Positive Solutions for Some Weighted Elliptic Problems
- 1 Taibah University, Saudi Arabia
Published On: 21 July 2020
Copyright: © 2020 Hanadi Zahed. This is an open access article distributed under the terms of the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original author and source are credited.
In this study, we study the existence and the nonexistence of positive solutions for the following nonlinear elliptic problems:
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.
- Asymptotically Linear Nonlinearity
- Mountain Pass Theorem
- Weighted Problem
- Palais Smale Condition