By Wenming Zou

ISBN-10: 038732965X

ISBN-13: 9780387329659

This publication provides the various most recent examine in severe element conception, describing tools and proposing the latest purposes. insurance contains extrema, even valued functionals, vulnerable and double linking, signal altering recommendations, Morse inequalities, and cohomology teams. purposes defined contain Hamiltonian structures, Schrödinger equations and structures, leaping nonlinearities, elliptic equations and platforms, superlinear difficulties and beam equations.

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This open problem was studied in T. Bartsch-M. Willem [48]. 7 is an improvement and generalization of the results in [48]. Another example is f{x,u) =i^|i^|^~^ln(2 + |i^|), a G (1,2); g{x,u) =/ii^ln(l + |i^|). Then (Di), (D2), (D3) and (D4)-(2) hold if /i < 0; (Di), (D2), (D3) and (D4)-(3) hold with o^ = 2 if /i > 0. If we choose g{x,u) = u'^ for |i^| < 1; g{x,u) = c|^|-i/2ln(l + 1^1) for |^| > 1, then (Di), (D2), (D3) and (D4)-(l) hold. 7. Assume that (DiJ-fD^) hold. Then equation (D) has infinitely many solutions {uk] satisfying ^{uk) '•=-\\uk\\^ — I F{x,Uk)dx — I G{x,Uk)dx ^ 0~ 2 JQ JQ as k ^ 00, where \\u\\ = (J^ | y i^p(ix)^/^.

1) {I\u^v)-I\u^w),v-w) > \\v - w\\h{\\v - w\\) for all u G Ei^v^w G ^2, then we have the following results. (1) There exists a continuous function (j) : Ei ^ E2 such that I{u -\- (j){u)) = min I{u + v). veE2 Moreover, (j){u) is the unique member of E2 such that {I'{u^(j){u)),v) =0, \JveE2. veEi. (3) An element u e Ei is a critical point of J if and only if u -\- (j){u) is a critical point of L Proof. (1) For each u G ^ 1 , define H^ : E2 ^ K by Hu{v) = I{u + v). 1), Hu is of C^ and has at most one critical point.

Li-Z. Q. Wang [217], sign-changing small energy solutions were obtained. 7 of the present chapter were obtained by W. Zou in [385]. Chapter 4 Linking and Homoclinic Type Solutions In this chapter, we first prove a weak finking tfieorem wfiicfi, to some extent, 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 Schrodinger equations.

### Critical Point Theory and Its Applications by Wenming Zou

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