Spectral extremal problems for the $(p,Q)$-spectral radius of hypergraphs

Let $Q$ be an $s$-vertex $r$-uniform hypergraph, and let $H$ be an $n$-vertex $r$-uniform hypergraph. Denote by $\mathcal{N}(Q,H)$ the number of isomorphic copies of $Q$ in $H$. For a hereditary family $\mathcal{P}$ of $r$-uniform hypergraphs, define $$\pi (Q,\mathcal{P}):=\underset{n\to \mathrm{\infty }}{\lim }{\left(\genfrac{}{}{0px}{}{n}{s}\right)}^{-1}\max \{\mathcal{N}(Q,H):H\in \mathcal{P}and\mathrm{\mid }V(H)\mathrm{\mid }=n\}\mathrm{.}$$ For $p\ge 1$, the $(p,Q)$-spectral radius of $H$ is defined as $${\lambda }^{(p)}(Q,H):=\underset{\mathrm{\parallel }\mathbf{x}{\mathrm{\parallel }}_p=1}{\max }s!\sum _{\{i_1,\dots ,i_s\}\in \left(\genfrac{}{}{0px}{}{[n]}{s}\right)}\mathcal{N}(Q,H[\{i_1,\dots ,i_s\}])x_{i_1}\cdots x_{i_s}\mathrm{.}$$ In this paper, we present a systematically investigation of the parameter ${\lambda }^{(p)}(Q,H)$. First, we prove that the limit $${\lambda }^{(p)}(Q,\mathcal{P}):=\underset{n\to \mathrm{\infty }}{\lim }{n}^{s\mathrm{/}p-s}\max \{{\lambda }^{(p)}(Q,H):H\in \mathcal{P}and\mathrm{\mid }V(H)\mathrm{\mid }=n\}$$ exists, and for $p>1$, it satisfies $$\pi (Q,\mathcal{P})={\lambda }^{(p)}(Q,\mathcal{P})\mathrm{.}$$ Second, we study spectral generalized Turán problems. Specifically, we establish a spectral stability result and apply it to derive a spectral version of the Erdős Pentagon Problem: for $p\ge 1$ and sufficiently large $n$, the balanced blow-up of $C_5$ maximizes ${\lambda }^{(p)}(C_5,H)$ among all $n$-vertex triangle-free graphs $H$, thereby improving a result of Liu \cite{Liu2025}. Furthermore, we show that for $p\ge 1$ and sufficiently large $n$, the $l$-partite Turán graph $T_l(n)$ attains the maximum ${\lambda }^{(p)}(K_s,H)$ among all $n$-vertex F-free graphs $H$, where $F$ is an edge-critical graph with $\chi (F)=l+1$. This provides a spectral analogue of a theorem due to Ma and Qiu \cite{MQ2020}.