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

## TL;DR

This paper studies saddle-shaped solutions to a class of semilinear integro-differential equations, characterizing the kernels for which a related operator is positive, and proving existence and energy estimates for these solutions.

## Contribution

It introduces a new operator for odd solutions with respect to the Simons cone, characterizes kernels ensuring positivity, and establishes existence and energy bounds for saddle-shaped solutions.

## Key findings

- Characterized kernels for positivity of the new operator.
- Proved energy estimates for saddle-shaped solutions.
- Established existence of saddle-shaped solutions.

## Abstract

This is the first of two papers concerning saddle-shaped solutions to the semilinear equation $L_K u = f(u)$ in $\mathbb{R}^{2m}$, where $L_K$ is a linear elliptic integro-differential operator and $f$ is of Allen-Cahn type. Saddle-shaped solutions are doubly radial, odd with respect to the Simons cone $\{(x', x'') \in \mathbb{R}^m \times \mathbb{R}^m \, : \, |x'| = |x''|\}$, and vanish only on this set. By the odd symmetry, $L_K$ coincides with a new operator $L_K^{\mathcal{O}}$ which acts on functions defined only on one side of the Simons cone, $\{|x'|>|x''|\}$, and that vanish on it. This operator $L_K^{\mathcal{O}}$, which corresponds to reflect a function oddly and then apply $L_K$, has a kernel on $\{|x'|>|x''|\}$ which is different from $K$. In this first paper, we characterize the kernels $K$ for which the new kernel is positive and therefore one can develop a theory on the saddle-shaped solution. 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 convex function. Assuming this, we prove an energy estimate for doubly radial odd minimizers and 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 linearized operator, and their uniqueness.