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Full-Text Articles in Applied Mathematics
Positive Solution Curves Of Semipositone Problems With Concave Nonlinearities, Alfonso Castro, Sudhasree Gadam, Ratnasingham Shivaji
Positive Solution Curves Of Semipositone Problems With Concave Nonlinearities, Alfonso Castro, Sudhasree Gadam, Ratnasingham Shivaji
All HMC Faculty Publications and Research
We consider the positive solutions to the semilinear equation:
-Δu(x) = λf(u(x)) for x ∈ Ω
u(x) = 0 for x ∈ ∂Ω
where Ω denotes a smooth bounded region in RN (N > 1) and λ > 0. Here f :[0, ∞)→R is assumed to be monotonically increasing, concave and such that f(0) < 0 (semipositone). Assuming that f'(∞) ≡ lim t→∞ f'(t) > 0, we establish the stability and uniqueness of large positive solutions in terms of (f(t)/t)'. When Ω is a ball, we determine the exact number of positive solutions for each λ > 0. We also obtain the geometry of the branches of positive solutions completely and establish how …
Existence Results For Semipositone Systems, V. Anuradha, Alfonso Castro, Ratnasingham Shivaji
Existence Results For Semipositone Systems, V. Anuradha, Alfonso Castro, Ratnasingham Shivaji
All HMC Faculty Publications and Research
We study existence of positive solutions to the coupled-system of boundary value problems of the form
-Δu(x) = λf(x,u,v); x ∈ Ω
-Δv(x) = λg(x,u,v); x ∈ Ω
u(x) = 0 = v(x); x ∈ ∂Ω
where λ > 0 is a parameter, Ω is a bounded domain in R^N; N ≥ 1 with a smooth boundary ∂Ω and f,g are C^1 function with at least one of f(x_0,0,0) or g(x_0,0,0) being negative for some x_0 ∈ Ω (semipositone). We establish our existence results using the method of sub-super solutions. We also discuss non-existence results for λ small.