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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 $f\in {C}^{2}\left(X,𝐑\right)$. Then ${x}_{0}$ may be classified by its critical groups ${c}_{k}\left(f,{x}_{0}\right):={H}_{k}\left(\left\{f\le c\right\},\left\{f\le c\right\}\setminus \left\{{x}_{0}\right\}\right)$, where $c=f\left({x}_{0}\right)$. 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 ${f}^{\text{'}\text{'}}\left(\infty \right)$ is nondegenerate and has Morse index $\mu$, then ${c}_{k}\left(f,\infty \right)\ne 0$ if and only if $k=\mu$. It is known that if ${f}^{\text{'}\text{'}}\left({x}_{0}\right)$ is degenerate and $f$ satisfies the so-called local linking condition at ${x}_{0}$, then ${c}_{k}\left(f,{x}_{0}\right)\ne 0$ for some $k$. A similar result is shown to hold for ${c}_{k}\left(f,\infty \right)$. 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 $-{\Delta }u=p\left(x,u\right)$ in ${\Omega }$, $u=0$ on $\partial {\Omega }$, where ${\Omega }$ is a bounded domain in ${𝐑}^{N}$, $p\left(x,0\right)=0$ and $p$ is asymptotically linear. It is shown that under different conditions ${c}_{k}\left(f,0\right)\ne {c}_{k}\left(f,\infty \right)$ 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