On partially free boundary solutions for elliptic problems with non-Lipschitz nonlinearities

Abstract

We show that the elliptic equation with a non-Lipschitz right-hand side, −Δu=λ|u|β−1u−|u|α−1u with λ>0 and 0<α<β<1, considered on a smooth star-shaped domain Ω subject to zero Dirichlet boundary conditions, might possess a nonnegative ground state solution which violates Hopf’s maximum principle only on a nonempty subset Γ of the boundary ∂Ω such that Γ≠∂Ω.