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 $$\pi(Q,\mathcal{P}):=\lim\limits_{n\to \infty}\binom{n}{s}^{-1}\max\{\mathcal{N}(Q,H): H\in \mathcal{P}~~\mbox{and}~~|V(H)|=n\}.$$ For $p\geq1$, the $(p,Q)$-spectral radius of $H$ is defined as $$\lambda^{(p)}(Q,H):=\max_{\|\mathbf{x}\|_{p}=1}s!\sum_{\{i_{1},\ldots,i_{s}\}\in \binom{[n]}{s}}\mathcal{N}(Q,H[\{i_{1},\ldots,i_{s}\}])x_{i_{1}}\cdots x_{i_{s}}.$$ %generalizing the concept of the $p$-spectral radius introduced by %Keevash, Lenz, and Mubayi \cite{KLM2014}. In this paper, we present a systematically investigation of the parameter $\lambda^{(p)}(Q,H)$. First, we prove that the limit $$\lambda^{(p)}(Q,\mathcal{P}):=\lim\limits_{n\to \infty}n^{s/p-s}\max\{\lambda^{(p)}(Q,H): H\in \mathcal{P}~~\mbox{and}~~|V(H)|=n\}$$ exists, and for $p>1$, it satisfies $$\pi(Q,\mathcal{P})=\lambda^{(p)}(Q,\mathcal{P}).$$ Second, we study spectral generalized Tur\'an problems. Specifically, we establish a spectral stability result and apply it to derive a spectral version of the Erd\H{o}s Pentagon Problem: for $p\geq1$ 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 $p\geq1$ and sufficiently large $n$, the $l$-partite Tur\'an 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}.