Positive Solutions for Some Weighted Elliptic Problems

摘要

In this study, we study the existence and the nonexistence of positive solutions for the following nonlinear elliptic problems:  (P)where, Ω is a regular bounded domain in a?? , N ≥ 2, a(x) is a smooth function on   and f(x, s) is asymptotically linear in s at infinity, that is  = a?? & ∞. We will prove that the problem (P) has a positive solution for a?? large enough and does not have positive solutions for a?? less than the first eigenvalue of the operator. We prove also that the method works for the case when f(x, s) is sub-critical and super-linear at +∞.2010 Mathematics Subject classification: 35J05, 35J65, 35J20, 35J60, 35K57, 35J70.