Antisymmetry of solutions for some weighted elliptic problems
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This article concerns the antisymmetry, uniqueness, and monotonicity properties of solutions to some elliptic functionals involving weights and a double well potential. In the one-dimensional case, we introduce the continuous odd rearrangement of an increasing function and we show that it decreases the energy functional when the weights satisfy a certain convexity-type hypothesis. This leads to the antisymmetry or oddness of increasing solutions (and not only of minimizers). We also prove a uniqueness result (which leads to antisymmetry) where a convexity-type condition by Berestycki and Nirenberg on the weights is improved to a monotonicity condition. In addition, we provide with a large class of problems where antisymmetry does not hold. Finally, some rather partial extensions in higher dimensions are also given.
CitationCabre, X., Lucia, M., Sanchón, M., Villegas, S. Antisymmetry of solutions for some weighted elliptic problems. "Communications in partial differential equations", 17 Març 2018, vol. 43, núm. 3, p. 506-547.