Let us consider for $0<\alpha\leq V(x)\leq \beta$ and $0\leq K(x)<\gamma$ the equation \begin{equation}\label{\star} \Delta^2u+V(x)u=g(x, u)+K(x)u, \end{equation} where $|g(x,s)|\leq \varepsilon|s|^{1+\sigma}+c|s|^{p}$ with $1<\sigma<p<\frac{N+4}{N-4}$ and $s\geq 0.$

Is the solution $u$ of equation above in ${H}^4(\mathbb{R}^N)?$