# Multiplicity of radial and nonradial solutions to equations with fractional operators

## TL;DR

This paper investigates the existence and multiplicity of both radial and nonradial solutions to scalar field equations involving fractional operators, providing new existence results and characterizations for various dimensions and mass cases.

## Contribution

It establishes the existence of infinitely many radial solutions, identifies least energy solutions, and proves the existence of multiple nonradial solutions for different dimensions and mass conditions.

## Key findings

- Infinitely many radial solutions for N ≥ 2
- Existence of least energy solutions with Pohozaev identity
- Multiple nonradial solutions for N ≥ 4 and N ≥ 6

## Abstract

In this paper, we study the existence of radial and nonradial solutions to the scalar field equations with fractional operators. For radial solutions, we prove the existence of infinitely many solutions under $N \geq 2$. We also show the existence of least energy solution (with the Pohozaev identity) and its mountain pass characterization. For nonradial solutions, we prove the existence of at least one nonradial solution under $N \geq 4$ and infinitely many nonradial solutions under either $N =4$ or $N \geq 6$. We treat both of the zero mass and the positive mass cases.