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Frank, Rupert L.; König, Tobias and Kovarik, Hynek (2020): Energy asymptotics in the Brezis-Nirenberg problem: The higher-dimensional case. In: Mathematics in Engineering, Vol. 2, No. 1: pp. 119-140

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Abstract

For dimensions N >= 4, we consider the Brezis-Nirenberg variational problem of finding S (epsilon V) := inf(0 not equivalent to u epsilon H01(Omega)) integral(Omega) vertical bar del u vertical bar(2) dx + epsilon integral(Omega) V vertical bar u vertical bar(2) dx/(integral(Omega) vertical bar u vertical bar(q) dx)(2/q) , where q = 2N/N-2 is the critical Sobolev exponent, Omega subset of R-N is a bounded open set and V : (Omega) over bar -> R is a continuous function. We compute the asymptotics of S (0) - S (epsilon V) to leading order as epsilon -> 0+. We give a precise description of the blow-up profile of (almost) minimizing sequences and, in particular, we characterize the concentration points as being extrema of a quotient involving the Robin function. This complements the results from our recent paper in the case N = 3.

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