Variational Methods for Potential Operator Equations: With Applications to Nonlinear Elliptic Equations

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Assuming some familiarity with functional analysis, nonlinear partial differential equations, and the theory of Sobolev spaces, Chabrowski (U. of Queensland) presents methods and problems of the variational calculus directly related to the theory of differential equations of elliptic type. These methods can be applied to nonlinear wave equations, Hamiltonian systems, and problems relevant to surfaces of prescribed mean curvature. Chapters cover: constrained minimization, applications of Lusternik-Schnirelman theory, nonhomogeneous potentials, potentials with covariance conditions, eigenvalues and level sets, generalizations of the (intriguingly named) mountain pass theorem, nondifferentiable functionals, and the concentration- compactness principle. An appendix helpfully furnishes some of the knowledge prerequisites. Annotation c. by Book News, Inc., Portland, Or.
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Table of Contents

1 Constrained minimization 1
2 Applications of Lusternik-Schnirelman theory 39
3 Nonhomogeneous potentials 74
4 Potentials with covariance condition 115
5 Eigenvalues and level sets 128
6 Generalizations of the mountain pass theorem 149
7 Nondifferentiable functionals 167
8 Concentration-compactness principle - subcritical case 198
9 Concentration-compactness principle - critical case 224
App. A.1 Sobolev spaces 253
App. A.2 Embedding theorems 254
App. A.3 Compact embeddings of spaces W[superscript 1,p](R[superscript n]) and D[superscript 1,p](R[superscript n]) 255
App. A.4 Conditions of concentration and uniform decay at infinity 259
App. A.5 Compact embedding for H[actual symbol not reproducible](R[superscript n]) 261
App. A.6 Schwarz symmetrization 264
App. A.7 Pointwise convergence 264
App. A.8 Gateaux derivatives 266
Bibliography 270
Glossary 287
Index 289
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