The effect of domain topology on the number of positive solutions for singular elliptic problems involving the Caffarelli-Kohn-Nirenberg inequalities

摘要

In this paper we consider the critical singular equation involving the Caffarelli-Kohn-Nirenberg inequalities of the type [GRAPHICS] Here ohm is a bounded domain with smooth boundary in R-N and contains 0 in its interior, 0 <= mu < (root(mu) over bar - a)(2), (mu) over bar = ((2)/(N - 2)2), N >= 3, a <= b < a + 1, a <= d < a + 1, = p (a, b) (triangle)(=) (N - 2(1 + a - b))/(2N), D = D(a, d) (triangle)(=) (N - 2(1 + a - d)) (2N), lambda is a positive parameter and 2 <= q < D. By Lusternik-Schnirelmann category theory, we prove that the problem (P-1) has at least cat(ohm) positive solutions. (c) 2007 Elsevier Inc. All rights reserved.