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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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