Existence and Symmetries for Elliptic Equations with Multipolar Potentials and Polyharmonic Operators

• Published in: Indiana University mathematics journal 2013-01, Vol.62 (6), p.1955-1982
• Main Authors: Ferreira, Lucas C. F., Mesquita, Cláudia Aline A. S.
• Format: Article
• Language: English
• Subjects: Boundary conditions; Differential equations; Elliptic equations; Greens function; Hematocrit; Infinity; Nonlinearity; Partial differential equations; Polygons
• ISSN: 0022-2518; 1943-5258
• DOI: 10.1512/iumj.2013.62.5131

This article concerns the existence and qualitative properties of solutions for semilinear elliptic equations with critical multipolar potentials and polyharmonic operators. The results are also new for the Laplacian case. They cover a large class of anisotropic potentials, and show that solutions are stable with respect to potential angular components. The analysis is performed in ${\mathcal{H}}_{\mathrm{k},\overrightarrow{\mathrm{\alpha }}}-\mathrm{s}\mathrm{p}\mathrm{a}\mathrm{c}\mathrm{e}\mathrm{s}$ which are a sum of weighted L∞-spaces and seem to be a minimal framework for the potential profile of interest. For this purpose, a theory with basic properties for ${\mathcal{H}}_{\mathrm{k},\overrightarrow{\mathrm{\alpha }}}$ is developed, including optimal (and positive) decompositions, Holder-type inequality, and quasi-monotonicity of the norm. We investigate a concept of symmetry for solutions which extends radial symmetry and carries out an idea of invariance around singularities. Our approach can be employed in a number of other situations like exterior domains and other types of PDEs.