Alternative Methods

Abstract

In Chapter II we showed how to construct critical sequences, i.e., sequences that satisfy

$$G\left( {u_k } \right) \to c, - \infty < c \leqslant \infty ,G'\left( {u_k } \right)/\left( {\left\| {u_k } \right\| + 1} \right)^\beta \to 0$$

((4.1.1))

for some β ≥ 0, where G is a C¹-functional on a Banach space E (cf. Section 2.7). For our applications, (4.1.1) leads to a critical point provided the sequence is bounded (cf. Theorem 3.4.1). In the present chapter we shall show that, by fine tuning our arguments, we can obtain an alternative of the form: Either

• (a) there exists a Palais-Smale sequence, i.e., a sequence satisfying

  $$G\left( {u_k } \right) \to c, - \infty < c < \infty ,G'\left( {u_k } \right) \to 0,$$

  ((4.1.2))

  Or

• (b) there is a sequence satisfying

  $$\begin{gathered}G\left( {u_k } \right) \to c, - \infty < c \leqslant \infty ,\rho _k = \left\| {u_k } \right\| \to \infty \hfill \\G\left( {u_k } \right)/\rho _{\rho _k }^{\beta + 1} \to 0,G'\left( {u_k } \right)/\rho _{\rho _k }^\beta \to 0. \hfill \\\end{gathered}$$

  ((4.1.3))