## 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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