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

By Wenming Zou

This ebook provides a few of the newest examine in severe element concept, describing tools and providing the latest functions. insurance comprises extrema, even valued functionals, vulnerable and double linking, signal altering suggestions, Morse inequalities, and cohomology teams. purposes defined contain Hamiltonian structures, Schrödinger equations and platforms, leaping nonlinearities, elliptic equations and platforms, superlinear difficulties and beam equations.

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Then {Uu}ueTix,k) U t/o is an open covering of ^(A, k):={ueE: \\u\\ < m(4(A)) + 4, ^ A M < Cfe(A) + £0}. 40 CHAPTERS. EVEN FUNCTIONALS Hence, there exists a refinement {Nj}j^j such that Nj C Uu or Sj C UQ and a locahy Lipschitz continuous partition of unity {Pj}jeJi where J is the index set. Define UJJ{U) = uj{u) for Nj C Uu] ^j{u) = 0 for Nj C t/o, and set J\f := U^GJ^j- Then ^(A, k) C A/". Let ^*(A, k):={ueE: ^x{u) < Ck{\) - 2(A - A,)} and . ^(-, u). Noting that for any u G J^£Q{X, A:), we have that either 2 2 cjo = 0 or \\iOn\\ = ||c(;(i^)|| = ,, ^, , ,,, < —.

Q. Wang [236], P. L. Lions [227], M. Schechter [316], M. Struwe [349] and other papers for results without the (PS) condition. The ideas adopted here and in the sequel essentially come from L. Jeanjean in [191] (see also the survey of I. Ekeland-N. Ghoussoub [146]). Further developments were made by A. Szulkin, M. Willem, and W. Zou in [361, 378, 386, 395] for homoclinic orbits of Hamiltonian systems and Schrodinger equations which shall be described in later chapters. 44 CHAPTERS. 2 EVEN FUNCTIONALS High Energy Solutions Consider the existence of multiple solutions to the nonlinear Schrodinger equations (S) -Au^b{x)u = f{x,u), xeK^.

2, we get the conclusion. 9. /'d(0). Proof. Since ^^^Jw^{u{Xn)) = 0, then h{^n)f= / (^Xnf{x,u{Xn))u{Xn)^g{x,u{Xn))u{Xn)jdx. If, up to a subsequence, ||i^(An)|| ^ oo as n ^ oo, then by (D2), ||^(A^)f(l + o ( l ) ) = I g{x,u{Xn)MXn)dx, 52 CHAPTERS. EVEN FUNCTIONALS where o(l) ^ 0 as n ^ oo. Evidently, it is a contradiction if (D4)-(l) holds. t) for for 22 < Otherwise, set Wn = .. e. x G O. If i(; ^ 0 in ^ , and lim |ii|^oo U —CO in (D4)-(2), then, for n large enough, by Fatou's Lemma, we have that 1 + 0(1) -g{x,u{X„))u{X„) a ^ '""' ' " >c+[ ^ 2J -g(a:^A)MA„)^^^^,^^ oo, a contradiction.