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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:
 58E05 Abstract critical point theory 34B15 Nonlinear boundary value problems for ODE 35J65 Nonlinear boundary value problems for linear elliptic equations
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##### 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