Dirichlet Problems for Fractional Laplace Equations with Singular Nonlinearity.

We consider positive solutions of the Dirichlet problem for the fractional Laplace equation with singular nonlinearity (- Δ) s u (x) = K (x) u - α (x) + μ u p - 1 (x) in Ω , u > 0 in Ω , u = 0 in Ω c : = R N \ Ω , <graphic href="12346_2023_900_Article_Equ42.gif"></graphic> where s ∈ (0 , 1) , α > 0 and Ω ⊂ R N is a bounded domain with smooth boundary ∂ Ω and N > 2 s. Under some appropriate assumptions of α , p , μ and K, we obtain the existence of multiple weak solutions, and among them, including the minimal solution and a ground state solution. Radial symmetry of C loc 1 , 1 ∩ L ∞ solutions are also established for subcritical exponent p when the domain is a ball. Nonexistence of C 1 , 1 ∩ L ∞ solutions are obtained for star-shaped domain under a condition of K. [ABSTRACT FROM AUTHOR]