By Maria do Rosário Grossinho, Stepan Agop Tersian
The publication is meant to be an advent to serious aspect conception and its functions to differential equations. even supposing the similar fabric are available in different books, the authors of this quantity have had the subsequent ambitions in brain:
- to provide a survey of current minimax theorems,
- to offer purposes to elliptic differential equations in bounded domain names,
- to contemplate the twin variational procedure for issues of non-stop and discontinuous nonlinearities,
- to give a few parts of serious element idea for in the community Lipschitz functionals and provides purposes to fourth-order differential equations with discontinuous nonlinearities,
- to review homoclinic ideas of differential equations through the variational equipment.
The contents of the ebook encompass seven chapters, every one divided into a number of sections.
Audience: Graduate and post-graduate scholars in addition to experts within the fields of differential equations, variational tools and optimization.
Read or Download An Introduction to Minimax Theorems and Their Applications to Differential Equations PDF
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Additional resources for An Introduction to Minimax Theorems and Their Applications to Differential Equations
Therefore (2) is satisfied and the Theorem is proved . • Another (PS) type condition has been introduced by M. Schechter [Sch1J, [Sch2]. Let W be the set of positive non-increasing functions 'l/J (s) : R+ -+ R+ such that i OO 'l/J (s) ds = 00. 8 (Schechter, 1991). Let c E Rand 'l/J E W. The C 1 functional f satisfies (PS)c,1/J if every sequence (Xj)j C X such that . _ lIff (Xj) - c and . ' (Xj)ll_ lIf 'l/J (1Ixjll) - 0, has a convergent subsequence in X. If'l/J is a constant we have the usual (PS)c condition, if'l/J (s) = are in the case of Cerami (PSC)c condition.
44) it follows (11Yjll) 11f' (Yj) II -+ O. By (PS)1jJ condition there exists a critical point Y such that every p such that r < p < R . 20 we obtain the following "three critical point theorem" (which we refer as TCPT). 8. Let fECI (X, R), 'I/J E \[! and suppose that f satisfies (PS)1jJ condition. Suppose that f has two local minima. Then f has at least one more critical point. Similar three critical points theorems with (PS) condition were proved by Mawhin & Willem [MW1], Figueredo & Sollimini [FS], Pucci & Serrin [PS1].
Moreover, there exists C1 < co such that if If (x) - cl ~ 2c1, Ilxll ~ R, d (x, Kc) ~ 6, Minimax Theorems 29 then 11/ (x) II ~ 4;1. 34) In fact, assuming the contrary, there exists a sequence (Xj)j such that 1 If (Xj) - ci ::; -:-, J d (Xj, Kc) ~ Ilxjll::; R, 0 and 11/ (Xj)11 < y. < 'P (~), J Since 'P is increasing, 'P (1Ixjll) ::; 'P (R). So 'P (1Ixjll) 11/ (Xj)11 ::; 'P (R) 11/ (Xj) II and it follows that lif'P (1lxjll) III (Xj)11 = o. Then limjxj = Xo, j' (xo) = 0 and f (xo) which contradicts to d (x, Kc) ~ O.