Uniqueness and Multiplicity of Solutions - Herreros - Cortázar; Pilar (2023)
PROJECT RESULTS: Describe the results of your research in reference to its original and/or modiﬁed Project objectives. The maximum length for this section is 5 pages (Arial or Verdana font, size 10). The project is divided into two main subjects. The ﬁrst “Scattering and Lens Bound- ary Rigidity” refers to the behavior of geodesics and magnetic geodesics on manifolds, and some rigidity of certain conﬁgurations. The problems described in the proposal have been studied, obtaining a better understanding of them and the behavior of the magnetic and geodesic ﬂows. There has been no signiﬁcant (publishable) results in these problems. A related problem is the behavior of the mediatrix: the set of points equidistant to two given sets. In the beginning of 2017, jointly with M. Ponce and JPP. Veerman, we ﬁnished the article “Equators have at most countable many singularities with bounded total angle” that was published later that year. At the end of November, 2017 JJP. Veerman visited P. Universidad Cat´olica to continue this work, we had interesting discussions but obtained no publishable results. The second part of the proyect “Bound State Solutions of a Nonlinear Equation” con- siders the problem of multiplicity and uniqueness of solutions of a nonlinear elliptic equation of the form ∆u + f(u) = 0, x ∈ RN, N ≥ 2, lim |x|→∞ u(x) = 0. where f is a prescribed function, satisfying appropriate conditions. And of the related parabolic problem ut = ∆u + f(u) + h(x, t), (x, t) ∈ RN × (0, ∞) u(x, 0) = u0(x) with u0 ∈ H1(RN), f ∈ C1(R) and h a suitable function on RN × (0, ∞). Jointly with C.Cort´azar and M. Garc´ıa-Huidobro, we approached this problem study- ing the weighted Laplace operator, establishing the uniqueness of the radial bound state solutions to div A ∇v + B f(v) = 0 , lim |x|→+∞ v(x) = 0, x ∈ Rn, 1 n > 2, where A and B are two positive, radial, smooth functions deﬁned on Rn \ {0}.We assume that the nonlinearity f ∈ C(−c, c), 0 < c ≤ ∞ is an odd function satis-fying some convexity and growth conditions, ...