Nonlinear Partial Differential Equations of Elliptic Type

Radulescu V.D.

The present book is based on a one semester course at the University of Craiova. The goal of this textbook is to provide the background which is necessary to initiate work on a Ph.D. thesis in Applied Nonlinear Analysis. My purpose is to provide for the student a broad perspective in the subject, to illustrate the rich variety of phenomena encompassed by it and to impart a working knowledge of the most important techniques of analysis of the solutions of the equations. The level of this book is aimed at beginning graduate students. Prerequisites include a truly advanced Calculus course, basic knowledge on Functional Analysis and PDE, as well as the necessary tools on Sobolev spaces.

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Table of contents :
Introduction……Page 3
Method of sub and super solutions……Page 6
Abstract theorems……Page 18
A basic bifurcation theorem……Page 21
Qualitative properties of the minimal solution in a neighbourhood of the bifurcation point……Page 23
Bifurcation problems associated to convex positive asymptotic linear functions……Page 28
Basic properties of locally Lipschitz functionals……Page 43
Ekeland’s Variational Principle……Page 45
Mountain Pass and Saddle Point type theorems……Page 48
Applications of the Mountain-Pass Theorem……Page 61
Critical points and coerciveness of locally Lipschitz functionals with the strong Palais-Smale property……Page 65
Basic results on the notions of genus and Lusternik-Schnirelmann category……Page 73
A finite dimensional version of the Lusternik-Schnirelmann theorem……Page 76
Critical points of locally Lipschitz Z-periodic functionals……Page 80
Multivalued variants of some results of Brezis-Nirenberg and Mawhin-Willem……Page 86
Multivalued problems of Landesman-Lazer type with strong resonance at infinity……Page 97
Multivalued problems of Landesman-Lazer type with mixed resonance……Page 105