This is the first monograph dealing with the analysis of nonlinear problems described by partial differential equations with variable exponent and nonhomogeneous differential operators. This analysis is developed by using variational and topological methods. The authors are interested in many problems of huge interest both for mathematicians and for physicists. We refer only to the spectral analysis of the associated differential operators. Due to the presence of nonstandard nonlinear terms, the authors point out several striking phenomena like concentration properties of the spectrum. It is showed that the spectrum is no longer discrete (as for the standard Laplace operator) and a continuous spectrum can appear either in a neighborhood of the origin or at infinity. The authors of this monograph also elaborate general methods for the bifurcation analysis of solutions, the study of discrete anisotropic problems with variable exponents, etc. The monograph is mainly based on the authors’ original results in the field but some important recent contributions of other authors have been also introduced in this volume. The study of this field has become extremely active in the last decade due to several powerful applications of models involving variable exponents to concrete problems arising in electrorheological and thermorheological fluids, robotics, and image reconstruction.

COBISS.SI-ID: 17325401

A solenoid is an inverse limit of circles. When a solenoid is embedded in three space, its complement is an open three manifold. We discuss the geometry and fundamental groups of such manifolds, and show that the complements of different solenoids (arising from different inverse limits) have different fundamental groups. Embeddings of the same solenoid can give different groups. In particular, the nicest embeddings are unknotted at each level, and give an Abelian fundamental group, while other embeddings have non-Abelian groups. We show using geometry that every solenoid has uncountably many embeddings with nonhomeomorphic complements.

COBISS.SI-ID: 17540697

We describe a new general method for the fibrewise extension of a given endofunctor on the category of pointed topological spaces to the category of fibrewise pointed spaces. We derive some properties of the construction and show how it can be profitably used to build the Whitehead-Ganea framework for the fibrewise Lusternik-Schnirelmann category and the topological complexity.

COBISS.SI-ID: 17282649

This paper deals with a perturbation of the eigenvalue problem for the p-Laplace operator on an arbitrary open subset of the Euclidean space. The main features of this paper are: (i) the presence of a singular indefinite potential; (ii) the influence of a critical nonlinearity (in the Sobolev sense) in relationship with a subcritical term; (iii) the possible lack of compactness of the problem; (iv) the analysis carried out in this paper is developed in relationship with a related nonlinear eigenvalue problem. The paper is concerned with the study of multiple perturbation effects due to the presence of several types of nonlinearities, potential terms and a real parameter. The problem studied in this paper is in relationship with the classical Brezis-Nirenberg problem for the critical exponent. The main result establishes the existence of at least one nontrivial solution for all parameters less than a certain range. The proof combines variational and analytic tools, such as Hardy’s inequality and a recent concentration-compactness lemma for singular problems. The methods developed in this paper can be extended to several classes of nonlinear partial differential equations, including problems with variable exponents or involving poly-harmonic operators. This paper appeared in an excellent journal, placed near the top of the SCI list and it has received attention from experts in this area of nonlinear analysis. We have already reported on our results at international conferences in EU and we got very positive feedback from the experts.

COBISS.SI-ID: 17086297

The Ekeland variational principle is one of the strongest tools in nonlinear analysis. Roughly speaking, this result extends the Fermat theorem in an infinite dimensional framework. A major role of the Ekeland variational principle is that it establishes the existence of “almost critical points”. This property combined with a suitable compactness condition (usually of Palais-Smale type) implies the existence of weak solutions for large classes of nonlinear problems having a variational structure. This is usually done by employing variational powerful results like the mountain pass theorem, linking-type theorems, etc. In the present paper, the authors are concerned with the use of Ekeland-type variational principles in the mathematical analysis of the equilibrium problems. These problems have many applications in several fields like game theory and mathematical physics. The central application of the problems of this type is the Nash equilibrium, which is also illustrated in the present paper. One of the authors of this work (V.Radulescu) has a signed contract with Academic Press (Mathematics in Science and Engineering book series) to complete until June 2017 a monograph dealing with the mathematical analysis of the equilibrium problems.

COBISS.SI-ID: 17204569