On a Brezis-Oswald-type result for degenerate Kirchhoff problems

Abstract

In the present note we establish an almost-optimal solvability result for Kirchhoff-type problems of the following form$ \begin{cases} -M\big(\|\nabla u\|^2_{L^2(\Omega)}\big)\Delta u = f(x, u) & \text{in } \Omega , \\ u \geq 0, \, u\not\equiv 0 & \text{in } \Omega , \\ u = 0 & \text{on } \partial \Omega . \end{cases} $where $ f $ has sublinear growth and $ M $ is a non-decreasing map with $ M(0)\geq 0 $. Our approach is purely variational, and the result we obtain is resemblant to the one established by Brezis and Oswald (Nonlinear Anal., 1986) for sublinear elliptic equations.