In this project we study properties of existence, non-existence, multiplicity and regularity of the viscosity solutions of some classes of elliptic problems with quadratic growth in the gradient. We propose a qualitative study of the solutions of equations of second and fourth orders, as well as of systems of equations of second order, aiming the characterization of the continuum of solutions generated by the family of a one parameter problem. We also propose the study of the respective phenomenon of multiplicity of solutions in the case that the coefficients of the equation are unbounded. For this, we need to apply topological methods combined to the C1-alpha and W2p regularity theory of solutions, in addition to the study of the first eigenvalue, which strongly influence the generality of the hypotheses of the problem. On the other hand, we intend to verify if the purely nonlinear techniques developed to obtain a priori estimates via blow-up for equations of order two are also applicable to order four and to systems of equations of second order, that is, Harnack type inequalities and generalization of the strong maximum principle of Vázquez for fully nonlinear equations. This problem inspires the study of other interesting properties about the solutions, such as symmetry, regularity of the minimal solution and regularity of the continuum of solutions obtained, asymptotic behavior, as well as problems with dual nature to the one of Vázquez. We also propose to analyze how the results can be improved when incorporating different function spaces, spectral and variational tools. Moreover, when introducing another methods for obtaining a priori estimates, such as the Alexandrov-Serrin moving planes method and its variants.
News published in Agência FAPESP Newsletter about the scholarship: