Symmetry and existence of solutions for nonlinear elliptic problems

Abstract

In this project we study properties of symmetry, existence, nonexistence, multiplicity and regularity of solutions of some classes of nonlinear elliptic problems with linear and superlinear growth in the gradient. We propose a study on the qualitative properties of the solutions for some relevant equations and systems, up to fully nonlinear structure, by exploiting the phenomena of symmetry and symmetry breaking for solutions in the case that the coefficients of the equation do not have the expected monotony in order to preserve the radiality. On the other hand, we aim to deal with symmetry properties, either radial or in the axial sense, for both equations and systems in unbounded domains, by improving the existing hypotheses so far.This problem inspires the study of other interesting properties about the solutions, such as the optimum assumptions that we need to impose in order to have partial symmetry; regularity and uniqueness of the continuum of solutions obtained; the asymptotic behavior of particular solutions when they approximate to some critical value, and so on. For this, it is our goal to treat alternative topological and spectral methods, even by investigating a possible Morse index formulation of the problems, or symmetrization techniques.We also propose to analyze how the results can be improved when we assume a priori symmetry of the solutions into the problem, or when we incorporate different function spaces; by treating alternative classes of solutions variationally, or by introducing new variants of the classical methods.