Global boundedness and normalized solutions to a $p$-Laplacian equation

In the paper, we prove the existence of radial solutions to \begin{equation}\notag%\label{main-eq-abstarct} %\begin{aligned} -\Delta_p u+({\rm sgn}(p-s)+V(x))|u|^{p-2}u+\lambda |u|^{s-2}u=|u|^{q-2}u\qquad\text{in}\,\R^N \\ %\int_{\R^N}|u|^sdx&=\rho^s %\end{aligned} \end{equation} with prescribed $L^s(\R^N)$-norm, where $N\ge 3,\,p\in[2,N),\,s\in(1,p],\,q\in(p\frac{N+s}{N},\frac{Np}{N-p})$ and $V:\R^N\to\R$ is a suitable radial potential. We stress that $V$ is required to be radial but not necessarily bounded, and there are no assumptions about its sign. The case $V\equiv 0$ is also included. The proof is variational and relies on a min-max argument. A key-tool is the Pohozaev identity, which is shown to be true for any solution under quite weak assumptions about the potential $V$. This identity is proved with the aid of a new global boundedness result for subsolutions to a suitable $p$-Laplace equation.