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In this chapter, we study the existence of sign-changing solutions for nonlinear elliptic equations via linking methods. A linking type theorem is established with the location of the critical point in terms of the cone structure of the space. The abstract theorem is applied to elliptic equations that have jumping nonlinearities. Under stronger conditions, we show that the existence of sign-changing critical points can be independent of the Fučík spectrum. [ABSTRACT FROM AUTHOR]

The theory of Chapter 7 was established to fit the systems with Hamiltonian type potentials. In this chapter, a linking theorem will be given which is designed for systems with gradient type potentials. [ABSTRACT FROM AUTHOR]

Let E be a real Hilbert space with an inner product 〈·, ·〉 and the associated norm ‖ · ‖. Let A and B be two bounded subsets of E such that A links B. We describe the situation in which there are two linear, bounded and invertible operators B1, B2 : E → E and a functional H ∈ C1 (E, R) whose values are separated by B1B and B2A, i.e., $$ \mathop {\sup }\limits_{B_2 A} H \leqslant \mathop {\inf }\limits_{B_1 B} H $$. Note that B1B and B2A become much more uncontrollable. We prove the existence of a critical point of H without assuming the (PS) type conditions. This theory fits some special elliptic systems. [ABSTRACT FROM AUTHOR]

We solve elliptic semilinear boundary value problems in which the nonlinear term is superlinear, but the nonlinearities need not satisfy the superquadracity condition. Because of this, we are able to include more equations than hitherto permitted. The region is not assumed to be bounded. [ABSTRACT FROM AUTHOR]

If A links B, does B link A? Can they yield two different critical points without the (PS) type compactness conditions? To find out, read on... [ABSTRACT FROM AUTHOR]

In this chapter, we first prove a weak linking theorem which, to some ex-tent, unifies the classical linking theorems. Moreover, it produces a bounded Palais-Smale sequence for a non-even functional. Applications will be given on the existence of homoclinic orbits for Hamiltonian systems and solutions to Schrödinger equations. [ABSTRACT FROM AUTHOR]

In this chapter we present some abstract theorems which concern the existence of infinitely many critical points for even functionals. The Palais-Smale type compactness condition is not necessary for the new results. By taking advantage of the abstract theorems, we study the existence of infinitely many large energy solutions for nonlinear Schrödinger equations and of infinitely many small energy solutions for semilinear elliptic equations with concave and convex nonlinearities. [ABSTRACT FROM AUTHOR]

If a functional has the same infimum on two balls Br ⊂ BR, we establish the existence result of a bounded (PS)-sequence. On the other hand, we give a new proof for classical results on functionals bounded below. [ABSTRACT FROM AUTHOR]

Let E be a real Hilbert space with an inner product 〈·, ·〉 and associated norm ‖ · ‖, and let I ∈ C1(E, R) be a strongly indefinite functional, i. e., I is unbounded from below and from above on any subspace of finite codimension. It is well known that the Morse index of any critical point of I must necessarily be infinite. In this case usually one can not expect to obtain any useful information from the usual Morse theory. In order to overcome this difficulty, a more advanced theory is needed. In sections 10.1-10.4 of this chapter, we first introduce the W. Kryszewski-A. Szulkin infinite-dimensional cohomology theory and a Morse theory associated with it (see [200]) and then we develop some methods to compute the groups precisely. Applications to Hamiltonian systems and beam equations will be given. [ABSTRACT FROM AUTHOR]

In this chapter, we present some classical results on nonlinear functional analysis and partial differential equation. Some of them are well known and we shall omit their proofs. For others, although their proofs may be found in many existing books, we make no apology for repeating them. [ABSTRACT FROM AUTHOR]