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arXiv · 2312.00998

On Dancer's conjecture for stable solutions with sign-changing nonlinearity

Yong Liu·Kelei Wang·Juncheng Wei·Ke Wu
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Abstract

We establish a Liouville type result for stable solutions for a wide class of second order semilinear elliptic equations in $\mathbb{R}^{n}$ with sign-changing nonlinearity $f$. Under the hypothesis that the equation does not have any nonconstant one dimensional stable solution, and a further nondegeneracy condition of $f$ at its zero points, we show that in any dimension, stable solutions of the equation must be constant. This partially answers a question raised by Dancer.

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Yong Liu, Kelei Wang, Juncheng Wei, Ke Wu. 2023-12-02. On Dancer's conjecture for stable solutions with sign-changing nonlinearity. https://arxiv.org/abs/2312.00998

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