Concentration and multiple normalized solutions for a class of biharmonic Schrödinger equations *

Abstract

In the present paper, we study the existence and concentration of multiple normalized solutions to the following nonlinear biharmonic Schrödinger equation: $$\{\begin{array}{ll}{\epsilon }^4{\mathrm{\Delta }}^{2}u+V(x)u=\lambda u+h(u),& x\in {\mathbb{R}}^N,\\ {\int }_{{\mathbb{R}}^N}|u{|}^2\mathrm{d}x=c^2{\epsilon }^N,& x\in {\mathbb{R}}^N,\end{array}$$ where $\epsilon >0$ is a positive parameter, $\lambda \in \mathbb{R}$ is unknown and appears as a Lagrange multiplier, and V is a positive potential such that $\underset{\mathrm{\Lambda }}{inf}V<\underset{\mathrm{\partial }\mathrm{\Lambda }}{inf}V$ for some open bounded subset $\mathrm{\Lambda }\subset {\mathbb{R}}^N(N⩾5)$ . Applying the penalization techniques and Ljusternik–Schnirelmann theory, we obtain multiple mormalized solutions $u_{\epsilon }$ . When $\epsilon \to 0$ , these solutions concentrates around a local minimum of V.

This paper extends the results of Alves and Thin (2023), which considered the nonlinear Schrödinger equations with general nonlinearities, to the biharmonic Schrödinger equations. We develop a truncated skill to obtain the minimum via careful analysis. Moreover, we also obtain orbital stability of the solutions.