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dc.contributor.authorAnuradha, V. ( )
dc.contributor.authorDickens, S. ( )
dc.contributor.authorShivaji, R. ( )
dc.identifier.citationAnuradha, V., Dickens, S., & Shivaji, R. (1994). Existence results for non-autonomous elliptic boundary value problems. Electronic Journal of Differential Equations, 1994(04), pp. 1-10.en_US

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.

dc.format.extent10 pages
dc.format.medium1 file (.pdf)
dc.publisherSouthwest Texas State University, Department of Mathematicsen_US
dc.sourceElectronic Journal of Differential Equations, 1994, San Marcos, Texas: Southwest Texas State University and University of North Texas.
dc.subjectElliptic boundary value problemsen_US
dc.titleExistence Results for Non-Autonomous Elliptic Boundary Value Problemsen_US
dc.rights.licenseCreative Commons License
This work is licensed under a Creative Commons Attribution 4.0 International License.



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