Non-existence of dead cores in fully nonlinear elliptic models

Abstract

We investigate non-existence of non-negative dead core solutions for the problem [Formula: see text] Here, [Formula: see text] is a bounded smooth domain, [Formula: see text] is a fully nonlinear elliptic operator, [Formula: see text] is a sign-changing weight, [Formula: see text], and [Formula: see text]. We show that this problem has no nontrivial dead core solutions if either [Formula: see text] is close enough to [Formula: see text] or the negative part of [Formula: see text] is sufficiently small. In addition, we obtain the existence and uniqueness of a positive solution under these conditions on [Formula: see text] and [Formula: see text]. Our results extend previous ones established in the semilinear case, and are new even for the simple model [Formula: see text], where [Formula: see text] is a uniformly elliptic and non-negative matrix.