The Existence And Nodal Character Of Solutions For Laplace Equation With Critical Sobolev And Hardy Exponents

This paper is concerned with the existence and nodal character of the nontrivial solutions for the following equations involving critical Sobolev and Hardy exponentswhere 2~* =2N/N-2 is the critical Sobolev exponent for the embedding H_r~1(R~N)→L~2~*(R~N). f(t) is a given function satisfying some assumptions. The main results obtained in this paper are as follows:(i) There exists a nontrivial solution of equation (1) for N ≥ 4.(ii) There exists at least a pair of nontrivial solutions u_k~+, u_k~- of equation (1) for each k ∈ N ∪ {0}, both u_k~+ and u_k~- possess exactly k nodes for suitable positive numbers μ for N ≥ 6.