Liouville-type theorems for unbounded solutions of elliptic equations in half-spaces

Boyan Sirakov; Philippe Souplet

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arXiv · 2002.07247

Liouville-type theorems for unbounded solutions of elliptic equations in half-spaces

Boyan Sirakov·Philippe Souplet

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Abstract

We prove that the Dirichlet problem for the Lane-Emden equation in a half-space has no positive solutions which grow at most like the distance to the boundary to a power given by the natural scaling exponent of the equation; in other words, we rule out {\it type~I grow-up} solutions. Such a nonexistence result was previously available only for bounded solutions, or under a restriction on the power in the nonlinearity. Instrumental in the proof are local pointwise bounds for the logarithmic gradient of the solution and its normal derivative, which we also establish.

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Boyan Sirakov, Philippe Souplet. 2020-02-17. Liouville-type theorems for unbounded solutions of elliptic equations in half-spaces. https://arxiv.org/abs/2002.07247

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