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Seminar at DISIM – Univ. of L’Aquila

REF. DATE:3 November 2015 CREATED/MODIFIED: 4 November 2015 39 views No Comment

SeminarTITLE: Multiplicity results for Schrödinger-Poisson systems

LOCATION: Department of Information Engineering, Computer Science and Mathematics – University of L’Aquila [url:01]

ABSTRACT: We discuss the core of some results about the multiplicity of solutions of Schrödinger-Poisson systems of the following type

     $$ \left\{ \begin{array}{ll} -\Delta u+u+l(x)\phi u = f(x,u)  \quad & \mbox{ in } \mathbb{R}^3,\\ -\Delta \phi = l(x)u^2\quad & \mbox{ in } \mathbb{R}^3, \end{array}\right. $$

for some classes of nonlinearities f (e.g. indefinite nonlinearity or involving the critical exponent). These systems can be seen as nonlocal equations where variational methods and critical point theory may be applied. A direct method will be shown which do not involve Palais-Smale condition or the Ekeland variational principle. Special phenomena of these systems, compared with semilinear elliptic equations, will be point out. We emphasize open questions about the extension of these systems, a numerical analysis approach based on a recent reproducing kernel Hilbert and Tikhonov regularization technique, and discuss their possible applications to superconductors and nanoscience (computational chemistry).
Joint work with L. Huang and J. Chen.

 

Based on some papers as: (a) Nonlin. Anal. Real World Appl. 28(2016), 1-19 (to appear); (b) Libertas Math. 34:2(2014), 91-104; (c) J. Math. Anal. Appl. 408(2013), 55-69; (d) J. Differential Equations 255(2013), 2463-2483.

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