Note on the multiplicity of solutions for nonlinear scalar field equations with a critical inverse-square potential

TL;DR

This paper investigates the existence of infinitely many solutions, both radial and non-radial, for a nonlinear scalar field equation with a critical inverse-square potential, highlighting the solution multiplicity in such equations.

Contribution

It proves the existence of infinitely many solutions to the scalar field equation with a critical inverse-square potential, including non-radial solutions, which was previously unestablished.

Findings

Existence of infinitely many radial solutions.

Existence of infinitely many non-radial solutions.

Solution multiplicity for equations with critical inverse-square potential.

Abstract

We are interested in the multiplicity of solutions to the following scalar field equation $$ -\Delta u - \frac{(N-2)^2}{4|x|^2} u = g(u), \quad \mbox{in } \mathbb{R}^N \setminus \{0\}. $$ We establish the existence of infinitely many radial and non-radial solutions.