关于具有共振次齐次项的$p$-拉普拉斯算子的广义Fredholm抉择
On the generalized Fredholm alternative for the $p$-Laplacian with resonant subhomogeneous terms
AI总结：
研究带共振次齐次项的p-拉普拉斯能量泛函，依据pκ与q的大小关系刻画其几何性质，并给出临界点个数条件，揭示解集的S形结构，应用于双相泛函与非线性Fredholm抉择。
AI中文摘要：
设$1 \leq q < p$，并设$\lambda_1$为有界域$\Omega$上$p$-拉普拉斯算子的第一特征值。我们研究能量泛函$$ E_\lambda(u)=\frac{1}{p}\left(\int_\Omega|\nabla u|^p\\,dx -\lambda\int_\Omega|u|^p\\,dx\right)-\mathcal{F}(u), \quad u\in W_0^{1,p}(\Omega), $$ 其中$\mathcal{F}$是正$q$-齐次的，并沿第一特征空间消失。假设$\mathcal{F}$与$E_{\lambda_1}$的主部在该特征空间附近的$\kappa$次幂之间存在适当关系，我们根据$p\kappa<q$、$p\kappa=q$或$p\kappa>q$的关系描述$E_{\lambda_1}$的行为。特别地，在第一种情形下泛函下方无界，在最后一种情形下具有负下确界。然后我们研究$\mathcal{F}$的充分小的$q$-齐次扰动如何影响$E_\lambda$的几何结构。通过这种方式，我们描述了保证在$\lambda_1$的左邻域存在三个临界点、在$\lambda_1$的右邻域存在两个临界点的条件，这表明解集具有$S$形结构。结果应用于双相泛函和非线性Fredholm抉择。
英文摘要：
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 $q$-homogeneous and vanishes along the first eigenspace. Assuming a suitable relation between $\mathcal{F}$ and the $κ$-th power of the principal part of $E_{λ_1}$ near this eigenspace, we describe the behavior of $E_{λ_1}$ according to the relations $pκ<q$, $pκ=q$, or $pκ>q$. 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 $E_λ$. In this way, we describe assumptions guaranteeing the existence of three critical points in a left neighborhood of $λ_1$ and two critical points in a right neighborhood of $λ_1$, which indicates an $S$-shaped structure of the solution set. The results are applied to double-phase functionals and the nonlinear Fredholm alternative.
