Abstract
In Chapter II we showed how to construct critical sequences, i.e., sequences that satisfy
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))
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© 1999 Springer Science+Business Media New York
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Schechter, M. (1999). Alternative Methods. In: Linking Methods in Critical Point Theory. Birkhäuser, Boston, MA. https://doi.org/10.1007/978-1-4612-1596-7_4
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DOI: https://doi.org/10.1007/978-1-4612-1596-7_4
Publisher Name: Birkhäuser, Boston, MA
Print ISBN: 978-1-4612-7210-6
Online ISBN: 978-1-4612-1596-7
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