Nonexistence of nonconstant global minimizers with limit at $\infty$ of semilinear elliptic equations in all of $R^N$

TL;DR

This paper proves that nonconstant global minimizers with a limit at infinity do not exist for a broad class of semilinear elliptic equations in all dimensions, highlighting fundamental constraints on solutions.

Contribution

It establishes the nonexistence of nonconstant global minimizers with limits at infinity for semilinear elliptic equations in all dimensions, generalizing previous results.

Findings

Nonconstant global minimizers with limit at infinity do not exist.

Nonconstant bounded radial global minimizers do not exist.

Results hold for general nonlinearities in all dimensions.

Abstract

We prove nonexistence of nonconstant global minimizers with limit at infinity of the semilinear elliptic equation $-\Delta u=f(u)$ in the whole $R^N$, where $f\in C^1(R)$ is a general nonlinearity and $N\geq 1$ is any dimension. As a corollary of this result, we establish nonexistence of nonconstant bounded radial global minimizers of the previous equation.