## Info

in C!o"(K+) as a —» oo, where A is as in the statement of Lemma 5.1. As a main consequence, du(-,u0) _ du(-,u0) __e±i dv(-,a) _

- V>i(0) + o(l)) = Au^ + o (uo^) , as wo —> oo. This finishes the proof of Theorem 5.1. □

### We proceed now to the proof of Lemma 5.1.

Proof of Lemma 5.1. Fix R,h positive and choose Dr,h any bounded C2,a sub domain of containing {x : \x'\ < R, 0 < xn < h}, for instance a regularization of the latter domain which contains it. Consider the auxiliary problem,

By using a standard weak comparison argument it follows that problem (24) admits at most a unique positive solution. Such solution can be obtained by the method of sub and supersolutions by taking v = 0 as subsolution, v = 1 as a supersolution. Thus (24) admits a unique positive solution VR,h € C2'a(Dntfl) which satisfies,

Now observe that if v = v(z) G Clo'°(R+) is any nonnegative bounded solution to (21) (and so satisfying (22)), then

defines a subsolution to problem (24) which, as a consequence of weak comparison, satisfies v(z) < vRth{z) in DrHence,

0 < v(z) < vR>h(z) z G DRih, where strong comparison has been also employed.

For n G N set Dn := nDR<h = {nz : z G DR,h} while vn(z) designates the solution to problem (24) in Dn. Just for the same argument as the one given above we arrive at:

0 < v{z) < vn+k{z) < vn(z) <1 z G Dn, for every n, k G N. In fact, the restriction of vn+k to Dn defines a strict subsolution to (24) in Dn that can be strongly compared with vn.

Using the Lq and Schauder's estimates as has been already done we get the convergence,