Symmetry and monotonicity of least energy solutions

By Jaeyoung Byeon, Louis Jeanjean, M. Mariş. 2008

105 citations recorded by the source corpus.

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. Our proofs rely on results in Brothers and Ziemer (J Reine Angew Math 384:153–179, 1988) and Mariş (Arch Ration Mech Anal, 192:311–330, 2009) and answer questions from Brézis and Lieb (Comm Math Phys 96:97–113, 1984) and Lions (Ann Inst H Poincaré Anal Non Linéaire 1:223–283, 1984).

Subfields: Applied Mathematics, Computational Theory and Mathematics, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/918783aed19105c0dc3d3d37ad6acd68f6a80d37

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In Applied Mathematics

In Computational Theory and Mathematics

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Citations in the corpus

Stability and instability results for standing waves of quasi-linear Schrödinger equations (2010, background): “7) which, in fact, satisfies a scaling property being the essence of the results of [7].”
Existence of groundstates for a class of nonlinear Choquard equations (2012, methodology): “Finally, we obtain qualitative properties of groundstates of ( P ), which are the counterpart of the properties obtained for solutions of the corresponding local equation [10,14,16].”