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Layer and stable solutions to a nonlocal model

In celebration of Professor Vladimír Šverák's 65th birthday

This research is partially supported by Simons travel grant (formerly collaboration grant) #947054 and NSF grant DMS 2306393

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  • We study the layer and stable solutions of nonlocal problem

    $ \begin{equation*} -\Delta u+F'(u)\left( -\Delta \right) ^{s}F(u)+G'(u) = 0\text{ in }\mathbb{R}^{n} \end{equation*} $

    where $ F\in C_{{\text{loc}}}^2( \mathbb R) $ satisfies $ F(0) = 0 $ and $ G $ is a double well potential. For $ n = 2,s>0 $ and $ n = 3, $ $ s\geq 1/2, $ we establish the 1-d symmetry of layer solutions for this equation. When $ n = 2 $ and $ F' $ is bounded away from zero, we prove the 1-d symmetry of stable solutions for this equation. Using a different approach, we also prove the 1-d symmetry of stable solutions for

    $ \begin{equation*} F'(u)\left( -\Delta \right) ^{s}F(u)+G'(u) = 0\text{ in }\mathbb{R}^{2}. \end{equation*} $

    Mathematics Subject Classification: Primary: 35B06, 35B38; Secondary: 35J15.

    Citation:

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