摘要 : We study the following minimization problem e beta(b) := inf({u is an element of H1(R2):integral R2 vertical bar u vertical bar 2dx=1}) E-beta(b)(u), (0.1) where E-beta(b)(.) is some kind of Kirchhoff functional (the precise form ... 展开 We study the following minimization problem e beta(b) := inf({u is an element of H1(R2):integral R2 vertical bar u vertical bar 2dx=1}) E-beta(b)(u), (0.1) where E-beta(b)(.) is some kind of Kirchhoff functional (the precise form is given by (1.2) below) with a periodic potential. We prove that problem (0.1) can be attained for all beta >= beta* provided that b > 0 is small enough, which is different from the case of b = 0. Moreover, we are surprised to discover that the limits of minimizers of (0.1) as b -> 0+ for the cases of beta = beta* and beta > beta* are totally different. Up to rescaling, in the former case the minimizers converge to Q(x) in H-1(R-2), where Q(x) is the unique positive solution of -Delta u + u - u(3) = 0. However, in the latter case the minimizers converge to a minimizer of (0.1) with b = 1 and potential V(x) equivalent to 0. (c) 2021 Elsevier Inc. All rights reserved. 收起