[0,00) is the where ipS is anY regular positive extension of


from (TV? n Q) U fis U U Ml'2 to 0, which is bounded away from zero 2 2 2 2

in Cl \ (Aft U Q,\ U Q2 U Ml'2). Note that ip5 exists, since the functions 2 2 2 2

VoleA^nn, Vi.Jan1. nn> Ian2, and vi\dN\<2nn are positive and bounded

5 2 2 f away from zero. By construction, $(x) > 0 for each x e fi and therefore, it is bounded away from zero in

To complete the proof of the result it remains to show that there exists M > 0 sufficiently large, such that u = provides us with a positive strict supersolution of Problem (l)v

Indeed, by construction we have that in M\ fl fi the following holds:

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