Abstract
Let T ⊂ ℝ be a time-scale, with a = inf T, b = sup T. We consider the nonlinear boundary value problem - [p(t)uΔ(t)]Δ + q(t) uσ(t) = λf(t, uσ(t)), on T, u(a) = u(b) = 0, where λ ∈ ℝ+ :=[0, ∞), and f : T × ℝ → ℝ satisfies the conditions f(t, ξ) > 0, (t, ξ) ∈ T × ℝ, f(t, ξ) > fξ (t, ξ) ξ, (t, ξ) ∈ T × ℝM+. We prove a strong maximum principle for the linear operator defined by the left-hand side of (1), and use this to show that for every solution (λ, u) of (1)-(2), u is positive on T \ {a,b}. In addition, we show that there exists λmax > 0 (possibly λmax = ∞), such that, if 0 ≤ λ < λmax then (1)-(2) has a unique solution u(λ), while if λ ≥ λ max then (1)-(2) has no solution. The value of λmax is characterised as the principal eigenvalue of an associated weighted eigenvalue problem (in this regard, we prove a general existence result for such eigenvalues for problems with general, nonnegative weights).
Original language | English |
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Pages (from-to) | 491-504 |
Number of pages | 14 |
Journal | Journal of Mathematical Analysis and Applications |
Volume | 300 |
Issue number | 2 |
DOIs | |
Publication status | Published - 15 Dec 2004 |
Keywords
- Nonlinear boundary value problem
- Positive solutions
- Strong maximum principle
- Time-scales
- Weighted eigenvalue problem
ASJC Scopus subject areas
- Analysis
- Applied Mathematics