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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 \(f\in C^2(X,{\mathbf R})\). 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.

MSC:
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
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