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

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 $\{(x^{\prime },x^{\prime \prime })\in {\mathbb{R}}^m\times {\mathbb{R}}^m:|x^{\prime }|=|x^{\prime \prime }|\}$, 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, $\{|x^{\prime }|>|x^{\prime \prime }|\}$, 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 $\{|x^{\prime }|>|x^{\prime \prime }|\}$ 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.