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Critical groups at infinity, saddle point reduction and elliptic resonant problems. (English) Zbl 1058.35075
Infinite dimensional Morse theory is a very useful tool in treating multiple solution problems in the study of nonlinear differential equations. One of the main concepts in this theory is the critical group at infinity $$C_q(f,\infty)$$, where $$f: X\to \mathbb{R}$$ is a $$C^1$$-functional and $$X$$ is a Banach space. The authors’ results give new computations of these critical groups. Moreover, they apply their results to the following semilinear elliptic boundary value problem $\begin{cases} -\Delta u= p(u)\quad &\text{in }\Omega,\\ u= 0\quad &\text{on }\partial\Omega,\end{cases}\tag{1}$ where $$\Omega\subset\mathbb{R}^n$$ is a bounded domain with smooth boundary and $$p\in C^1(\mathbb{R},\mathbb{R})$$. They prove in the case $$p'(0)< \lambda_1$$ the existence of at least 4-nontrivial solutions of (1). Here $$\lambda_1$$ is the first eigenvalue of the Laplacian.

##### MSC:
 35J65 Nonlinear boundary value problems for linear elliptic equations 58E05 Abstract critical point theory (Morse theory, Lyusternik-Shnirel’man theory, etc.) in infinite-dimensional spaces
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