Sufficient conditions for the existence and asymptotic behaviour of solution to a quasi-linear elliptic problem

Abstract

In this article, we combine the existing regularity theory, perturbation method and the lower and upper solutions method to study the existence and asymptotic behaviour of positive solution to a boundary value problem for the p-Laplacian operator. More exactly, we study the existence and asymptotic behaviour of the positive solution to a quasi-linear elliptic problem of the form−Δ _p u=λ a(x)g(u) in D′(Ω), u>0 in Ω, lim _{x→∂ Ω} u(x)=0. Under some conditions on a and g, we show that there is a non-negative number Λ₀ such that for all λ∈(0, Λ₀], the problem has a solution u _λ in the sense of distribution, which is bounded from above by some positive numbers μ(λ). Such estimates and the asymptotic behaviour are important in computer programs when we know an algorithm for determining the solution.

Keywords:

• quasi-linear problem
• existence
• asymptotic behaviour

2010 Mathematics Subject Classifications :

• 35J25
• 35J60
• 35J62
• 35J75
• 35J66

Acknowledgements

The author would like to thank the referee for valuable suggestions. The work is supported by CNCSIS-UEFISCSU, project number PN II – IDEI 1080/2008 from West University of Timisoara.