Problemas no lineales de tipo Ambrosetti-Prodi.

Abstract

Finding solutions to Ambrosetti-Prodi type problems remains an open problem. This study addresses the existence of at least two solutions to this type of problem with non homogeneous Dirichlet boundary conditions. In this case, it is necessary to manipulate the non-homogeneous problem in order to turn it into a homogeneous one. Different resolution methods are taken into account for each solution. For the first one, we use the method of sub-supersolutions and the method of monotone iteration to find a minimal solution. While the second solution is found through a variational approach with the help of the Palais-Smale condition and the Mountain Pass Theorem. Finally, it is important to note that there is a shift due to the nonhomogeneity of the boundary conditions, which leads to provide some hypothesis to nonlinearity and observe that the trivial solution is not a solution to this problem.