Abstract
We investigate the symmetry properties of several radially symmetric minimization problems. The minimizers which we obtain are nodal solutions of superlinear elliptic problems, or eigenfunctions of weighted asymmetric eigenvalue problems, or they lie on the first curve in the Fucik spectrum. In all instances, we prove that the minimizers are foliated Schwarz symmetric. We give examples showing that the minimizers are in general not radially symmetric. The basic tool which we use is polarization, a concept going back to Ahlfors. We develop this method of symmetrization for sign changing functions.
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References
L. V. Ahlfors,Conformal Invariants, McGraw-Hill, New York, 1973.
M. Arias, J. Campos, M. Cuesta and J. P. Gossez,Asymmetric elliptic problems with indefinite weights, Ann. Inst. H. Poincaré Anal. Non Linéaire19 (2002), 581–616.
T. Bartsch, Z. Liu and T. Weth,Sign changing solutions to superlinear Schrödinger equations, Comm. Partial Differential Equations29 (2004), 25–42.
T. Bartsch and T. Weth,A note on additional properties of sign changing solutions to superlinear Schrödinger equations, Topol. Methods Nonlinear Anal.22 (2003), 1–14.
T. Bartsch and T. Weth,Three nodal solutions of singularly perturbed elliptic equations on domains without topology, Ann. Inst. H. Poincaré Anal. Non Linéaire, to appear.
T. Bartsch and M. Willem,Infinitely many radial solutions of a semilinear elliptic problem on ℝ N, Arch. Rational Mech. Anal.124 (1993), 261–276.
H. Brezis and L. Nirenberg,Positive solutions of nonlinear elliptic equations involving critical Sobolev exponents, Comm. Pure Appl. Math.36 (1983), 437–477.
F. Brock and A. Yu. Solynin,An approach to symmetrization via polarization, Trans. Amer. Math. Soc.352 (2000), 1759–1796.
A. Castro, J. Cossio and J. M. Neuberger,A sign-changing solution for a superlinear Dirichlet problem, Rocky Mountain J. Math.27 (1997), 1041–1053.
D. G. de Figueiredo and J. P. Gossez,On the first curve of the Fučik spectrum of an elliptic operator, Differential Integral Equations7 (1994), 1285–1302.
B. Gidas, W. M. Ni and L. Nirenberg,Symmetry and related properties via the maximum principle, Comm. Math. Phys.68 (1979), 209–243.
F. Pacella,Symmetry results for solutions of semilinear elliptic equations with convex non-linearities, J. Funct. Anal.192 (2002), 271–282.
J. Serrin,A symmetry property in potential theory, Arch. Rational Mech. Anal.43 (1971), 304–318.
D. Smets, J. Su and M. Willem,Nonradial ground states for the Hénon equation, Comm. Contemp. Math.4 (2002), 467–480.
D. Smets and M. Willem,Partial symmetry and asymptotic behaviour for some elliptic variational problems, Calc. Var. Partial Differential Equations18 (2003), 37–75.
M. Willem,Minimax Theorems, Birkhäuser, Basel, 1996.
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Supported by NATO grant PST.CLG.978736.
Supported by DFG grant WE 2821/2-1.
Supported by NATO grant SPT.CLG.978736.
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Bartsch, T., Weth, T. & Willem, M. Partial symmetry of least energy nodal solutions to some variational problems. J. Anal. Math. 96, 1–18 (2005). https://doi.org/10.1007/BF02787822
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DOI: https://doi.org/10.1007/BF02787822