Ambrosetti-Prodi type problem with non-homogeneous Dirichlet boundary conditions

Abstract

This article demonstrates the existence of at least three solutions to the Ambrosetti-Prodi type problem with non-homogeneous Dirichlet boundary conditions. For the search for the first solution, which will also be minimal, we employ the sub-supersolutions method combined with a monotone iteration scheme. In addition, we utilize the variational method and the steepest descent technique, along with the Palais-Smale conditions and the Mountain Pass Theorem, to find the second and third solutions. It should be noted that the non-homogeneity of the boundary conditions generates a shift; therefore, it is important to impose some growth hypotheses on non-linearity. Furthermore, it is observed that the trivial function is not a solution to this problem.