*Published Paper*

**Inserted:** 26 nov 2009

**Last Updated:** 10 feb 2015

**Journal:** Adv. in Diff. Eq.

**Year:** 2010

**Abstract:**

This paper is devoted to the analysis of the asymptotic behaviour when the parameter $\lambda$ goes to infinity for operators of the form $-Delta + \lambda a $ or more generally, cooperative systems operators, where the potentials vanish in some subregions of the domain.
We use the theory of Gamma-convergence, even for the non-variational cooperative system, to prove that for any reasonable bounded potentials $a$ and $d$ those operators converge in the strong resolvent sense to the operator
in the vanishing regions of the potentials, so does the spectrum.
The class of potentials considered here is fairly large substantially improving previous results, allowing in particular ones that vanish on Cantor sets, and forcing us to enlarge the class of domains to the so-called *quasi-open sets*. For the system various situations are considered applying our general result to the interplay of the vanishing regions of the potentials of both equations.

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