Periodic solutions for time-dependent subdifferential evolution inclusions
Nikolaos S. Papageorgiou Vicenţiu D. Rădulescu
Evolution Equations & Control Theory 2017, 6(2): 277-297 doi: 10.3934/eect.2017015

We consider evolution inclusions driven by a time dependent subdifferential plus a multivalued perturbation. We look for periodic solutions. We prove existence results for the convex problem (convex valued perturbation), for the nonconvex problem (nonconvex valued perturbation) and for extremal trajectories (solutions passing from the extreme points of the multivalued perturbation). We also prove a strong relaxation theorem showing that each solution of the convex problem can be approximated in the supremum norm by extremal solutions. Finally we present some examples illustrating these results.

keywords: Convex subdifferential multivalued perturbation extremal solutions strong relaxation
Noncoercive elliptic equations with subcritical growth
Vicenţiu D. Rădulescu
Discrete & Continuous Dynamical Systems - S 2012, 5(4): 857-864 doi: 10.3934/dcdss.2012.5.857
We study a class of nonlinear elliptic equations with subcritical growth and Dirichlet boundary condition. Our purpose in the present paper is threefold: (i) to establish the effect of a small perturbation in a nonlinear coercive problem; (ii) to study a Dirichlet elliptic problem with lack of coercivity; and (iii) to consider the case of a monotone nonlinear term with subcritical growth. This last feature enables us to use a dual variational method introduced by Clarke and Ekeland in the framework of Hamiltonian systems associated with a convex Hamiltonian and applied by Brezis to the qualitative analysis of large classes of nonlinear partial differential equations. Connections with the mountain pass theorem are also made in the present paper.
keywords: Mountain pass geometry. Nonlinear elliptic equation Dirichlet boundary condition Dual variational method
Bifurcation of positive solutions for nonlinear nonhomogeneous Robin and Neumann problems with competing nonlinearities
Nikolaos S. Papageorgiou Vicenţiu D. Rădulescu
Discrete & Continuous Dynamical Systems - A 2015, 35(10): 5003-5036 doi: 10.3934/dcds.2015.35.5003
In this paper we deal with Robin and Neumann parametric elliptic equations driven by a nonhomogeneous differential operator and with a reaction that exhibits competing nonlinearities (concave-convex nonlinearities). For the Robin problem and without employing the Ambrosetti-Rabinowitz condition, we prove a bifurcation theorem for the positive solutions for small values of the parameter $\lambda>0$. For the Neumann problem with a different geometry and using the Ambrosetti-Rabinowitz condition we prove bifurcation for large values of $\lambda>0$.
keywords: nonhomogeneous differential operator bifurcation phenomena positive solutions Competing nonlinearities Robin and Neumann problems.
On a new fractional Sobolev space and applications to nonlocal variational problems with variable exponent
Anouar Bahrouni VicenŢiu D. RĂdulescu
Discrete & Continuous Dynamical Systems - S 2018, 11(3): 379-389 doi: 10.3934/dcdss.2018021

The content of this paper is at the interplay between function spaces $L^{p(x)}$ and $W^{k, p(x)}$ with variable exponents and fractional Sobolev spaces $W^{s, p}$. We are concerned with some qualitative properties of the fractional Sobolev space $W^{s, q(x), p(x, y)}$, where $q$ and $p$ are variable exponents and $s∈ (0, 1)$. We also study a related nonlocal operator, which is a fractional version of the nonhomogeneous $p(x)$-Laplace operator. The abstract results established in this paper are applied in the variational analysis of a class of nonlocal fractional problems with several variable exponents.

keywords: Fractional $p(x)$-Laplace operator density integral functional Gagliardo seminorm variational method
A tribute to Professor Philippe G. Ciarlet on his 70th birthday
Hervé Le Dret Vicenţiu D. Rădulescu Roderick S. C. Wong
Communications on Pure & Applied Analysis 2009, 8(1): 1-4 doi: 10.3934/cpaa.2009.8.1
Professor Philippe G. Ciarlet was born on October 14, 1938, in Paris. He obtained his undergraduate degree in 1961 from the celebrated École Polytechnique in Paris, followed by graduate studies (1962-1964) at the École Nationale des Ponts et Chaussées in Paris. Professor Ciarlet received in 1966 his Ph.D. at the Case Institute of Technology, Cleveland, U.S.A., under the guidance of Professor Richard S. Varga. The title of his Ph.D. thesis is Variational Methods for Non-Linear Boundary-Value Problems. He continued with a Doctorat d'État Fonctions de Green Discrètes et Principe du Maximum Discret) at the University of Paris in 1971 and his advisor was Professor Jacques-Louis Lions.

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Positive solutions for perturbations of the Robin eigenvalue problem plus an indefinite potential
Nikolaos S. Papageorgiou Vicenţiu D. Rădulescu Dušan D. Repovš
Discrete & Continuous Dynamical Systems - A 2017, 37(5): 2589-2618 doi: 10.3934/dcds.2017111

We study perturbations of the eigenvalue problem for the negative Laplacian plus an indefinite and unbounded potential and Robin boundary condition. First we consider the case of a sublinear perturbation and then of a superlinear perturbation. For the first case we show that for $ λ<\widehat{λ}_{1}$ ($ \widehat{λ}_{1}$ being the principal eigenvalue) there is one positive solution which is unique under additional conditions on the perturbation term. For $ λ≥q\widehat{λ}_{1}$ there are no positive solutions. In the superlinear case, for $ λ<\widehat{λ}_{1}$ we have at least two positive solutions and for $ λ≥q\widehat{λ}_{1}$ there are no positive solutions. For both cases we establish the existence of a minimal positive solution $ \bar{u}_{λ}$ and we investigate the properties of the map $ λ\mapsto\bar{u}_{λ}$.

keywords: Indefinite and unbounded potential Robin eigenvalue problem sublinear perturbation superlinear perturbation maximum principle positive solution minimal positive solution
Perturbations of nonlinear eigenvalue problems
Nikolaos S. Papageorgiou Vicenţiu D. Rădulescu Dušan D. Repovš
Communications on Pure & Applied Analysis 2019, 18(3): 1403-1431 doi: 10.3934/cpaa.2019068

We consider perturbations of nonlinear eigenvalue problems driven by a nonhomogeneous differential operator plus an indefinite potential. We consider both sublinear and superlinear perturbations and we determine how the set of positive solutions changes as the real parameter $λ$ varies. We also show that there exists a minimal positive solution $\overline{u}_λ$ and determine the monotonicity and continuity properties of the map $λ\mapsto\overline{u}_λ$. Special attention is given to the particular case of the $p$-Laplacian.

keywords: Nonhomogeneous differential operator sublinear and superlinear perturbation nonlinear regularity nonlinear maximum principle comparison principle minimal positive solution

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