The nonlinear Schrödinger equation: electrostatic self-interaction and interplay with geometric contexts - Ramos, Gustavo de Paula (2024)
This thesis consists of a compilation of the contributions proposed by its author during his doctorate, all of them related in some degree to the Nonlinear Schrödinger Equation (NLSE) -epsilon^2 Delta u + V u = u |u|^(p-2) in Omega (whose properties vary from chapter to chapter), where epsilon > 0; V denotes a positive effective external potential in Omega and 2 < p < 2^* , 2^* denoting the critical Sobolev exponent. Throughout the thesis, we employ the variational method, which consists in associating the critical points of an energy functional J_epsilon to the weak solutions of the respectively considered problems. The obtained results are concerned with the existence and properties of weak solutions, the influence of the domain in their multiplicity and their asymptotic profile in limit situations. We divide the thesis in two independent parts: (i) electrostatic self-interaction of quantum matter and (ii) the NLSE in geometric contexts. In the first part, we study systems which model the electrostatic self-interaction of charged quantum matter in R 3 , all of them of the form -epsilon^2 Delta u + (V + phi) u = u |u|^(p-2); P phi = 4 pi u^2, where P is an elliptic differential operator whose definition depends on the considered theory for electromagnetism. The second part is focused on the NLSE in domains more usually studied in differential geometry, considering the nondegeneracy of solutions in Riemannian manifolds and the multiplicity of solutions in a Riemannian orbifold.