Existence of normalized positive solution of nonhomogeneous biharmonic Schrödinger equations: mass-supercritical case

We deal with the following biharmonic nonlinear Schrödinger equations with a nonhomogeneous perturbation: \(\begin{aligned} \left\{ \begin{array}{ll} \Delta ^2u+\lambda u=|u|^{p-2}u+g(x), ~\text {in}~{\mathbb {R}}^N,\\ \int _{{\mathbb {R}}^N}|u|^2\mathrm{{d}}x=c^2, ~u\in ~H^2({\mathbb {R}}^N), \end{array}\right. \end{aligned}\) $$\begin{array}{c}\hfill \left\{\begin{array}{c}{\mathrm{\Delta }}^{2}u+\lambda u={|u|}^{p-2}u+g(x),\text{in}{\mathbb{R}}^N,\hfill \\ {\int }_{{\mathbb{R}}^N}{|u|}^2\mathrm{d}x=c^2,u\in H^2({\mathbb{R}}^N),\hfill \end{array}\right)\end{array}$$ where \(N\ge 1\) $N\ge 1$ , \(c\ge 0\) $c\ge 0$ , \({\bar{p}}<p<4^*\) $\overline{p}<p<4^{\ast }$ , \({\bar{p}}=2+\frac{8}{N}\) $\overline{p}=2+\frac{8}{N}$ , \(4^*=\frac{2N}{N-4}\) $4^{\ast }=\frac{2N}{N-4}$ if \(N\ge 5\) $N\ge 5$ and \(4^*=\infty \) $4^{\ast }=\infty$ if \(1\le N\le 4\) $1\le N\le 4$ , \(g(x)>0\) $g(x)>0$ is a perturbation. For small positive radial function g, the existence of a mountain pass normalized solution with positive energy is established.