Elliptic Boundary Value Problems Involving Critical Sobolev Exponents And Singular Terms

In this paper, we discuss the existence of solutions for the boundary value problems where is a bounded smooth domain, div is the p-Laplacian andp is the critical Sobolev exponent, is a parameter, R is a function some conditions to be specified. Problem (Q)p has the following characters: it is singular at 0 and it has a term with critical Sobolev exponent.Some scholars have studied Problem (Q)p for the case p = 2 (see [1, 2, 3] and the references therein). For example, in [1], H. Brezis and L. Nirenberg studied the existence of the solutions for Problem (Q)i in the case of \JL = 0, h(x,u) = Xu in 1983, and they pointed out that the existence of the solutions depends on the dimensions of the spaces and the range of A (see [1]). In [2], E. Jannelli obtained that the existence of the positive solutions for Problem (Q)z, when f2(x,u) = where is the first eigenvalue of the linear operator (I is the identical operator) under the 0-Dirichlet boundary value condition, and the author extend the corresponding results in paper [1] to the case n ^ 0 for Problem (Q)2- In 2001, A. Ferrero and F. Gazzola' 1 obtained the existence and non-existence results of nontrivial solutions for Problem (Q)? under conditions that f2(x,u) and A > 0. They also cosidered the more general case where g is a function satisfying some conditions. In [3], the authors left an open question: does Problem (Q)z have a nontrivial solution in the case n < 0 (see [3, p.519])For the case 1 < p < N and p 2, the work on Problem (Q)p is still bound in the following two cases: i) fp(x,u) = 0. In this case, if fi is a star-shaped domain, then there isn't nontrivial solution for Problem In this case, the existence of nontrivial solution for Problem (Q)p depends on the perturbative term fp(x,u) (see [5, 20]).In this paper, we obtain the existence of positive solutions for Problem (Q)2 by using the different ways from papers [1, 2, 3]; in particular, we give a positive answer for the open question in [3] (or see above). Moreover, by using the Hardy inequality, variational method and the concentration compactess principle of P. L. Lions ( see [8]), we also obtain the existence of nontrivial solutions for Problem (Q}p when fp(x,u) ^ 0 and p > 1. As far as we know, (Q)p in the case fp(x, u) ^ 0 and p > l,p ^ 2 has not been studied.