Normalized Solutions of Nonautonomous Kirchhoff Equations: Sub- and Super-critical Cases

In this paper, we establish the existence of normalized solutions to the following Kirchhoff-type equation {-(a + b integral(R3) vertical bar del u vertical bar(2)dx) Delta u - lambda u = K(x) f(u), x is an element of R-3; u is an element of H-1(R-3) where a, b > 0, lambda is unknown and appears as a Lagrange multiplier, K is an element of C(R-3, R+) with 0 < lim(vertical bar y vertical bar ->infinity) K(y) <= inf(R3) K, and f is an element of C(R, R) satisfies general L-2-supercritical or L-2-subcritical conditions. We introduce some new analytical techniques in order to exclude the vanishing and the dichotomy cases of minimizing sequences due to the presence of the potential K and the lack of the homogeneity of the nonlinearity f. This paper extends to the nonautonomous case previous results on prescribed L-2-norm solutions of Kirchhoff problems.