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## Exploratory Research Project

## "NONSMOOTH PHENOMENA IN NONLINEAR ELLIPTIC PROBLEMS"

## PN II ID PCE 2008 No. 501, ID 2162 (2009, 2010, 2011)

## Project summary

## Members of the research team

## Published/Accepted papers within this project

## Talks at conferences and research seminars

### Cs. Varga

### A. Kristály

### H. Lisei

### A. Molnar

The present project proposes the study of certain elliptic differential inclusion problems, as well as a class of hemivariational inequalities defined on unbounded domains and on Riemann manifolds by using recent variational methods. This study is motivated by several nonsmooth phenomena which appear in the nature, for instance, in fluid mechanics, quantum mechanics and the theory of fields. These problems lead us to differential inclusions (hemivariational inequalities) in various frameworks. The following two types of problems will be treated:

- Elliptic problems with discontinuities on unbounded domains
- Nonsmooth eigenvalue problems with and without constraints

In both cases the solutions will be found as critical points of the energy functional associated to the studied problem, they represent the equilibrium states of certain mechanical systems. In the case of quantum mechanics and the theory of fields, these problems describe the elementary particles' state, when the energy level varies. In order to achieve the theoretical results, we will apply recent variational methods, such as the nonsmooth version of the principle of symmetric criticality, as well the variational principles of Ricceri. Our purpose is to discuss the existence, multiplicity and the asymptotical behavior of the solutions. These problems will be considered on unbounded domains and on Riemann manifolds in both cases A and B.

Project manager: | Assoc. Prof. Dr. Hannelore Lisei |

Experienced researcher: | Prof. Dr. Csaba Varga (Curriculum Vitae, List of Publications) |

Assoc. Prof. Dr. Alexandru Kristály (Curriculum Vitae, List of Publications) | |

Young researcher: | Dr. Ioana Lazăr (Curriculum Vitae, List of Publications) member in the team in the period 01.06.2009 – 31 .05.2010 |

Andrea-Éva Molnár (PhD Student, Curriculum Vitae, List of Publications) member in the team since 01.06.2010 |

- Alexandru Kristály, Ioana Lazăr, Nikolaos S. Papageorgiou,
*A variational inequality on the half line, Nonlinear Analysis*. Theory, Methods & Applications, Volume 71, Issue 10, (2009), 5003-5009 (ISI) - Michael Filippakis, Alexandru Kristály, Nikolaos S. Papageorgiou,
*Existence of five nonzero solutions with exact sign for a $p$-Laplacian equation*. Discrete Contin. Dyn. Syst. 24 (2009), no. 2, 405-440 (ISI) - Hannelore Lisei, Csaba Varga,
*Multiple Solutions for a Differential Inclusion Problem with Nonhomogeneous Boundary Conditions*. Numerical Functional Analysis and Optimization, 30 (2009) 566-581 (ISI) - Alexandru Kristály, Csaba Varga,
*Variational-Hemivariational Inequalities on Unbounded Domains*. Studia Univ. Babes-Bolyai, Mathematica, LV, Nr. 2 (2010) 3-87 - Hannelore Lisei, Ioana Lazăr,
*Application of a Three Critical Point Theorem for a Class of Inclusion Problems*. Mathematica (Romanian Academy), Vol. 53 (76), No. 2 (2011) - Alexandru Kristály, Waclaw Marzantowicz, Csaba Varga,
*A non-smooth three critical points theorem with applications in differential inclusions*. J. Glob. Optim. 46 (2010), 49–62, DOI 10.1007/s10898-009-9408-0 (ISI) - Alexandru Kristály,
*On singular elliptic equations involving oscillatory terms*. Nonlinear Analysis 72 (2010) 1561-1569 (ISI) - Hannelore Lisei, Csaba Varga,
*Multiple Solutions for Gradient Elliptic Systems with Nonsmooth Boundary Conditions*. Mediterranean Journal of Mathematics 8 (2010), 69-79, DOI 10.1007/s00009-010-0052-1. (ISI) - Wilfried Grecksch, Hannelore Lisei,
*Stochastic Nonlinear Equations of Schrödinger Type*. To appear in: Stochastic Analysis and Applications (ISI) - Monica Bota, Andrea E. Molnár, Csaba Varga,
*On Ekeland’s variational principle in b-metric spaces*. To appear in: International Journal on Fixed Point Theory, Computation and Applications (ISI) - Andrea E. Molnár,
*A Nonsmooth Sublinear Elliptic Problem in R^N with Perturbations*. To appear in: Studia Univ. Babes-Bolyai, Ser. Mathematica - Wilfried Grecksch, Hannelore Lisei,
*Stochastic Schrödinger Equation Driven by Cylindrical Wiener Process and Fractional Brownian Motion*. To appear in: Studia Univ. Babes-Bolyai, Ser. Mathematica Volume LVI, Number 2, 381-391 (2011) - Hannelore Lisei, Andrea E. Molnár, Csaba Varga,
*On a class of inequality problems with lack of compactness*. Journal of Mathematical Analysis and Applications 378, 741-748 (2011), DOI 10.1016/j.jmaa.2010.12.041 (ISI) - Francesca Faraci,Antonio Iannizzoto, Alexandru Kristaly,
*Low dimensional compact embeddings of symmetric Sobolev spaces and applications*. Proceedings of the Royal Society of Edinburgh 141 A, 383–395 (2011) (ISI) - Dusan Repovs, Csaba Varga,
*Nash Type Solution for Hemivariational Inequality Systems*. Nonlinear Analysis TMA 74, 5585–5590 (2011) (ISI) - Alexandru Kristály, Ildiko Mezei,
*Multiple Solutions for a Perturbed System on Strip-Like Domains*. To appear in: Discrete and Continuous Dynamical Systems Series S, Vol. 5, No. 4, August 2012

- Universita degli Studi di Perugia (Perugia, Italia), Dipartimento di Matematica e Informatica:
*A non-smooth three critical points theorem with applications in differential inclusions*. (23.09.2009) - la Universita degli Studi di Perugia (Perugia, Italia), Dipartimento di Matematica e Informatica:
*Infinitely many solutions for a class of inhomogeneous Neumann problems*. (30.09.2009)

- Faculty of Mathematics and Informatics, Ovidius University Constanta, Romania:
*Weber-type problems and Nash equilibria: a geometrical approach*. (21.05.2010) - IMAR Montly Lectures, Bucharest, Romania:
*Elliptic problems involving oscillatory nonlinearities*. (17.11.2010) - 7th Bolyai-Gauss-Lobachevsky Conference, Cluj-Napoca, Romania:
*Nash-Stampacchia equilibrium points on Riemannian manifolds*. (05.-09.07.2010)

- Seminarul de Cercetare al Catedrei de Calcul Numeric şi Statistic de la Facultatea de Matematică şi Informatică a Universităţii Babeş-Bolyai din Cluj-Napoca:
*Soluţii multiple pentru ecuaţii neliniare generate de forme Dirichlet pe domenii fractale*. (10.03.2009) - GAMM-Workshop "Stochastische Modelle und Steuerung" Lutherstadt Wittenberg, Germania:
*Multiple Solutions for Nonlinear Equations Involving Dirichlet Forms*. (16 - 19.03.2009) - Romanian-German Symposium on Mathematics and Its Applications -Workshop in Nonlinear Analysis and Mathematical Physics, Sibiu:
*Multiple Solutions for Nonlinear Equations Involving Dirichlet Forms*. (14-17.05.2009) - Faculty of Mathematics and Computer Science Babes-Bolyai University, Cluj-Napoca, Romania:
*Stochastic Nonlinear Equations of Schrödinger Type*. (28.04.2010) - 8th AIMS Conference on Dynamical Systems, Differential Equations and Applications, Technical University Dresden, Germany:
*Solutions for Stochastic Equations of Schrödinger Type*. (25.–28.05.2010) - Deterministic and stochastic variational methods and applications - Workshop, Martin-Luther Universität Halle-Wittenberg, Germany:
*Stochastic Schrödinger Equations*. (07.-20.11.2010) - Invited Talk: Institute for Mathematics Paderborn University, Germany:
*Stochastic Schrödinger Equations.*. (19.07.2010)

- Transilvanian Student Conference on Science, Cluj-Napoca, Romania:
*Variational principles and applications*. (14-16.05.2010) - Deterministic and stochastic variational methods and applications - Workshop, Martin-Luther Universität Halle-Wittenberg, Germany:
*Ekeland’s variational principle in b-metric spaces*. (07.-20.11.2010)