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Let $u: (M, g)\to (N, h)$ be a map between Riemannian manifolds $(M, g)$ and$(N, h)$. The $p$-bienergy of $u$ is defined by$E_p(u)=\int_M|\tau(u)|^pd\nu_g$, where $\tau(u)$ is the tension field of $u$and $p>1$. Critical points of $E_p(\cdot)$ are called $p$-biharmonic maps. Inthis paper we will prove nonexistence result of proper $p$-biharmonic maps when$p\geq2$. In particular when $M=\mathbb{R}^m$, we get Liouville type resultsunder proper integral conditions , which extend the related results of Baird,Fardoun and Ouakkas \cite{BFO}.

Nonexistence of proper $p$-biharmonic maps and Liouville type theorems I: case of $p\geq2$