On semilinear inequalities involving the Dunkl Laplacian and an inverse-square potential outside a ball

Abstract

Let Delta kbe the Dunkl generalized Laplacian operator associated with a root systemRofN,>= N2, anda nonnegative multiplicity functionkdefined onRand invariant by thefinite reflection groupW. In this study,we study the existence and nonexistence of weak solutions to the semilinear inequality-+>=Delta uu uk lambda xp2 divided by divided by divided by divided by inB\N1under the boundary condition >= u0on partial derivative B1, where>p1,>=-- +/lambda N gamma 22 42(), andB1is the open unitball ofN. Namely, we show that the dividing line with respect to existence and nonexistence is given by acritical exponent that depends on lambda,N, and gamma k(), where=& sum;is an element of+gamma kk alpha alpha R()()and+Ris the positive subsystem.