Loop type subcontinua of positive solutions for indefinite concave-convex problems

TL;DR

This paper proves the existence of loop-shaped solution continua for indefinite concave-convex elliptic problems using bifurcation and topological methods, even when nonlinearities are not differentiable at zero.

Contribution

It introduces a novel approach combining regularization, a priori bounds, and topological methods to analyze non-differentiable nonlinearities in elliptic problems.

Findings

Existence of loop type subcontinua of solutions.

Positivity properties of solution subcontinua.

Extension of previous results to more general nonlinearities.

Abstract

We establish the existence of loop type subcontinua of nonnegative solutions for a class of concave-convex type elliptic equations with indefinite weights, under Dirichlet and Neumann boundary conditions. Our approach depends on local and global bifurcation analysis from the zero solution in a non-regular setting, since the nonlinearities considered are not differentiable at zero, so that the standard bifurcation theory does not apply. To overcome this difficulty, we combine a regularization scheme with a priori bounds, and Whyburn's topological method. Furthermore, via a continuity argument we prove a positivity property for subcontinua of nonnegative solutions. These results are based on a positivity theorem for the associated concave problem proved in [15], and extend previous results established in the powerlike case.