Symmetry and monotonicity of least energy solutions. (English) Zbl 1226.35041
Summary: 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 [J. E. Brothers and W. P. Ziemer, J. Reine Angew. Math. 384, 153–179 (1988; Zbl 0633.46030); M. Mariş, Arch. Ration. Mech. Anal. 192, No. 2, 311–330 (2009; Zbl 1159.49005)] and answer questions from H. Brézis and E. H. Lieb [Commun. Math. Phys. 96, 97–113 (1984; Zbl 0579.35025)] and P.-L. Lions [Ann. Inst. Henri Poincaré, Anal. Non Linéaire 1, 223–283 (1984; Zbl 0704.49004)].
MSC:
| 35J62 | Quasilinear elliptic equations |
| 35B06 | Symmetries, invariants, etc. in context of PDEs |
| 35J20 | Variational methods for second-order elliptic equations |
| 35J50 | Variational methods for elliptic systems |
| 35J60 | Nonlinear elliptic equations |
Keywords:
quasilinear elliptic systems; radially symmetric solutions; quasilinear elliptic equations; least energy solutionsReferences:
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