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

Abstract

We deal with the following biharmonic nonlinear Schrödinger equations with a nonhomogeneous perturbation:

$$\begin{aligned} \left\{ \begin{array}{ll} \Delta ^2u+\lambda u=|u|^{p-2}u+g(x), ~\text {in}~{\mathbb {R}}^N,\\ \int _{{\mathbb {R}}^N}|u|^2\mathrm{{d}}x=c^2, ~u\in ~H^2({\mathbb {R}}^N), \end{array}\right. \end{aligned}$$

where $N\ge 1$, $c\ge 0$, ${\bar{p}}<p<4^*$, ${\bar{p}}=2+\frac{8}{N}$, $4^*=\frac{2N}{N-4}$ if $N\ge 5$ and $4^*=\infty$ if $1\le N\le 4$, $g(x)>0$ is a perturbation. For small positive radial function g, the existence of a mountain pass normalized solution with positive energy is established.