Noninear Partial Differential Equations: Bubbles, Layers and Stability

Project title: Nonlinear Partial Differential Equations: Bubbles, Layers and Stability

Principal investigators: Prof. Shusen Yan (UNE), Prof. Norm Dancer (Univ of Sydney), Prof. Yihong Du (UNE) and Prof. Chang-Shou Lin (National Taiwan Univ.)

Funding body:   Australian Research Council (2017-2019)

Project description:
In this project we investigate several nonlinear elliptic partial differential equations that are of great concern in the recent study of the elliptic problems. They arise from well-established models in various applied fields, and the treatment of them poses great challenges to the current mathematical theory. We want to answer a number of important questions on these equations through deep analysis of the properties of their solutions, therefore enriching and expanding the existing mathematical theory in this area. One of the properties of the solutions under our investigation is known as bubbling, which describes the situation that as a key parameter in the equation approaches a certain critical value, the solution \mass" concentrates more and more at some points in the underlying space, like a Dirac delta function (in the limit). Such solutions are called "bubbling solutions", or "blowup solutions", and they frequently arise in various applied sciences, such as condensed matter physics and fluid mechanics.  Many important phenomena in the natural sciences are also described by solutions with sharp layers, representing sharp transitions of phases in physics, chemistry or biology. In the limit, a solution with a sharp layer converges to a discontinuous function. Since bubbling and sharp layered solutions do not converge to a continuous function in the limit, they pose great difficulties in partial differential equations, and have been a central topic of research of many groups of first rate mathematicians around the world. Another important property of the solutions that we will study in this project is characterized by the level of stability, measured by the Morse indices of the solutions. The bigger the Morse index, the less stable the solution, with Morse index 0 meaning the solution is stable under small perturbations (or is a stable equilibrium solution of the corresponding parabolic problem). We want to show that the behavior of solutions with finite Morse index is usually not beyond reach and can sometimes be completely classified.

Related publications:

  1. Musso, Monica; Wei, Juncheng; Yan, Shusen; Infinitely many positive solutions for a nonlinear field equation with super-critical growth. Proc. Lond. Math. Soc. (3) 112 (2016), no. 1, 1-26.
  2. Deng, Yinbin; Peng, Shuangjie; Yan, Shusen; Critical exponents and solitary wave solutions for generalized quasilinear Schrödinger equations. J. Differential Equations 260 (2016), no. 2, 1228-1262.
  3. Cao, Daomin; Peng, Shuangjie; Yan, Shusen; Planar vortex patch problem in incompressible steady flow. Adv. Math. 270 (2015), 263-301.
  4. Deng, Yinbin; Lin, Chang-Shou; Yan, Shusen; On the prescribed scalar curvature problem in R^N, local uniqueness and periodicity. J. Math. Pures Appl. (9) 104 (2015), no. 6, 1013-1044.
  5. Du, Yihong; Guo, Zongming; Finite Morse index solutions of weighted elliptic equations and the critical exponents. Calc. Var. Partial Differential Equations 54 (2015), no. 3, 3161-3181.
  6. Du, Yihong; Wei, Lei; Boundary behavior of positive solutions to nonlinear elliptic equations with Hardy potential. J. Lond. Math. Soc. (2) 91 (2015), no. 3, 731-749.
  7. Dancer, E. N.; A remark on stable solutions of nonlinear elliptic equations on R^3 or R^4. Differential Integral Equations 27 (2014), no. 5-6, 483-488.
  8. Du, Yihong; Guo, Zongming; Wang, Kelei; Monotonicity formula and e-regularity of stable solutions to supercritical problems and applications to finite Morse index solutions. Calc. Var. Partial Differential Equations 50 (2014), no. 3-4, 615-638.
  9. Lin, Chang-Shou; Yan, Shusen; Bubbling solutions for the SU(3) Chern-Simons model on a torus. Comm. Pure Appl. Math. 66 (2013), no. 7, 991-1027.
  10. Lin, Chang-Shou; Yan, Shusen; Existence of bubbling solutions for Chern-Simons model on a torus. Arch. Ration. Mech. Anal. 207 (2013), no. 2, 353-392.
  11. Du, Yihong; Guo, Zongming; Finite Morse-index solutions and asymptotics of weighted nonlinear elliptic equations. Adv. Differential Equations 18 (2013), no. 7-8, 737-768.
  12. Dancer, E. N.; Du, Yihong; Efendiev, Messoud; Quasilinear elliptic equations on half- and quarter-spaces. Adv. Nonlinear Stud. 13 (2013), no. 1, 115-136.