Energy asymptotics in the three-dimensional Brezis-Nirenberg problem

Frank, Rupert L. ; Konig, Tobias and Kovarik, Hynek (2021): Energy asymptotics in the three-dimensional Brezis-Nirenberg problem. In: Calculus of Variations and Partial Differential Equations, Vol. 60, No. 2, 58

Abstract

For a bounded open set Omega subset of R-3 we consider the minimization problem S(a + epsilon V) = inf(0 not equivalent to u is an element of H01(Omega)) integral(Omega)(vertical bar del u vertical bar(2) + (a + epsilon V)vertical bar u vertical bar(2))dx/(f(Omega)u(6)dx)(1/3) involving the critical Sobolev exponent. The function a is assumed to be critical in the sense of Hebey and Vaugon. Under certain assumptions on a and V we compute the asymptotics of S(a+epsilon V) - S as epsilon -> 0 +, where S is the Sobolev constant. (Almost) minimizers concentrate at a point in the zero set of the Robin function corresponding to a and we determine the location of the concentration point within that set. We also show that our assumptions are almost necessary to have S(a+epsilon V) < S for all sufficiently small epsilon > 0.

• Item Type: Journal article
• Faculties: Mathematics, Computer Science and Statistics > Mathematics > Analysis, Mathematical Physics and Numerics
• Subjects: 500 Science > 510 Mathematics
• ISSN: 0944-2669
• Language: English