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Seminars and Talks

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[3 Nov 2015 | No Comment | 46 views]

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.

[29 Oct 2009 | No Comment | 665 views]

I gave the talk “Linear and nonlinear elliptic equations with critical exponents, Hardy-Sobolev terms and combined singular terms” at the regular seminar of the Group Functional Analysis and Applications of UIMA.

[24 Jul 2009 | No Comment | 357 views]

I gave the talk “Four solutions for an elliptic equation with critical exponent and singular term” at Equadiff 12, July 20-24, 2009, Brno, Czech Republic.

We show, under some conditions, that the Dirichlet problem

-\Delta u - {\lambda\over {|x|^2}}u = |u|^{2^{\ast}-2}u  + \mu|x|^{\alpha-2}u + f(x) a.e. on \Omega\backslash\{0\} with u\in H_0^1(\Omega),

has four nontrivial solutions where at least one of solutions is sign changing. We assume that 0\in \Omega\subset\mathbb{R}^N (with N\geq 3) is a bounded domain with smoothboundary, 2^{\ast}:=2N/(N-2) is the critical Sobolev exponent, 0\leq \lambda < \Lambda:=((N-2)/2)^2 and f\in L^\infty(\Omega). These results extend some previous works on the literature, as [1-3]. This is a joint work with Jianqing Chen.
[1] N. Hirano and N. Shioji, A multiplicity result including a sign changing solution for an inhomogeneous Neumann problem with critical exponent, Proc. Roy. Soc. Edinburgh 137A(2007) 333-347.
[2] D. Kang and Y. Deng, Multiple solutions for inhomogeneous elliptic problems involving critical Sobolev-Hardy exponents, J. Math. Anal. Appl. 60(2005) 729-753.
[3] G. Tarantello, Multiplicity results for an inhomogeneous Neumann problem with critical exponent, Manuscripta Math. 81(1993) 51-78.

[29 Nov 2008 | No Comment | 295 views]


I participated in the 5th JEM workshopImpact of ICT on the Teaching of Mathematics and on the Mathematics Curriculum“, 26-27 Nov 2008, Paris, giving the talk “ICT in the Portuguese Education, the initiatives EECM, PmatE, TexMat, and IntBooks, and the importance of Abstract Abilities”. JEM kindly supported the traveling and living expenses.

You find the presentation slides HERE (18.81MB).

[31 Oct 2008 | No Comment | 302 views]

diaeqThe “Centro de Matemática” of the University of Minho organized the “Dia das Equações” on 31th Oct 2008, where I presented the seminar “Existence of solutions of elliptic problems: a general nonlinear eigenvalue problem“.

[19 May 2007 | No Comment | 373 views]

Participation in the Conference Variational and Topological Methods, 23-26 May 2007, Flagstaff, USA, giving the talk “A Multiplicity theorem for hemivariational inequalities with a p-Laplacian-like differential operator“.