Resonance at the first eigenvalue for first-order systems in the plane: vanishing Hamiltonians and the Landesman-Lazer condition

Abstract

The concept of resonance with the first eigenvalue ($\lambda = 0$) of the scalar $T$-periodic problem $$ x''+\lambda x =0, \ \ \ x(0)=x(T), \; x'(0)=x'(T) $$ is considered for first-order planar systems, by dealing with positively homogeneous Hamiltonians which can vanish at some points on $\mathbb{S}^1$. By means of degree methods, an existence result at double resonance for a planar system of the kind $$ Ju'=F(t, u), \quad J= \left( \begin{array}{cc} 0 & -1 \\ 1 & 0 \\ \end{array} \right), $$ is then proved, under the assumption that $F(t, u)$ is controlled from below by the gradient of one of such Hamiltonians described above, complementing the main theorem in [7] and including some classical results for the scalar case.