2024/03/07 by López-Gómez, Julián, Rabinowitz, Paul H., Zanolin, Fabio
#35B09 #35B25 #35B32 #35J15 #Analysis of PDEs (math.AP) #FOS: Mathematics
paper · doi:10.48550/arxiv.2403.04396
In this paper the existence of solutions, (λ,u), of the problem -Δu=λu -a(x)|u|p-1u \hboxin Ω, u=0 \hboxon ∂Ω, is explored for 0 < p < 1. When p>1, it is known that there is an unbounded component of such solutions bifurcating from (σ1, 0), where σ1 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 (σ1, ∞), unbounded outside of a neighborhood of (σ1, ∞), and having u \gneq 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.