Entire solutions to semilinear nonlocal equations in $R^2$

Xavier Ros-Oton (1) , Yannick Sire (2)

1 University of Texas at Austin [Austin]
2 I2M - Institut de Mathématiques de Marseille

We consider entire solutions to $L u= f(u)$ in $R^2$, where $L$ is a general nonlocal operator with kernel $K(y)$. Under certain natural assumtions on the operator $L$, we show that any stable solution is a 1D solution. In particular, our result applies to any solution $u$ which is monotone in one direction. Compared to other proofs of the De Giorgi type results on nonlocal equations, our method is the first successfull attempt to use the Liouville theorem approach to get flatness of the level sets.

Domaines: Equations aux dérivées partielles [math.AP]

Xavier Ros-Oton, Yannick Sire. Entire solutions to semilinear nonlocal equations in $R^2$. 2015. ⟨hal-01341881⟩