Positive Solutions for Some Weighted Elliptic Problems

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 ℝ , N ≥ 2, a(x) is a smooth function on   and f(x, s) is asymptotically linear in s at infinity, that is  = ℓ < ∞. We will prove that the problem (P) has a positive solution for ℓ large enough and does not have positive solutions for ℓ 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.