On the generalized Fredholm alternative for the $p$-Laplacian with resonant subhomogeneous terms

Let $1 \leq q < p$ and let $λ_1$ be the first eigenvalue of the $p$-Laplacian in a bounded domain $Ω$. We study the energy functional $$ E_λ(u)=\frac{1}{p}\left(\int_Ω|\nabla u|^p\,dx -λ\int_Ω|u|^p\,dx\right)-\mathcal{F}(u), \quad u\in W_0^{1,p}(Ω), $$ where $\mathcal{F}$ is positively $q$-homogeneo