Existence and asymptotic behavior of high energy normalized solutions for the Kirchhoff type equations in R3

In this paper, we study the multiplicity of solutions with a prescribed L-2-norm for a class of nonlinear Kirchhoff type problems in R-3 - (a + b integral(R3) vertical bar del u vertical bar(2)) Delta u - lambda u = vertical bar u vertical bar(p-2)u, where a, b > 0 are constants, lambda is an element of R, p is an element of(14/3, 6). To get such solutions we look for critical points of the energy functional I-b(u) = a/2 integral(R3) vertical bar del u vertical bar(2) + b/4 (integral(R3) vertical bar del u vertical bar(2))(2) - 1/p integral(R3) vertical bar u vertical bar(p) restricted on the following set S-r(c) ={u is an element of H-r(1) (R-3) : parallel to u parallel to(2)(L2(R3)) = c}, c > 0. For the value p is an element of(14/3, 6) considered, the functional. I-b is unbounded from below on S-r (c). By using a minimax procedure, we prove that for any c > 0, there are infinitely many critical points {u(n)(b)}(n is an element of N+) of I-b restricted on S-r (c) with the energy I-b (u(n)(b)) -> +infinity(n -> +infinity). Moreover, we regard b as a parameter and give a convergence property of u(n)(b) as b -> 0(+). (C) 2016 Elsevier Ltd. All rights reserved.