1. Non-negative solutions of a sublinear elliptic problem.
- Author
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López-Gómez, Julián, Rabinowitz, Paul H., and Zanolin, Fabio
- Abstract
In this paper, the existence of solutions, (λ , u) , of the problem - Δ u = λ u - a (x) | u | p - 1 u in Ω , u = 0 on ∂ Ω , is explored for 0 < p < 1 . When p > 1 , it is known that there is an unbounded component of such solutions bifurcating from (σ 1 , 0) , where σ 1 is the smallest eigenvalue of - Δ in Ω under Dirichlet boundary conditions on ∂ Ω . These solutions have u ∈ P , the interior of the positive cone. The continuation argument used when p > 1 to keep u ∈ P fails if 0 < p < 1 . Nevertheless when 0 < p < 1 , we are still able to show that there is a component of solutions bifurcating from (σ 1 , ∞) , unbounded outside of a neighborhood of (σ 1 , ∞) , and having u ⪈ 0 . This non-negativity for u cannot be improved as is shown via a detailed analysis of the simplest autonomous one-dimensional version of the problem: its set of non-negative solutions possesses a countable set of components, each of them consisting of positive solutions with a fixed (arbitrary) number of bumps. Finally, the structure of these components is fully described. [ABSTRACT FROM AUTHOR]
- Published
- 2024
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