POSITIVE SOLUTIONS FOR ELLIPTIC EQUATIONS RELATED TO THE CAFFARELLI-KOHN-NIRENBERG INEQUALITIES

In this paper, we are concerned with the following elliptic problems which are related to the well-known Caffarelli-Kohn-Nirenberg inequalities: -div(vertical bar x vertical bar-(2a)del u) - lambda vertical bar x vertical bar(-eE)u = vertical bar x vertical bar(-bp)vertical bar u vertical bar(p-2)u + eta vertical bar x vertical bar(-dD)vertical bar u vertical bar(q-2)u in Omega u = 0 on partial derivative Omega, (0.1) where a = b < 0,p = 2N/N-2 (N >= 3), a <= d <= a + 1, a <= e <= a +1, D = 2N/N-2-2(a-d), E = 2N/N-2-2(a-e), 2 < q < D, lambda and eta are real constants. We obtain positive solutions for problem (0.1). Moreover, we establish a corresponding Pohozaev identity for problem (0.1), from which, the nonexistence of positive solutions for problem ( 0.1) is obtained.