On the weighted dirichlet eigenvalues of hardy operators involving critical gradient terms

The purpose of this paper is to show the existence of Dirichlet eigenvalues {lambda mu 1,mu 2,i}i and their lower bounds for the Dirichlet Hardy-Leray operator with a critical gradient term, i.e. -Delta u + mu 1|x|-2x center dot backward difference u + mu 2|x|-2u = lambda V u in S2, u = 0 on partial differential S2, under the one of the assumptions: (i) mu 2 - N-2 2 mu 1 > - (N-2)2 4 , S2 is a connected C2 domain containing the origin (may be unbounded) and V is a positive potential in C alpha loc(S2 over bar \{0})boolean AND Lqloc(S2) boolean AND LN2 (S2) with alpha is an element of (0,1) and q > N2 ; (ii) mu 2- N-2 2 mu 1 =- (N-2)2 4 , S2 is a connected C2 bounded domain containing the origin and V is a positive potential in C alpha loc( over bar S2 \ {0}) boolean AND Lqloc(S2) with alpha is an element of (0,1) and q > N2 . Our lower bounds of {lambda mu 1,mu 2,i}i is motivated by the Li-Yau's method for mu 1 = mu 2 = 0. When mu 2 - N-2 2 mu 1 > - (N-2)2 4 and by the Karachalios' method for mu 2 - N-2 2 mu 1 >= - (N-2)2 4 in a general domain which could be bounded or unbounded.