Keller-Osserman and Harnack type results for nonlinear elliptic PDE with unbounded ingredients

Abstract

We show that the classical Keller-Osserman theorem on the solvability of the equation $\mathcal{L}[u] = f(u)$ is valid when $\mathcal{L}$ is a general operator in divergence form with unbounded coefficients in the natural regime of local integrability. This has been open up to now, earlier results concerned operators with locally bounded ingredients. We also settle an open question from \cite{SS21} about the validity of the strong maximum principle for supersolutions of $\mathcal{L}[u] = f(u)$ under the optimal integral condition of V\'azquez. More generally, we obtain a Harnack inequality for positive solutions of this equation, which extends a result by V. Julin.