1st Edition

# Partial Differential Equations with Variable Exponents Variational Methods and Qualitative Analysis

324 Pages 11 B/W Illustrations
by Chapman & Hall

323 Pages
by Chapman & Hall

Also available as eBook on:

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

Bibliography

Index

A Glossary is included at the end of each chapter.

### Biography

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.