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arXiv · 2004.01284

Past and recent contributions to indefinite sublinear elliptic problems

Uriel Kaufmann·Humberto Ramos Quoirin·Kenichiro Umezu
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Abstract

We review the indefinite sublinear elliptic equation $-\Delta u=a(x)u^{q}$ in a smooth bounded domain $\Omega\subset\mathbb{R}^{N}$, with Dirichlet or Neumann homogeneous boundary conditions. Here $0<q<1$ and $a$ is continuous and changes sign, in which case the strong maximum principle does not apply. As a consequence, the set of nonnegative solutions of these problems has a rich structure, featuring in particular both dead core and/or positive solutions. Overall, we are interested in sufficient and necessary conditions on $a$ and $q$ for the existence of positive solutions. We describe the main results from the past decades, and combine it with our recent contributions. The proofs are briefly sketched.

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Uriel Kaufmann, Humberto Ramos Quoirin, Kenichiro Umezu. 2020-04-02. Past and recent contributions to indefinite sublinear elliptic problems. https://arxiv.org/abs/2004.01284

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