Non-negative and positive solutions for some indefinite sublinear elliptic problems

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.