Resumen de Dirichlet Problems for Fractional Laplace Equations with Singular Nonlinearity

Jian Wang, Zhuoran Du

• We consider positive solutions of the Dirichlet problem for the fractional Laplace equation with singular nonlinearity ⎧ ⎪⎨ ⎪⎩ (−)s u(x) = K(x)u−α(x) + μu p−1(x) in , u > 0 in , u = 0 in c := RN \ , where s ∈ (0, 1), α > 0 and ⊂ RN is a bounded domain with smooth boundary ∂ and N > 2s. 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 C1,1 loc ∩ L∞ solutions are also established for subcritical exponent p when the domain is a ball. Nonexistence of C1,1 ∩ L∞ solutions are obtained for star-shaped domain under a condition of K.