Research Article Open Access

Positive Solutions for Some Weighted Elliptic Problems

Hanadi Zahed1
  • 1 Taibah University, Saudi Arabia
Journal of Mathematics and Statistics
Volume 16 No. 1, 2020, 125-132

DOI: https://doi.org/10.3844/jmssp.2020.125.132

Submitted On: 5 March 2020
Published On: 21 July 2020

How to Cite: Zahed, H. (2020). Positive Solutions for Some Weighted Elliptic Problems. Journal of Mathematics and Statistics, 16(1), 125-132. https://doi.org/10.3844/jmssp.2020.125.132

Abstract

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  = < ∞. We will prove that the problem (P) has a positive solution for 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.


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Keywords

  • Asymptotically Linear Nonlinearity
  • Mountain Pass Theorem
  • Weighted Problem
  • Palais Smale Condition