Steady states of FitzHugh-Nagumo-type systems with sign-changing coefficients

Abstract

We establish existence and multiplicity results for steady-state solutions of spatially heterogeneous FitzHugh-Nagumo-type systems, extending the existing theory from constant to variable coefficients that may change sign. Specifically, we study the system Specifically, we study the system \begin{align*} -\Delta u + a(x)v &= f(x,u) && \text{in } \mathbb{R}^N, \\ -\Delta v + b(x)v &= c(x)u && \text{in } \mathbb{R}^N. \end{align*} where $N \geqslant 3$, the coefficients $a,b,c : \mathbb{R}^N \to \mathbb{R}$ are $L^\infty_{\mathrm{loc}}$-functions bounded from below, and $f:\mathbb{R}^N \times \mathbb{R} \to \mathbb{R}$ is a Carath\'eodory function with subcritical growth. For assumptions permitting sign changes and non-coercivity of the coefficients, we prove the existence of a mountain pass solution. In the case where $a,b,c$ do not change sign, still allowing non-coercive behavior, we additionally establish the existence of componentwise positive and negative solutions.