Normalized ground states of nonlinear biharmonic Schr\"odinger equations with Sobolev critical growth and combined nonlinearities

TL;DR

This paper investigates normalized ground states of a nonlinear biharmonic Schrödinger equation with Sobolev critical growth and combined nonlinearities, establishing existence and variational properties of solutions in high dimensions.

Contribution

It introduces a new analysis of ground state solutions for biharmonic Schrödinger equations with critical Sobolev growth and combined nonlinearities, including existence and minimization properties.

Findings

Existence of normalized ground state solutions for the equation.

Ground states are local minima of the energy functional.

Behavior of ground state energy with respect to prescribed mass analyzed.

Abstract

This paper is devoted to studying the following nonlinear biharmonic Schr\"odinger equation with combined power-type nonlinearities \begin{equation*} \begin{aligned} \Delta^{2}u-\lambda u=\mu|u|^{q-2}u+|u|^{4^*-2}u\quad\mathrm{in}\ \mathbb{R}^{N}, \end{aligned} \end{equation*} where $N\geq5$, $\mu>0$, $2<q<2+\frac{8}{N}$, $4^*=\frac{2N}{N-4}$ is the $H^2$-critical Sobolev exponent, and $\lambda$ appears as a Lagrange multiplier. By analyzing the behavior of the ground state energy with respect to the prescribed mass, we establish the existence of normalized ground state solutions. Furthermore, all ground states are proved to be local minima of the associated energy functional.