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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.

MSC:

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
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References:

[1] Bartsch, T.; Li, S.J., Critical point theory for asymptotically quadratic functionals and applications to problems with resonance, Nonlinear anal., 28, 3, 419-441, (1997) · Zbl 0872.58018
[2] Chang, K.C., Infinite-dimensional Morse theory and multiple solution problems, progress in nonlinear differential equations and their applications, vol. 6, (1993), Birkhäuser Boston, MA
[3] Li, S.J.; Liu, J.Q., Computations of critical groups at degenerate critical point and applications to nonlinear differential equations with resonance, Houston J. math., 25, 3, 563-582, (1999) · Zbl 0981.58011
[4] Liu, J.Q., The Morse index of a saddle point, Systems sci. math. sci., 2, 1, 32-39, (1989) · Zbl 0732.58011
[5] K. Perera, M. Schechter, Double resonance problems with respect to the Fučı́k spectrum, preprint. · Zbl 1030.35079
[6] Perera, K.; Schechter, M., Nontrivial solutions of elliptic semilinear equations at resonance, Manuscripta math., 101, 3, 301-311, (2000) · Zbl 0958.35051
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