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arXiv · 2204.10161

On the nodal set of solutions to some sublinear equations without homogeneity

Nicola Soave·Giorgio Tortone
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Abstract

We investigate the structure of the nodal set of solutions to an unstable Alt-Phillips type problem \[ -\Delta u = \lambda_+(u^+)^{p-1}-\lambda_-(u^-)^{q-1} \] where $1 \le p 0$, $\lambda_- \ge 0$. The equation is characterized by the sublinear inhomogeneous character of the right hand-side, which makes difficult to adapt in a standard way classical tools from free-boundary problems, such as monotonicity formulas and blow-up arguments. Our main results are: the local behavior of solutions close to the nodal set; the complete classification of the admissible vanishing orders, and estimates on the Hausdorff dimension of the singular set, for local minimizers; the existence of degenerate (not locally minimal) solutions.

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Nicola Soave, Giorgio Tortone. 2022-04-21. On the nodal set of solutions to some sublinear equations without homogeneity. https://doi.org/10.1007/s00205-024-01970-4

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