Normalized solutions of mass subcritical Schrodinger equations in exterior domains

In this paper, we study the nonlinear Schrodinger equation with L-2-norm constraint: {-Delta u = lambda u + vertical bar u vertical bar(p-2) u in Omega, u = 0 on partial derivative Omega, integral(Omega) vertical bar u vertical bar(2) dx = a(2), where N >= 3, Omega subset of R-N is an exterior domain, i.e., Omega is an unbounded domain with R-N\(Omega) over bar non-empty and bounded, a > 0, 2 < p < 2 + 4/N, and lambda is an element of R is Lagrange multiplier, which appears due to the mass constraint parallel to u parallel to(L2(Omega)) = a. We use Brouwer degree, barycentric functions and minimax method to prove that for any a > 0, there is a positive solution u is an element of H-0(1)(Omega) for some lambda < 0 if R-N\Omega is contained in a small ball. In addition, if we remove the restriction on Omega but impose that a > 0 is small, then we also obtain a positive solution u is an element of H-0(1)(Omega) for some lambda < 0. If Omega is the complement of unit ball in R-N, then for any a > 0, we get a positive radial solution u is an element of H-0(1) (Omega) for some lambda < 0 by Ekeland variational principle. Moreover, we use genus theory to obtain infinitely many radial solutions {(u(n), lambda(n))} with lambda(n) < 0, I-p(u(n)) < 0 for n >= 1 and I-p(u(n)) -> 0(-) as n -> infinity, where I-p is the corresponding energy functional.