Existence results for nonlinear elliptic problems with free nonlinearities

Abstract

The main objective of the present work is to study, within the field of partial differential equations, elliptic problems where we can identify some form of criticality in the behavior of the nonlinear function present and, at the end of each of the three appointed problems, to prove the existence of strictly positive solutions to such. In the first chapter, we present to the reader a brief historical vision of the problems we seek to study and the notions of critical growth in the sense of Sobolev and in the sense of Trudinger-Moser, which differ from one another mainly by the considered elliptic operator, by the function spaces in which we look for solutions and, additionally, by the methods we employ. This are the factors that summon the main complications we have encountered while resolving the proposed problems. In the second chapter, we look at our first problem considered, namely the mixed boundary condition problem, (Formula available in the original file) where B(u) is a Dirichlet-Neumann mixed boundary operator, which combines the two different notions of boundary condition. In this case, the critical behavior of the function f is given by the Sobolev critical exponent, (Formula available in the original file), where N is the dimension of the space where Ω resides. Following that, in our tird chapter, we look at an elliptic system highly coupled, (Formula available in the original file) and the fact that we still treat the Laplacian operator implies once more that the critical growth condition is given by the Sobolev critical exponent, so that we take f below (but still able to achieve the growth of) the curve (Formula available in the original file). One may notice that this growth condition is also seen in the first equation, joined with the presence of a singular term as part of the nonlinearity. At last, in the fourth and final chapter, it is considered a problem with the nonlinear elliptic operator, the N-Laplacian, (Formula available in the original file) We see that once more we treat a system, quite similar even to the first. The operator, however, forces us to consider the condition of criticality of Trudinger-Moser, whereas we also incorporate the same function f to the first equation. Additional details about the treated problems, their operators and the two notions of critical growth will be given in time, as will be done for the methods used to solve them. We will make use, here, of the Galerkin Method and the Schayder Fixed Point Theorem, both comprising a non variational approach to the resolution of elliptic problems. Furthermore, the important results of each chapter.