Normalized solutions to polyharmonic equations with Hardy-type potentials and exponential critical nonlinearities

TL;DR

This paper establishes the existence of normalized solutions for polyharmonic equations with Hardy-type potentials and exponential critical nonlinearities using constrained minimization techniques.

Contribution

It introduces a novel approach to find normalized solutions to complex polyharmonic equations with critical exponential nonlinearities and Hardy potentials.

Findings

Existence of solutions with prescribed L^2 norm.

Application of constrained minimization to high-order PDEs.

Handling of exponential critical growth nonlinearities.

Abstract

Via a constrained minimization, we find a solution $(\lambda,u)$ to the problem \begin{equation*} \begin{cases} (-\Delta)^m u+\frac{\mu}{|x|^{2m}}u + \lambda u = \eta u^3 + g(u)\\ \int_{\mathbb{R}^{2m}} u^2 \, dx = \rho \end{cases} \end{equation*} with $1 \le m \in \mathbb{N}$, $\mu,\eta \ge 0$, $\rho > 0$, and $g$ having exponential critical growth at infinity and mass supercritical growth at zero.