Progress in Fractional Differentiation & Applications
New Regularity Estimates for the Extremal Solution of Nonlinear Elliptic Problems Involving the Fractional Laplacian with General Nonlinearity
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 .
Digital Object Identifier (DOI)
"New Regularity Estimates for the Extremal Solution of Nonlinear Elliptic Problems Involving the Fractional Laplacian with General Nonlinearity,"
Progress in Fractional Differentiation & Applications: Vol. 9:
2, Article 8.
Available at: https://digitalcommons.aaru.edu.jo/pfda/vol9/iss2/8