Computation of critical groups in resonance problems where the nonlinearity may not be sublinear. (English) Zbl 1088.58005

The authors compute critical groups at infinity (a notion introduced by Bartsch-Li in [T. Bartsch and S. J. Li, Nonlinear Anal., Theory Methods Appl. 28, No. 3, 419–441 (1997; Zbl 0872.58018)]) of the asymptotically quadratic functional \(F(u) ={1 \over 2}(Au,u) +G(u)\) on a Hilbert space \(H\), where \(A\) is a bounded self-adjoint operator on \(H\) and \(G\in C^ 1(H,{\mathbb R})\) satisfies \({{| G(u)| }\over {\| u\|^ 2}} \to 0\) as \(\| u\| \to \infty\). They allow the nonlinearity to grow faster than sublinearly as they require the compact subdifferential \(g\) of \(G\) to satisfy \({{\| g(u)\| }\over {\| u\|}} \to 0\) as \(\| u\| \to \infty\). The case of sublinear growth at infinity \(\| g(u)\| \leq (\| u\|^ \alpha + 1)\) for \(\alpha \in (0,1)\) was treated by Li-Liu in [S. J. Li and J.Q. Liu, Houston J. Math. 25, No. 3, 563–582 (1999; Zbl 0981.58011)].
As applications, some existence theorems for asymptotically linear elliptic boundary value problems at resonance are proved.


58E05 Abstract critical point theory (Morse theory, Lyusternik-Shnirel’man theory, etc.) in infinite-dimensional spaces
35J20 Variational methods for second-order elliptic equations
47J30 Variational methods involving nonlinear operators
49J35 Existence of solutions for minimax problems
Full Text: DOI


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