Nonlocal elliptic PDEs with general nonlinearities. arXiv:2402.08338
Preprint, arXiv:2402.08338 [math.AP] (2024).
Summary: In this thesis we investigate how the nonlocalities affect the study of different PDEs coming from physics, and we analyze these equations under almost optimal assumptions of the nonlinearity. In particular, we focus on the fractional Laplacian operator and on sources involving convolution with the Riesz potential, as well as on the interaction of the two, and we aim to do it through variational and topological methods. We examine both quantitative and qualitative aspects, proving multiplicity of solutions for nonlocal nonlinear problems with free or prescribed mass, showing regularity, positivity, symmetry and sharp asymptotic decay of ground states, and exploring the influence of the topology of a potential well in presence of concentration phenomena. On the nonlinearities we consider general assumptions which avoid monotonicity and homogeneity: this generality obstructs the use of classical variational tools and forces the implementation of new ideas. Throughout the thesis we develop some new tools: among them, a Lagrangian formulation modeled on Pohozaev mountains is used for the existence of normalized solutions, annuli-shaped multidimensional paths are built for genus-based multiplicity results, a fractional chain rule is proved to treat concave powers, and a fractional center of mass is defined to detect semiclassical standing waves. We believe that these tools could be used to face problems in different frameworks as well.
MSC:
35A15 | Variational methods applied to PDEs |
35B06 | Symmetries, invariants, etc. in context of PDEs |
35B09 | Positive solutions to PDEs |
35B25 | Singular perturbations in context of PDEs |
35B33 | Critical exponents in context of PDEs |
35B38 | Critical points of functionals in context of PDEs (e.g., energy functionals) |
35B40 | Asymptotic behavior of solutions to PDEs |
35B65 | Smoothness and regularity of solutions to PDEs |
35D30 | Weak solutions to PDEs |
35D40 | Viscosity solutions to PDEs |
35J20 | Variational methods for second-order elliptic equations |
35J60 | Nonlinear elliptic equations |
35J61 | Semilinear elliptic equations |
35Q55 | NLS equations (nonlinear Schrödinger equations) |
35R09 | Integro-partial differential equations |
35R11 | Fractional partial differential equations |
45K05 | Integro-partial differential equations |
45M05 | Asymptotics of solutions to integral equations |
45M20 | Positive solutions of integral equations |
46M20 | Methods of algebraic topology in functional analysis (cohomology, sheaf and bundle theory, etc.) |
47J30 | Variational methods involving nonlinear operators |
49J35 | Existence of solutions for minimax problems |
58E05 | Abstract critical point theory (Morse theory, Lyusternik-Shnirel’man theory, etc.) in infinite-dimensional spaces |
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