The existence and the concentration behavior of normalized solutions for the L2-critical Schrodinger-Poisson system

In this paper, we study the existence and the concentration behavior of critical points for the following functional derived from the Schrodinger-Poisson system: E(u) = 1/2 integral(R3) vertical bar del u vertical bar(2) + 1/4 integral(R3) (vertical bar x vertical bar(-1) * u(2))u(2) - 3/10 integral(R3) vertical bar u vertical bar(10/3) constrained on the L-2-spheres S(c) = {u is an element of H-1 (R-3)vertical bar vertical bar u vertical bar(2) =c} when c > c(*) = vertical bar Q vertical bar(2), where Q is up to translations, the unique positive of -Delta Q + 2/3 Q = vertical bar Q vertical bar(4/3) Q in R-3. As such constrained problem is L-2-critical, E(u) is unbounded from below on S(c) when c > c>(*) and the existence of critical points constrained on S(c) is obtained by a mountain pass argument on S(c). We show that there exists c(1) > (9/7)(3/4) c(*) such that E(u) has at least one positive critical point restricted to S(c) for c(*) < c <= c(1). As c approaches c(*) from above, we obtain that the critical point u(c) behaves like (1/vertical bar del u(c)vertical bar(2))(3/2) u(c) (1/vertical bar del u(c)vertical bar(2) (x+y(c))) approximate to (1/c(*))3/2 Q ((1/c(*))x) for some y(c) is an element of R-3. (C) 2017 Elsevier Ltd. All rights reserved.