Normalized solutions to biharmonic Schrödinger equation with critical growth in ${\mathbb {R}}^N$

Abstract

This paper is concerned with the following class of elliptic problems

$$\begin{aligned} \left\{ \begin{array}{ll} {\Delta }^2u=\lambda u + \mu |u|^{p-2}u+ |u|^{4^*-2}u\ \ \text{ in }\ {\mathbb {R}}^N, \\ \int _{{\mathbb {R}}^N} u^2 dx = c, \\ \end{array} \right. \end{aligned}$$

where $N \ge 5$, $\mu , c>0$, $2+\frac{8}{N}<p<4^{*}:=\frac{2N}{N-4}$ and $\lambda \in {\mathbb {R}}$ is a Lagrange multiplier. We show the existence of normalized solutions for $\mu$ large enough by verifying the (PS) condition at the corresponding mountain-pass level. In this sense, we extend the recent results obtained by Ma and Chang (Appl Math. Lett 135:108388, 2022) to the $L^2$-supercritical perturbation. Moreover, we discuss the asymptotic behavior of the energy to the mountain pass solution when $c\rightarrow +\infty$.