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On \(\Gamma\)-convergence for problems of jumping type. (English) Zbl 1038.49023

The paper is concerned with the convergence of the critical values of a \(\Gamma\)-converging sequence of functionals. The functionals taken into account are related to a classical “jumping problem”, and are perturbations of \[ f_\infty(u)={1\over 2}\int_\Omega\sum_{i,j=1}^nA_{ij}^{(\infty)}(x)D_iuD_ju\,dx- {\alpha\over2}\int_\Omega(u^+)^2dx- {\beta\over2} \int_\Omega(u^-)^2\,dx+ \int_\Omega\phi_1u\,dx \] of the type \[ f_h(u)={1\over 2}\int_\Omega\sum_{i,j=1}^na_{ij}^{(h)}(x,u)D_iuD_ju\,dx- {\alpha\over2} \int_\Omega(u^+)^2\,dx- {\beta\over2} \int_\Omega(u^-)^2\,dx+\int_\Omega\phi_1u\,dx, \] where \(t_h\to+\infty\), \(\beta<\alpha\), and \(\phi_1\) is a positive eigenfunction of \(-\sum_{i,j=1}^nD_j(A_{ij}^{(\infty)}(x)D_i)\) with homogeneous Dirichlet conditions.
The existence of at least three critical values for \(f_h\) is proved when \(\alpha\) and \(\beta\) satisfy the usual assumptions with respect to \(f_\infty\), but not with respect to \(f_h\).
Examples are also discussed.

MSC:

49J45 Methods involving semicontinuity and convergence; relaxation
58E05 Abstract critical point theory (Morse theory, Lyusternik-Shnirel’man theory, etc.) in infinite-dimensional spaces
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