On some elliptic problems involving powers of the laplace operator

Resumen

This PhD Thesis is devoted to the study of some elliptic problems involving powers of the positive Laplace operator, namely, the operator $(-\Delta)^s$, referred to as the spectral fractional Laplacian. This operator is one of the so called fractional Laplace operators and, as its very name suggests, is the one defined via the spectral decomposition of the Laplace operator under the particular boundary condition imposed. The common thread among the problems studied in this work is, at one hand, their critical nature, in the sense that we will deal with: I. Problems with a lack of regularity. II. Problems with a lack of compactness. On the other hand, the problems studied in this work can be also classified according to the imposed boundary conditions, namely, 1. Problems with Mixed Dirichlet-Neumann Boundary data. 2. Problems with Dirichlet Boundary data. Much is known about Dirichlet and Neumann boundary problems associated with nonlinear elliptic equations involving the classical Laplace operator. In contrast, nonlinear mixed Dirichlet-Neumann boundary value problems have been much less investigated. Nevertheless, some important results dealing with such nonlinear mixed Dirichlet-Neumann boundary problems have been proved over the years. Nonlinear fractional elliptic problems, substituting the Laplace operator by the fractional Laplacian, have been extensively investigated in the last years, with Dirichlet or Neumann boundary conditions, however, these nonlinear fractional elliptic problems, once again, have not been so much investigated with mixed Dirichlet-Neumann boundary data. Actually, up to our knowledge, there are no references for mixed Dirichlet-Neumann boundary problems involving the spectral fractional Laplacian operator, which is the main object of study of this PhD Thesis. On the other hand, although mixed Dirichlet-Neumann boundary value problems and Dirichlet boundary value problems share some important qualitative properties, the study of mixed problems presents unique particularities that make them of considerable interest. Examples of these particularities are: -Solutions of mixed boundary data problems are less regular than solutions of the same problems under Dirichlet boundary conditions. Indeed, there is an upper limit for the regularity in terms of Hölder continuity, namely, the optimal regularity for solutions of fractional elliptic problems with mixed boundary data is 1/2-Hölder continuous. -Moving the boundary condition so that the Dirichlet part of the boundary becomes small enough, the existence of solutions to a certain critical problem can be proved in contrast to non-existence results for the same critical problems under Dirichlet boundary condition. In addition to the boundary condition, the specific problems studied here are determined by the chosen nonlinearity or reaction term. In particular, in this work we will consider the following, a) A summable function. b) A concave-convex function involving subcritical powers. c) A critical power function and a linear term. d) A power function involving up to critical powers together with an inverse fractional operator. The main purpose of this work is then outlined as follows. First, we study the fractional mixed boundary value problems for the subcritical range of exponents. To this end we focus on: 1.1. Study the regularity properties of solutions of linear fractional elliptic problems with mixed Dirichlet-Neumann boundary condition. Study the behavior of such solutions when we move the boundary condition in a way to be specified along this work. Prove uniform estimates on the Hölder norm of solutions of such mixed linear fractional problems even when we move the boundary condition. 1.2. Study the existence and some qualitative properties of positive solutions of the mixed concave-convex problem obtained by considering problems with mixed Dirichlet-Neumann boundary condition and a concave-convex nonlinearity. Characterize the existence of such positive solutions. Study the multiplicity of positive solutions and the behavior of some class of solutions when we move the boundary condition. Next, we turn our attention to the study of the mixed critical problems obtained by considering problems with mixed Dirichlet-Neumann boundary condition and a nonlinearity given as in item c). Our main objective at this point is, 2. Characterize the existence of positive solutions for these mixed critical problems. Study the behavior of the positive solutions of the pure critical power problem when we move the boundary condition. Once we have completed these steps, we continue with the study of nonlinear problems involving an inverse operator and Dirichlet boundary condition. To accomplish this step, we now focus on: 3.1. Study the corresponding local problem, i.e., the problem involving the classical Laplace operator. Characterize the existence of solutions for both, the subcritical and the critical exponent cases. 3.2 Generalize the former results to the fractional framework and prove the existence of solutions to problems with a nonlinearity involving an inverse operator nd a power function for both, the subcritical and the critical exponent cases. As we will see, nonlinear problems involving an inverse operator arise when one studies the steady-states of certain high-order parabolic equations. To get closer to future extensions and analysis of similar high-order problems we conclude this work performing an homotopic study of a nonlinear high-order parabolic problem in divergence form. This PhD Thesis dissertation is then divided into the following main parts: Part 1. Subcritical Problems with Mixed Dirichlet-Neumann Boundary data. -Chapter 1: Regularity of solutions of a linear fractional elliptic problem with mixed Dirichlet-Neumann boundary conditions. -Chapter 2: Semilinear fractional elliptic problems with mixed Dirichlet-Neumann boundary conditions. Part 2. Critical Problems with Mixed Dirichlet-Neumann Boundary data. -Chapter 3: The Brezis-Nirenberg problem for the fractional Laplacian with mixed Dirichlet-Neumann boundary conditions. Part 3. Critical Problems involving inverse operators and Dirichlet Boundary data. -Chapter 4: Existence of positive solutions for a Brezis-Nirenberg--type problem involving an inverse operator. -Chapter 5: Existence of positive solutions for a semilinear fractional elliptic equation involving an inverse fractional operator. -Chapter 6: Homotopy Regularization for a High-Order parabolic equation.