Solutions Of Nonlinear Differential Equations: Existence Results Via The Variational Approach

by Lin Li, Shu-zhi Song

Overview

Variational methods are very powerful techniques in nonlinear analysis and are extensively used in many disciplines of pure and applied mathematics (including ordinary and partial differential equations, mathematical physics, gauge theory, and geometrical analysis).In our first chapter, we gather the basic notions and fundamental theorems that will be applied throughout the chapters. While many of these items are easily available in the literature, we gather them here both for the convenience of the reader and for the purpose of making this volume somewhat self-contained. Subsequent chapters deal with how variational methods can be used in fourth-order problems, Kirchhoff problems, nonlinear field problems, gradient systems, and variable exponent problems. A very extensive bibliography is also included.

Product Details

ISBN-13:	9789813108608
Publisher:	World Scientific Publishing Company, Incorporated
Publication date:	06/14/2016
Series:	Trends In Abstract And Applied Analysis , #3
Pages:	364
Product dimensions:	6.10(w) x 9.00(h) x 1.00(d)

Table of Contents

Preface vii

Some Notations and Conventions xi

1 Preliminaries and Variational Principles 1

1.1 Sobolev Spaces 1

1.2 Differentiable Functionals 7

1.3 Ekeland's Variational Principle 10

1.4 Minimax Principles 12

1.5 Ricceri's Variational Results 18

1.5.1 Three Critical Point Results 18

1.5.2 A General Variational Principle 22

2 Quasilinear Fourth-Order Problems 25

2.1 Introduction 25

2.2 Multiple Solutions of Quasilinear Fourth-Order Problems 25

2.3 Infinitely Many Solutions of Quasilinear Fourth-Order Problems 33

2.4 Quasilinear Fourth-Order Problems with Singular Term 44

2.5 Semilinear Fourth-Order Problems on R^N 51

3 Kirchhoff Problems 57

3.1 Introduction 57

3.2 Radial Solutions of Kirchhoff Problems 58

3.3 Multiple Solutions of Nonhomogeneous Problems 65

3.4 Multiple Solutions with Superlinear Nonlinearities 74

3.5 Multiple Solutions with Asymptotically Linear Nonlinearities 85

3.6 Infinitely Many- Solutions with Sublinear Nonlinearities 99

3.7 Multiple Solutions with Combined Nonlinearities 112

4 Nonlinear Field Problems 121

4.1 Introduction 121

4.2 Schrodinger-Maxwell Equations 122

4.2.1 Infinitely Many Solutions with Superlinear Nonlinearities 122

4.2.2 Multiple Solutions with Asymptotically Linear Nonlinearities 130

4.2.3 Quasilinear Schrodinger-Maxwell Equations 151

4.3 Klein-Gordon-Maxwell Systems 174

4.3.1 Infinitely Many Solutions 174

4.3.2 Sign-Changing Potential 190

4.3.3 Multiple Solutions without Odd Nonlinearities 199

4.3.4 Partially Sublinear Nonlinearities 209

4.3.5 Klein-Gordon Equation Coupled with Born-lnfcld Theory 213

5 Gradient Systems 225

5.1 Introduction 225

5.2 One Dimension Systems 225

5.2.1 Two-Point Boundary Value Systems 225

5.2.2 Hamiltonian Systems 233

5.2.3 Quasilinear Hamiltonian Systems 247

5.3 N Dimension Systems 256

5.3.1 Resonance Elliptic Systems 256

5.3.2 Quasilinear Fourth-Order Elliptic Systems 263

6 Variable Exponent Problems 277

6.1 Introduction 277

6.2 P(x)-Laplacian Problems 277

6.3 P(x)-Laplacian-Like Problems 283

6.4 P(x)-Biharmonic Problems 291

6.5 Two Parameter p(x)-Biharmonic Problems 304

Bibliography 313

Index 347