Non-existence of ground states and gap of variational problems for combined power-type nonlinear scalar field equations involving the Sobolev critical exponent in three space dimensions

TL;DR

This paper investigates the existence and non-existence of ground states for certain nonlinear scalar field equations with Sobolev critical exponent in three dimensions, revealing a threshold frequency phenomenon.

Contribution

It establishes the non-existence of ground states above a certain frequency and clarifies differences between variational characterizations in three-dimensional cases.

Findings

Existence of a threshold frequency for ground states in 3D

Non-existence of ground states above this threshold

Differences between variational problems for ground state characterization

Abstract

In this paper, we consider minimization problems related to the combined power-type nonlinear scalar field equations involving the Sobolev critical exponent in three space dimensions. In four and higher space dimensions, it is known that for any frequency and any power of the subcritical nonlinearity, there exists a ground state. In contrast to those cases, when the space dimension is three and the subcritical power is three or less, we can show that there exists a threshold frequency, above which no ground state exists, and below which the ground state exists. Furthermore, we prove the difference between two typical variational problems used to characterize the ground states.