions on the non-linearity f , in this case, proving the results require neither convexity assumption on the non-linearity f nor the domain Ω .

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