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This is the first of two papers concerning saddle-shaped solutions to thesemilinear equation $L_K u = f(u)$ in $\mathbb{R}^{2m}$, where $L_K$ is alinear elliptic integro-differential operator and $f$ is of Allen-Cahn type.  Saddle-shaped solutions are doubly radial, odd with respect to the Simonscone $\{(x', x'') \in \mathbb{R}^m \times \mathbb{R}^m \, : \, |x'| = |x''|\}$,and vanish only on this set. By the odd symmetry, $L_K$ coincides with a newoperator $L_K^{\mathcal{O}}$ which acts on functions defined only on one sideof the Simons cone, $\{|x'|>|x''|\}$, and that vanish on it. This operator$L_K^{\mathcal{O}}$, which corresponds to reflect a function oddly and thenapply $L_K$, has a kernel on $\{|x'|>|x''|\}$ which is different from $K$.  In this first paper, we characterize the kernels $K$ for which the new kernelis positive and therefore one can develop a theory on the saddle-shapedsolution. The necessary and sufficient condition for this turns out to be that$K$ is radially symmetric and $\tau\mapsto K(\sqrt \tau)$ is a strictly convexfunction.  Assuming this, we prove an energy estimate for doubly radial odd minimizersand the existence of saddle-shaped solution. In a subsequent article, part II,further qualitative properties of saddle-shaped solutions will be established,such as their asymptotic behavior, a maximum principle for the linearizedoperator, and their uniqueness.

Semilinear integro-differential equations, I: odd solutions with respect to the Simons cone