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Finally, taking (N>0) large enough, we deduce that (3.8).

Moreover, by a Sobolev embedding, (W^{s,q}( mathbb {R}^{N})hookrightarrow C^{0,alpha}( mathbb{R}^{N})) (for q large enough), we deduce that (v_{varepsilon}-urightarrow0) in (C^{0,alpha}( mathbb{R}^{N})), it follows from the decay of u that (|v_{varepsilon}(x)|rightarrow0) as (|x|rightarrowinfty) uniformly in (varepsilon>0).

Knowing that (alpha ^{2.3^{n-1}}A' sim A) and (alpha ^{4.3^{n-1}}=1) for n large enough, we deduce that (alpha ^{2.3^{n-1}}=pm,1) alsoalso that (A=pm,A'.) Let us discuss those cases: A=A′ We already know that (B sim B' alpha ^{3^{n}}) and also that (alpha ^{4.3^{n}}sim 1) for n large enough.

Combining the estimates of and selecting a small enough constant, we deduce that (3.18).

Since ( u n ) is bounded, q < 2 < α + β and − J ( u n, v n ) + 1 α + β 〈 J ′ ( u n, v n ), ( u n, v n ) 〉 ≤ d + ∥ ( u n, v n ) ∥. for n big enough, we easily deduce that ( v n ) is bounded.

As a direct consequence of the two previous properties, we deduce that, for large enough, function has constant sign and it is strictly monotone.

By Lemmas 3.4 and 3.5, we deduce that for n large enough, I n has a nontrivial critical point w n by using the mountain pass lemma and Theorem 1 in [24].

From the combination of 1. and 2., we deduce that one can choose β small enough such that P [ sup 0 ≤ t ≤ T ∧ τ ϵ ∥ w t ϵ − w ˜ t ϵ, β ∥ > δ ] ≤ ( C 1 + C 2 ) η. (25).

Further, we deduce an optimized EGS well layout must ensure enough long major flow path and less preferential flow in the reservoir, and the injection well is located close to the edge of the reservoir.

From (3.12), for, we deduce that there exists a positive integer number large enough, when, (3.32).

we deduce that must change its sign on if is large enough.