Positivity results for indefinite sublinear elliptic problems via a continuity argument

TL;DR

This paper proves positivity of solutions for certain indefinite sublinear elliptic problems where the strong maximum principle fails, using a continuity argument combined with variational and sub-supersolution methods.

Contribution

It introduces a novel continuity-based approach to establish positivity in problems lacking the strong maximum principle, including both Dirichlet and Neumann boundary conditions.

Findings

Nontrivial nonnegative solutions are positive under the studied conditions.

Existence and uniqueness results are derived for the class of problems.

Positivity results are extended to indefinite concave-convex problems.

Abstract

We establish a positivity property for a class of semilinear elliptic problems involving indefinite sublinear nonlinearities. Namely, we show that any nontrivial nonnegative solution is positive for a class of problems the strong maximum principle does not apply to. Our approach is based on a continuity argument combined with variational techniques, the sub and supersolutions method and some a priori bounds. Both Dirichlet and Neumann homogeneous boundary conditions are considered. As a byproduct, we deduce some existence and uniqueness results. Finally, as an application, we derive some positivity results for indefinite concave-convex type problems.