Critical Point Theory and Its Applications - download pdf or read online

By Wenming Zou

ISBN-10: 038732965X

ISBN-13: 9780387329659

ISBN-10: 0387329684

ISBN-13: 9780387329680

This booklet offers the various most recent learn in serious aspect concept, describing equipment and providing the most recent purposes. assurance comprises extrema, even valued functionals, vulnerable and double linking, signal altering recommendations, Morse inequalities, and cohomology teams. purposes defined contain Hamiltonian platforms, Schrödinger equations and platforms, leaping nonlinearities, elliptic equations and structures, superlinear difficulties and beam equations.

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HIGH ENERGY SOLUTIONS 47 estimates: $AW > l\\uf-^Ml-^\\u\\; > \hf-c\\u\\i > \hf-cal\\ur. If we choose rk := (4cpQ^^)^/^^~^\ then for u e Zk with ||i^|| = r/c, we get that ^xiu)>iAcpalf/(^-''\\-^):=h. It follows that bkW '= inf >bk^oo ueZk,\\u\\=rk as A: ^ oo uniformly for A. e. A G [1,2], there exists a sequence {u^{X)}'^^i such that sup||«^(A)||<^,$',(«n(A))-0 n and ^A(4(A))^Cfe(A)>6fe(A)>6fe, as n ^ oo. 41, {u^{X)}'^^i has a convergent subsequence. Hence, there exists a z^{X) such that ^^(z^(A)) = 0 and ^x{z^{X)) G [bk^Ck].

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. It is similar if lim |ii|^oo ^— = oo in (D4)-(3). Therefore, U w = 0. Define ^x (tnu(Xn)) := max ^x (tu(Xn)), then lim ^x (^n'^(^n)) = OO, {^Xn{tnU{Xn)),tnU{Xn)) OO = < tG[0,l] "^ = 0. I t folloWS t h a t . lim^ ( ^x^{tnU{Xn)) - l i m Xn I ( -f{x,tnU{Xn))tnU{Xn) n^oo /o V 2 n^oo -{^xA^nU{Xn)),tnU{Xn)) - F{x,tnU{Xn)) ) dx In + / I -g{x,tnU{Xn))tnU{Xn) - G{x,tnU{Xn)) ] dx.

Zou [326]. Possibly, it can be proved by other methods such as the degree theory or the contraction mapping principle. 4 has far more extended applications. We would like to leave them to the readers. Chapter 3 Even Functionals In this chapter we present some abstract theorems which concern the existence of infinitely many critical points for even functionals. The Palais-Smale type compactness condition is not necessary for the new results. By taking advantage of the abstract theorems, we study the existence of infinitely many large energy solutions for nonlinear Schrodinger equations and of infinitely many small energy solutions for semilinear elliptic equations with concave and convex nonlinear it ies.

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Critical Point Theory and Its Applications by Wenming Zou


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