Qualitative properties for nonnegative solutions to strongly coupled fourth order systems

Abstract

This thesis studies qualitative properties for nonnegative solutions to fourth or- der systems driven by a GrossPitaevskii type nonlinear coupling term on a punctured

domain with dimensions bigger than four. More accurately, we provide classication

results and a description of the local behavior near an isolated (non-removable) singu- larity. We divide our analysis into two cases. Namely, the underlying domain is either

the punctured space or a punctured ball. First, we classify the solutions in the whole space, called the blow-up limit (or EmdenFowler) solutions. Second, we show that these limiting solutions are the local models of our system near the origin. The growth of the nonlinear coupling term alters our analysis. In this fashion, we divide our approach into the (upper) critical and subcritical cases, which are also split into more sub-cases with respect to the so-called Serrin (or lower critical) exponent. Our analysis is based on cylindrical logarithm coordinates, Liouville-type results, integral representation formulas, sliding techniques, Pohozaev functionals, analytic Fredholm theory, and asymptotic analysis. In the critical setting, our system is closely related to conformal geometry, being the most natural vectorial extension of the conformally at Q-curvature equation. In this case, a delicate study of the geometric Jacobi elds in the kernel of the linearized operator around blow-up limit solutions is also required. The results in this thesis extend to the context of fourth order coupled systems the celebrated asymptotics due to J. Serrin [193], P.-L. Lions [146], P. Aviles [16], B. Gidas and J. Spruck [83], L. A. Caarelli et al. [130], and N. Korevaar et al. [31].