$p$-nuclearity of $L^p$-operator crossed products

Let $(X,\mathcal{B},\mu )$ be a measure space and $A$ be a norm closed subalgebra of $\mathcal{B}(L^p(X,\mu ))$, where $p\in [1,\mathrm{\infty })$. Let $(G,A,\alpha )$ be an $L^p$-operator algebra dynamical system, where $G$ is a countable discrete amenable group. We prove that the full $L^p$-operator crossed product $F^p(G,A,\alpha )$ is $p$-nuclear if and only if $A$ is $p$-nuclear {provided the action} $\alpha$ of $G$ on $A$ is $p$-completely isometric. As applications, we prove that $L^p$-Cuntz algebras and rotation $L^p$-operator algebras are $p$-nuclear. Our results solve { a problem raised by N. C. Phillips concerning {$p$-nuclearity} for $L^p$-Cuntz algebras.}