Blow-up estimates and a priori bounds for the positive solutions of a class of superlinear indefinite elliptic problems

Abstract

In this paper we find out some new blow-up estimates for the positive explosive solutions of a paradigmatic class of elliptic boundary value problems of superlinear indefinite type. These estimates are obtained by combining the scaling technique of Gidas–Spruck together with a generalized De Giorgi–Moser weak Harnack inequality found, very recently, by Sirakov (2020; 2022). In a further step, based on a comparison result of Amann and López-Gómez (1998), we will show how these bounds provide us with some sharp a priori estimates for the classical positive solutions of a wide variety of superlinear indefinite problems. It turns out that this is the first general result where the decay rates of the potential in front of the nonlinearity ($a\left(x\right)$ in (1.1)) do not play any role for getting a priori bounds for the positive solutions when $N\ge 3$.