Antisymmetry of solutions for some weighted elliptic problems

• Published in: Communications in partial differential equations 2018-03, Vol.43 (3), p.506-547
• Main Authors: Cabré, Xavier, Lucia, Marcello, Sanchón, Manel, Villegas, Salvador
• Format: Article
• Language: English
• Subjects: 35 Partial differential equations; 35B Qualitative properties of solutions; 35J Partial differential equations of elliptic type; 35Q Equations of mathematical physics and other areas of application; Antisymmetric solutions; Antisymmetry; bistable nonlinearity; Classificació AMS; continuous odd rearrangement; Convexity; Differential equations, Partial; Equacions diferencials parcials; Functionals; Matemàtiques i estadística; monotonicity; Uniqueness; weights; Àrees temàtiques de la UPC
• ISSN: 0360-5302; 1532-4133
• DOI: 10.1080/03605302.2018.1446168

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.