Let $J \in C(\mathbb{R})$, $J\ge 0$, $\int_{\tiny$\mathbb{R}$} J = 1$ andconsider the nonlocal diffusion operator $\mathcal{M}[u] = J \star u - u$. Westudy the equation $\mathcal{M} u + f(x,u) = 0$, $u \ge 0$, in $\mathbb{R}$,where $f$ is a KPP-type nonlinearity, periodic in $x$. We show that theprincipal eigenvalue of the linearization around zero is well defined and thata nontrivial solution of the nonlinear problem exists if and only if thiseigenvalue is negative. We prove that if, additionally, $J$ is symmetric, thenthe nontrivial solution is unique.
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机译：令$ J \ in C（\ mathbb {R}）$，$ J \ ge 0 $，$ \ int _ {\ tiny $ \ mathbb {R} $} J = 1 $并考虑非局部扩散算子$ \ mathcal {M } [u] = J \ star u-u $。对方程$ \ mathcal {M} u + f（x，u）= 0 $，$ u \ ge 0 $，在$ \ mathbb {R} $中进行计算，其中$ f $是KPP型非线性，周期为$ x $。我们表明，围绕零的线性化的主要特征值已得到很好的定义，并且当且仅当该特征值为负时，才存在非线性问题的非平凡解。我们证明如果另外$ J $是对称的，那么非平凡解是唯一的。
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