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Using numerical methods, we will show the existence of multiple solutions for the equation −∆u = λf (u) with Dirichlet boundary condition in a bounded domain Ω, where λ > 0 and f (u) is a superlinear function of u.
In this work we present a numerical approach for finding positive solutions for −∆u = λ(u + u 2 + u 3) for x ∈ Ω with Dirichlet boundary condition. We will show that in which range of λ, this problem achieves a numerical solution and what is the behavior of the branch of this solutions.
In a recent result (see Jaffar Ali and shivaji ), it was shown via the method of sub-super solutions that a semipositone problem with a sign changing weight has at least one positive solution. In this paper we want to investigate that solution numerically.
Using a numerical method based on sub-super solution, we will obtain positive solution to the coupled-system of boundary value problems of the form −∆u = λ 1 f (v) + µ 1 h(u) in Ω −∆v = λ 2 g(u) + µ 2 γ(v) in Ω u = 0 = v on ∂Ω where −∆ is the Laplacian operator λ 1 , λ 2 , µ 1 , µ 2 are nonnegative parameters, and Ω is a bounded region in R n , with smooth… (More)
Using a numerical method based on sub-super solution, we will obtain positive solution to the coupled-system of boundary value problems of the form −∆u(x) = λf (x, u, v) x ∈ Ω −∆v(x) = λg(x, u, v) x ∈ Ω u(x) = 0 = v(x) x ∈ ∂Ω where f , g are C 1 functions with at least one of f (x 0 , 0, 0) or g(x 0 , 0, 0) being negative for some x 0 ∈ Ω (semipositone).
Using two numerical methods, we will show the existence of multiple solutions for the well-known logistic equation Du 1⁄4 kgðxÞuð1 uÞ for x 2 X, with Dirichlet boundary condition. 2007 Elsevier Inc. All rights reserved.