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

Abstract

Let $1 \leq q<p$ and let $\lambda_1$ be the first eigenvalue of the $p$-Laplacian in a bounded domain $\Omega$. We study the energy functional $$ E_\lambda(u)=\frac{1}{p}\left(\int_\Omega|\nabla u|^p\,dx -\lambda\int_\Omega|u|^p\,dx\right)-\mathcal{F}(u), \quad u\in W_0^{1,p}(\Omega), $$ where $\mathcal{F}$ is positively $q$-homogeneous and vanishes along the first eigenspace. Assuming a suitable relation between $\mathcal{F}$ and the $\kappa$-th power of the principal part of $E_{\lambda_1}$ near this eigenspace, we describe the behavior of $E_{\lambda_1}$ according to the relations $p\kappa<q$, $p\kappa=q$, or $p\kappa>q$. In particular, the functional is unbounded from below in the first case, and has a negative infimum in the last case. We then study how sufficiently small $q$-homogeneous perturbations of $\mathcal{F}$ influence the geometry of $E_\lambda$. In this way, we describe assumptions guaranteeing the existence of three critical points in a left neighborhood of $\lambda_1$ and two critical points in a right neighborhood of $\lambda_1$, which indicates an $S$-shaped structure of the solution set. The results are applied to double-phase functionals and the nonlinear Fredholm alternative.