Symmetry and monotonicity of least energy solutions

TL;DR

This paper proves that least energy solutions to certain quasilinear elliptic equations are radially symmetric, with scalar solutions also being monotone and of constant sign, using a novel approach that does not rely on the moving planes method.

Contribution

It introduces a simple proof for symmetry and monotonicity of least energy solutions without requiring cooperative conditions or the moving planes method.

Findings

Least energy solutions are radially symmetric.

Scalar least energy solutions have constant sign.

Scalar solutions are monotone in the radial variable.

Abstract

We give a simple proof of the fact that for a large class of quasilinear elliptic equations and systems the solutions that minimize the corresponding energy in the set of all solutions are radially symmetric. We require just continuous nonlinearities and no cooperative conditions for systems. Thus, in particular, our results cannot be obtained by using the moving planes method. In the case of scalar equations, we also prove that any least energy solution has a constant sign and is monotone with respect to the radial variable.