Dušan Repovš

Partial differential equations with variable exponents : variational methods and qualitative analysis

Isotropic and Anisotropic Function Spaces Lebesgue and Sobolev Spaces with Variable Exponent History of function spaces with variable exponent Lebesgue spaces with variable exponent Sobolev spaces with variable exponent Dirichlet energies and Euler-Lagrange equations Lavrentiev phenomenon Anisotropic function spaces Orlicz spaces Variational Analysis of Problems with Variable Exponents Nonlinear Degenerate Problems in Non-Newtonian Fluids Physical motivation A boundary value problem with nonhomogeneous differential operator Nonlinear eigenvalue problems with two variable exponents A sublinear perturbation of the eigenvalue problem associated to the Laplace operator Variable exponents versus Morse theory and local linking The Caffarelli-Kohn-Nirenberg inequality with variable exponent Spectral Theory for Differential Operators with Variable Exponent Continuous spectrum for differential operators with two variable exponents A nonlinear eigenvalue problem with three variable exponents and lack of compactness Concentration phenomena: the case of several variable exponents and indefinite potential Anisotropic problems with lack of compactness and nonlinear boundary condition Nonlinear Problems in Orlicz-Sobolev Spaces Existence and multiplicity of solutions A continuous spectrum for nonhomogeneous operators Nonlinear eigenvalue problems with indefinite potential Multiple solutions in Orlicz-Sobolev spaces Neumann problems in Orlicz-Sobolev spaces Anisotropic Problems: Continuous and Discrete Anisotropic Problems Eigenvalue problems for anisotropic elliptic equations Combined effects in anisotropic elliptic equations Anisotropic problems with no-flux boundary condition Bifurcation for a singular problem modelling the equilibrium of anisotropic continuous media Difference Equations with Variable Exponent Eigenvalue problems associated to anisotropic difference operators Homoclinic solutions of difference equations with variable exponents Low-energy solutions for discrete anisotropic equations Appendix A: Ekeland Variational Principle Appendix B: Mountain Pass Theorem Bibliography Index A Glossary is included at the end of each chapter.

Stationary waves of Schrödinger-type equations with variable exponent

Abstract. We are concerned with a class of nonlinear Schrödinger-type equations with a reaction term and a differential operator that involves a variable exponent. By using related variational methods, we establish several existence results.

On intersections of compacta of complementary dimensions in Euclidean space

A pair of maps f:X → Rn and g: Y → Rn of compacta X and Y into the Euclidean n-space is said to have a stable intersection if there exist e>0 such that for any other pair of maps f′:X → Rn and g′:Y → Rn, satisfying ρ(f,f′) <e and ρ(g,g′)<e, it follows that f′(X) ∩ g′(Y) ≠ 0. The main result of this paper is the following theorem: Let X and Y be compacta and let n = dim X + dim Y. Then there exists a pair of maps f:X → Rn and g:Y → Rn with stable intersection if and only if dim(X × Y) = n.

Higher nonlocal problems with bounded potential

Abstract The aim of this paper is to study a class of nonlocal fractional Laplacian equations depending on two real parameters. More precisely, by using an appropriate analytical context on fractional Sobolev spaces due to Servadei and Valdinoci, we establish the existence of three weak solutions for nonlocal fractional problems exploiting an abstract critical point result for smooth functionals. We emphasize that the dependence of the underlying equation from one of the real parameters is not necessarily of affine type.

Combined effects in nonlinear problems arising in the study ofanisotropic continuous media

We are concerned with the Lane-Emden-Fowler equation −∆u = λk(x)u ± h(x)u in Ω, subject to the Dirichlet boundary condition u = 0 on ∂Ω, where Ω is a smooth bounded domain in R , k and h are variable potential functions, and 0 < q < 1 < p. Our analysis combines monotonicity methods with variational arguments.

On a non-homogeneous eigenvalue problem involving a potential: An Orlicz–Sobolev space setting

In this paper we study a non-homogeneous eigenvalue problem involving variable growth conditions and a potential V . The problem is analyzed in the context of Orlicz-Sobolev spaces. Connected with this problem we also study the optimization problem for the particular eigenvalue given by the infimum of the Rayleigh quotient associated to the problem with respect to the potential V when V lies in a bounded, closed and convex subset of a certain variable exponent Lebesgue space. 2000 Mathematics Subject Classification: 35D05, 35J60, 35J70, 58E05, 68T40, 76A02.

On Countably Compact 0-Simple Topological Inverse Semigroups

We describe the structure of 0-simple countably compact topological inverse semigroups and the structure of congruence-free countably compact topological inverse semigroups.

A deleted product criterion for approximability of maps by embeddings

Abstract We prove the following theorem: Suppose that m ⩾ 3(n + 1) 2 and that ƒ : K → R m is a PL map of an n -dimensional finite polyhedron K . Then ƒ is approximable by embeddings if and only if there exists an equivariant homotopical extension Φ : K → S m−1 of the map \ tf : K ƒ → S m−1 , defined by f (x,y) = (ƒ(x) − ƒ(y)) (∥ƒ(x) − ƒ(y)∥) , where K ƒ = {(x, y) e K × K ¦ ƒ(x) ≠ ƒ(y)} . Our result is a controlled version of the classical deleted product criterion of embeddability of n -dimensional polyhedra in R m . The proof requires additional (compared with the classical result) general position arguments, for which the restriction m ⩾ 3(n + l) 2 is again necessary. We also introduce the van Kampen obstruction for approximability by embeddings.

On sequences of solutions for discrete anisotropic equations

Taking advantage of a recent critical point theorem, the existence of infinitely many solutions for an anisotropic problem with a parameter is established. More precisely, a concrete interval of positive parameters, for which the treated problem admits infinitely many solutions, is determined without symmetry assumptions on the nonlinear data. Our goal was achieved by requiring an appropriate behavior of the nonlinear terms at zero, without any additional conditions.

Nonlinear algebraic systems with discontinuous terms

Abstract Using a multiple critical points theorem for locally Lipschitz continuous functionals, we establish the existence of at least three distinct solutions for a parametric discrete differential inclusion problem involving a real symmetric and positive definite matrix. Applications to tridiagonal, fourth-order and partial difference inclusions are presented.