Linking Methods in Critical Point Theory

As is well known, The Great Divide (a.k.a. The Continental Divide) is formed by the Rocky Mountains stretching from north to south across North America. It creates a virtual "stone wall" so high that wind, rain, snow, etc. cannot cross it. This keeps the weather distinct on both sides. Since railroad trains cannot climb steep grades and tunnels through these mountains are almost formidable, the Canadian Pacific Railroad searched for a mountain pass providing the lowest grade for its tracks. Employees discovered a suitable mountain pass, called the Kicking Horse Pass, el. 5404 ft., near Banff, Alberta. (One can speculate as to the reason for the name.) This pass is also used by the Trans-Canada Highway. At the highest point of the pass the railroad tracks are horizontal with mountains rising on both sides. A mountain stream divides into two branches, one flowing into the Atlantic Ocean and the other into the Pacific. One can literally stand (as the author did) with one foot in the Atlantic Ocean and the other in the Pacific. The author has observed many mountain passes in the Rocky Mountains and Alps. What connections do mountain passes have with nonlinear partial dif­ ferential equations? To find out, read on ...

1 Critical Point Theory.- 1.1 Introduction.- 1.2 Other Geometries.- 1.3 Semilinear Boundary Value Problems.- 1.4 The Critical Point Alternative.- 1.5 The Mountain Cliff Theorem.- 1.6 Estimates on Eigenspaces.- 1.7 Asymptotic Limits.- 1.8 Types of Resonance.- 1.9 Multiple Solutions.- 1.10 Eigenvalues.- 2 Linking.- 2.1 The Basic Concept.- 2.2 The Flow.- 2.3 Weaker Conditions.- 2.4 Another Form.- 2.5 Some Consequences.- 2.6 Examples of Linking.- 2.7 Critical Sequences.- 2.8 The Compact Case.- 2.9 A Sandwich Theorem.- 2.10 Appendix I: Pseudo-Gradients.- 2.11 Appendix II: Differential Equations.- 3 Semilinear Boundary Value Problems.- 3.1 Introduction.- 3.2 Mountain Pass Geometry.- 3.3 Finding a Critical Sequence.- 3.4 Obtaining a Solution.- 3.5 Solving the Problem.- 3.6 Resonance.- 3.7 Appendix I: The Sobolev Inequality.- 4 Alternative Methods 73.- 4.1 Introduction.- 4.2 The Saddle Point Alternative.- 4.3 An Alternate Form.- 4.4 Some Corollaries.- 4.5 An Application.- 4.6 Superlinear Problems.- 4.7 Some Modifications.- 5 Bounded Saddle Point Methods.- 5.1 Introduction.- 5.2 A Bounded Mountain Pass Lemma.- 5.3 The Mountain Pass Alternative.- 5.4 A Compactness Condition.- 5.5 Dual Situations.- 5.6 Combined Results.- 5.7 Nonlinear Eigenvalues.- 5.8 Double Resonance.- 5.9 Appendix I: Generalized Pseudo-Gradients.- 6 Estimates on Subspaces.- 6.1 Introduction.- 6.2 Some Important Quantities.- 6.3 The Estimates.- 6.4 Nontrivial Solutions.- 6.5 A Variation.- 7 The Fu?ík Spectrum.- 7.1 Introduction.- 7.2 Jumping Nonlinearities.- 7.3 Quantities Related to the Fu?í k Spectrum.- 7.4 Applications.- 8 Resonance.- 8.1 Introduction.- 8.2 More on Double Resonance.- 8.3 Resonance Involving Many Eigenvalues.- 8.4 Landesman—Lazer Resonance.- 8.5 Equal Limits at Infinity.- 8.6 Nonvanishing Solutions.- 8.7 Unequal Limits at Infinity.- 9 Boundary Conditions.- 9.1 Introduction.- 9.2 Bounded Linking.- 9.3 Reverse Boundary Conditions.- 9.4 An Application.- 9.5 Sufficient Conditions.- 10 Multiple Solutions.- 10.1 Introduction.- 10.2 The Abstract Theory.- 10.3 Some Applications.- 10.4 Additional Solutions.- 11 Nonlinear Eigenvalues.- 11.1 Introduction.- 11.2 The Hampwile Theorem.- 11.3 Applications.- 12 Strong Resonance.- 12.1 Introduction.- 12.2 Simple Solutions.- 12.3 A Different Approach.- 12.4 Resonance at the First Eigenvalue.- 12.5 Additional Solutions.- 13 Notes, Remarks and References.