Non-negative and positive solutions for some indefinite sublinear elliptic problems
Year: 2026
Type: article
Abstract: We study the existence of non-negative and positive solutions to the indefinite sublinear elliptic problem \(\ -\Delta u=\lambda u+m(x)\left \vert u\right \vert ^{\alpha -1}u\) in Ω, \(u=0\) on ∂ Ω, where Ω is a smooth bounded domain in \(\mathbb{R}^{N}\) , \(0<\alpha <1\) , m is a bounded changing sign weight and λ is a real parameter. Existence of non-negative solutions was considered by Brown. When \(\lambda =0\) existence of positive solutions was studied by Hernández-Mancebo-Vega and Godoy-Kaufmann. We extend and improve these results, obtaining linear stability results as well. We use variational methods (Nehari manifold) for the existence of non-negative solutions and bifurcation at infinity for the existence of positive solutions. In addition, we prove the existence of solutions with compact support under suitable additional assumptions: mainly by assuming that \(m(x)<0\) in a large region near the boundary ∂ Ω. We apply the method of local super and subsolutions to obtain suitable barrier functions, which now have some constraint on the radius of the ball \(B_{R}(x_{0})\) and on the maximum height when λ ≥ \(\lambda _{1}\) , the first eigenvalue for the Laplacian operator on Ω with zero Dirichlet boundary conditions. We also construct some global super and subsolutions with compact support for the case in which Ω is a ball. Finally, we analyze some applications of the Pohozaev identity to determine the non-existence of such solutions.
Source: Boundary Value Problems
Authors: J. I. Díaz, J. Hernández, Y. Ilyasov
Language: English
Subfield: Applied Mathematics
Field: Mathematics
Domain: Physical Sciences
Open Access status: gold
Estimated APC paid: $1,890
OpenAlex ID: W7118192448