Existence Results for Non-Autonomous Elliptic Boundary Value Problems
Date
1994-07-08
Authors
Anuradha, V.
Dickens, S.
Shivaji, R.
Journal Title
Journal ISSN
Volume Title
Publisher
Southwest Texas State University, Department of Mathematics
Abstract
We study solutions to the boundary value problems
−∆u(x) = λf (x, u); x ∈ Ω
u(x) + α(x) ∂u(x) / ∂n = 0; x ∈ ∂Ω
where λ > 0, Ω is a bounded region in ℝN ; N ≥ 1 with smooth boundary ∂Ω, α(x) ≥ 0, n is the outward unit normal, and f is a smooth function such that it has either sublinear or restricted linear growth in u at infinity, uniformly in x. We also consider f such that f (x, u)u ≤ 0 uniformly in x, when |u| is large. Without requiring any sign condition on f (x, 0), thus allowing for both positone as well as semipositone structure, we discuss the existence of at least three solutions for given λ ∈ (λn, λn+1) where λk is the k-th eigenvalue of −∆ subject to the above boundary conditions. In particular, one of the solutions we obtain has non-zero positive part, while another has non-zero negative part. We also discuss the existence of three solutions where one of them is positive, while another is negative, for λ near λ1, and for λ large when f is sublinear. We use the method of sub-super solutions to establish our existence results. We further discuss non-existence results for λ small.
Description
Keywords
Elliptic boundary value problems, Semipositone
Citation
Anuradha, V., Dickens, S., & Shivaji, R. (1994). Existence results for non-autonomous elliptic boundary value problems. <i>Electronic Journal of Differential Equations, 1994</i>(04), pp. 1-10.
Rights
Attribution 4.0 International