1st Edition

Partial Differential Equations with Variable Exponents Variational Methods and Qualitative Analysis

    324 Pages 11 B/W Illustrations
    by Chapman & Hall

    Partial Differential Equations with Variable Exponents: Variational Methods and Qualitative Analysis provides researchers and graduate students with a thorough introduction to the theory of nonlinear partial differential equations (PDEs) with a variable exponent, particularly those of elliptic type.

    The book presents the most important variational methods for elliptic PDEs described by nonhomogeneous differential operators and containing one or more power-type nonlinearities with a variable exponent. The authors give a systematic treatment of the basic mathematical theory and constructive methods for these classes of nonlinear elliptic equations as well as their applications to various processes arising in the applied sciences.

    The analysis developed in the book is based on the notion of a generalized or weak solution. This approach leads not only to the fundamental results of existence and multiplicity of weak solutions but also to several qualitative properties, including spectral analysis, bifurcation, and asymptotic analysis.

    The book examines the equations from different points of view while using the calculus of variations as the unifying theme. Readers will see how all of these diverse topics are connected to other important parts of mathematics, including topology, differential geometry, mathematical physics, and potential theory.

    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 OrliczSobolev 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



    A Glossary is included at the end of each chapter.


    Vicenţiu D. Rădulescu is a distinguished adjunct professor at the King Abdulaziz University of Jeddah, a professorial fellow at the "Simion Stoilow" Mathematics Institute of the Romanian Academy, and a professor of mathematics at the University of Craiova. He is the author of several books and more than 200 research papers in nonlinear analysis. He is a Highly Cited Researcher (Thomson Reuters) and a member of the Accademia Peloritana dei Pericolanti. He received his Ph.D. from the Université Pierre et Marie Curie (Paris 6).

    Dušan D. Repovš is a professor of geometry and topology at the University of Ljubljana and head of the Topology, Geometry and Nonlinear Analysis Group at the Institute of Mathematics, Physics and Mechanics in Ljubljana. He is the author of several books and more than 300 research papers in topology and nonlinear analysis. He is a member of the New York Academy of Sciences, the European Academy of Sciences, and the Engineering Academy of Slovenia. He received his Ph.D. from Florida State University.