zbMATH — the first resource for mathematics

Examples
Geometry Search for the term Geometry in any field. Queries are case-independent.
Funct* Wildcard queries are specified by * (e.g. functions, functorial, etc.). Otherwise the search is exact.
"Topological group" Phrases (multi-words) should be set in "straight quotation marks".
au: Bourbaki & ti: Algebra Search for author and title. The and-operator & is default and can be omitted.
Chebyshev | Tschebyscheff The or-operator | allows to search for Chebyshev or Tschebyscheff.
"Quasi* map*" py: 1989 The resulting documents have publication year 1989.
so: Eur* J* Mat* Soc* cc: 14 Search for publications in a particular source with a Mathematics Subject Classification code (cc) in 14.
"Partial diff* eq*" ! elliptic The not-operator ! eliminates all results containing the word elliptic.
dt: b & au: Hilbert The document type is set to books; alternatively: j for journal articles, a for book articles.
py: 2000-2015 cc: (94A | 11T) Number ranges are accepted. Terms can be grouped within (parentheses).
la: chinese Find documents in a given language. ISO 639-1 language codes can also be used.

Operators
a & b logic and
a | b logic or
!ab logic not
abc* right wildcard
"ab c" phrase
(ab c) parentheses
Fields
any anywhere an internal document identifier
au author, editor ai internal author identifier
ti title la language
so source ab review, abstract
py publication year rv reviewer
cc MSC code ut uncontrolled term
dt document type (j: journal article; b: book; a: book article)
Critical point theory for asymptotically quadratic functionals and applications to problems with resonance. (English) Zbl 0872.58018
The paper is concerned with Morse theory for asymptotically quadratic functionals on a Hilbert space $X$. Let $x_0$ be an isolated critical point and $f\in C^2(X,{\bold R})$. Then $x_0$ may be classified by its critical groups $c_k(f,x_0):=H_k(\{f\le c\},\{f\le c\}\setminus\{x_0\})$, where $c=f(x_0)$. In this paper the notion of critical group at infinity is introduced and it is shown that such groups have similar properties to those at finite points. In particular, if $f''(\infty)$ is nondegenerate and has Morse index $\mu$, then $c_k(f,\infty)\ne 0$ if and only if $k=\mu$. It is known that if $f''(x_0)$ is degenerate and $f$ satisfies the so-called local linking condition at $x_0$, then $c_k(f,x_0)\ne 0$ for some $k$. A similar result is shown to hold for $c_k(f,\infty)$. Moreover, a new “angle condition” is introduced under which the $c_k$’s behave as in the case of nondegenerate critical point. Applications of the above results are given to the Dirichlet problem $-\Delta u=p(x,u)$ in $\Omega$, $u=0$ on $\partial\Omega$, where $\Omega$ is a bounded domain in ${\bold R}^N$, $p(x,0)=0$ and $p$ is asymptotically linear. It is shown that under different conditions $c_k(f,0)\ne c_k(f,\infty)$ for some $k$, and therefore this problem has a nontrivial solution. Particular attention is paid to the resonant case.

MSC:
58E05Abstract critical point theory
34B15Nonlinear boundary value problems for ODE
35J65Nonlinear boundary value problems for linear elliptic equations
WorldCat.org
Full Text: DOI
References:
[1] Gromoll, D.; Meyer, W.: On differentiable functions with isolated critical points. Topology 8, 361-369 (1969) · Zbl 0212.28903
[2] Amann, H.; Zehnder, E.: Nontrivial solutions for a class of non-resonance problems and applications to nonlinear differential equations. Annali scu. Norm. sup. Pisa 7, 539-603 (1980) · Zbl 0452.47077
[3] Amann, H.; Zehnder, E.: Periodic solutions of asymptotically linear Hamiltonian systems. Manuscripta math. 32, 149-189 (1980) · Zbl 0443.70019
[4] Bartolo, P.; Benci, V.; Fortunato, D.: Abstract critical point theorems and applications to some nonlinear problems with ”strong” resonance at infinity. Nonlinear analysis 7, 981-1012 (1983) · Zbl 0522.58012
[5] Chang, K.C.: Infinite dimensional Morse theory and multiple solution problems. (1993) · Zbl 0779.58005
[6] Landesman, E.A.; Lazer, A.C.: Nonlinear perturbations of linear eigenvalue problems at resonance. J. math. Mech. 19, 609-623 (1970) · Zbl 0193.39203
[7] Li, S.; Liu, J.Q.: Nontrivial critical points for asymptotically quadratic functions. J. math. Analysis applic. 165, 333-345 (1992) · Zbl 0767.35025
[8] Mawhin, J.; Willem, M.: Critical point theory and Hamiltonian systems. (1989) · Zbl 0676.58017
[9] Conley, C.; Zehnder, E.: Morse type index theory for flows and periodic solutions to Hamiltonian systems. Communs pure appl. Math. 37, 207-253 (1984) · Zbl 0559.58019
[10] Liu, J.Q.: The Morse index of a saddle point. Syst. sc. & math. Sc. 2, 32-39 (1989) · Zbl 0732.58011
[11] Szulkin, A.: Cohomology and Morse theory for strongly indefinite functionals. Math. Z. 209, 375-418 (1992) · Zbl 0735.58012
[12] Li, S.; Szulkin, A.: Periodic solutions of an asymptotically linear wave equation. Top. methods in nonlinear analysis 1, 211-230 (1993) · Zbl 0798.35103
[13] Cerami, G.: Un criteria de esistenzia per i punti critici su varietà illimitate. Rc. ist. Lomb. sci. Lett. 112, 332-336 (1978)
[14] Lazer, A.C.; Solimini, S.: Nontrivial solutions of operator equations and Morse indices of critical points of MIN-MAX type. Nonlinear analysis 12, 761-773 (1988) · Zbl 0667.47036
[15] Chang, K.C.: Solutions of asymptotically linear operator equation via Morse theory. Communs pure appl. Math. 34, 693-712 (1981) · Zbl 0444.58008
[16] Brézis, H.; Nirenberg, L.: Characterizations of the ranges of some nonlinear operators and applications to the boundary value problems. Annali scu. Norm. sup. Pisa 5, 225-326 (1978) · Zbl 0386.47035
[17] Mizoguchi, N.: Asymptotically linear elliptic equations without nonresonance conditions. J. diff. Eqns 113, 150-165 (1994) · Zbl 0806.35040
[18] Chang, K.C.: Morse theory on Banach spaces and its applications. Chinese ann. Math. ser. 64, 381-399 (1983) · Zbl 0534.58020
[19] Capozzi, A.; Lupo, D.; Solimini, S.: On the existence of a nontrivial solution to nonlinear problems at resonance. Nonlinear analysis 13, 151-163 (1989) · Zbl 0684.35038