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arXiv · 1710.07802

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

Uriel Kaufmann·Humberto Ramos Quoirin·Kenichiro Umezu
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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.

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Uriel Kaufmann, Humberto Ramos Quoirin, Kenichiro Umezu. 2017-10-21. Loop type subcontinua of positive solutions for indefinite concave-convex problems. https://arxiv.org/abs/1710.07802

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