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 \(f\in C^2(X,{\mathbf R})\). Then \(x_0\) may be classified by its critical groups \(c_k(f,x_0):=H_k(\{f\leq c\},\{f\leq c\}\setminus\{x_0\})\), 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 \(c_k(f,\infty)\neq 0\) 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 \(c_k(f,x_0)\neq 0\) 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 \({\mathbf R}^N\), \(p(x,0)=0\) and \(p\) is asymptotically linear. It is shown that under different conditions \(c_k(f,0)\neq c_k(f,\infty)\) for some \(k\), and therefore this problem has a nontrivial solution. Particular attention is paid to the resonant case.


58E05 Abstract critical point theory (Morse theory, Lyusternik-Shnirel’man theory, etc.) in infinite-dimensional spaces
34B15 Nonlinear boundary value problems for ordinary differential equations
35J65 Nonlinear boundary value problems for linear elliptic equations
Full Text: DOI


[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), Birkhäuser
[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), Springer Boston · 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) · Zbl 0436.58006
[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
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.