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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.

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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