Groli, Alessandro On \(\Gamma\)-convergence for problems of jumping type. (English) Zbl 1038.49023 Electron. J. Differ. Equ. 2003, Paper No. 60, 16 p. (2003). 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. Reviewer: Riccardo De Arcangelis (Napoli) MSC: 49J45 Methods involving semicontinuity and convergence; relaxation 58E05 Abstract critical point theory (Morse theory, Lyusternik-Shnirel’man theory, etc.) in infinite-dimensional spaces Keywords:\(\Gamma\)-convergence; jumping problem; nonsmooth critical point theory; Gamma-convergence PDFBibTeX XMLCite \textit{A. Groli}, Electron. J. Differ. Equ. 2003, Paper No. 60, 16 p. (2003; Zbl 1038.49023) Full Text: EuDML EMIS