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Qualitative properties of positive solutions for mixed integro-differential equations

  • * Corresponding author: Y. Wang

    * Corresponding author: Y. Wang

P. Felmer is supported by Fondecyt Grant 1110291 and BASAL-CMM projects. Y. Wang is supported by NNSF of China, No: 11661045 and by the Jiangxi Provincial Natural Science Foundation, No: 20181BAB211002

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  • This paper is concerned with the qualitative properties of the solutions of mixed integro-differential equation

    $\begin{equation}\left\{ \begin{array}{l} (-\Delta)_x^{α} u+(-\Delta)_y u+u = f(u) \ \ \ \ {\rm in}\ \ {\mathbb{R}}^N×{\mathbb{R}}^M,\\ u>0\ \ {\rm{in}}\ {\mathbb{R}}^N\times{\mathbb{R}}^M,\ \ \ \lim_{|(x,y)|\to+\infty}u(x,y) = 0, \end{array} \right.\;\;\;\left( {0.1} \right)\end{equation}$

    with $N≥ 1$, $M≥ 1$ and $α∈ (0,1)$. We study decay and symmetry properties of the solutions to this equation. Difficulties arise due to the mixed character of the integro-differential operators. Here, a crucial role is played by a version of the Hopf's Lemma we prove in our setting. In studying the decay, we construct appropriate super and sub solutions and we use the moving planes method to prove the symmetry properties.

    Mathematics Subject Classification: 35R11, 35B06, 35B40, 35B50.


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