Existence and regularity of solutions for a class of fractional Laplacian problems

In this paper, we first establish a modified Marcinkiewicz interpolation result, and using this result, we obtain a new regularity result for a fractional Laplacian problem on a bounded C1,1-domain Ω in RN, and we also obtain a new regularity result for then fractional problem on unbounded Ω=RN by using the Stampacchia truncation method. Next, by the Leray-Schauder fixed point theorem we obtain the existence of solutions for a class of fractional Laplacian problems with weak growth conditions on the nonlinearities. Finally, as an application, we prove the existence of positive solutions for a fractional Laplacian problem which has an exponential growth nonlinearity.