Loop Type Subcontinua of Positive Solutions for Indefinite Concave-Convex Problems

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 nonregular 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 by us, and extend previous results established in the powerlike case.

Keywords

• concave-convex problem
• positive solution
• indefinite nonlinearity
• loop bifurcation
• 35j25
• 35j61
• 35b32