Normalized multibump solutions to nonlinear Schrodinger equations with steep potential well

We are concerned with the existence of multibump solutions to the nonlinear Schrodinger equation -Delta u + lambda a(x)u + mu u = vertical bar u vertical bar(2 sigma)u in R-N with an L-2-constraint parallel to u parallel to(2)(L2(RN)) = rho in the L-2-subcritical case sigma is an element of (0, 2/N) and the L-2-supercritical case sigma is an element of (2/N, 2*/N), where the usual critical Sobolev exponent is 2* = +infinity if N = 1, 2 and 2* = 2N/(N - 2) if N >= 3. Here mu is an element of R will arise as a Lagrange multiplier, and 0 <= a is an element of L-loc(infinity)(R-N) has a bottom int a(-1)(0) composed of l(0) (l(0) >= 1) connected components {Omega(i)}(i=1)(l0), where int a(-1)(0) is the interior of the zero set a(-1)(0) {x is an element of R-N vertical bar a(x) = 0} of a. When rho is fixed either large in the L-2-subcritical case or small in the L-2-supercritical case, we construct a l-bump (1 <= l <= l(0)) positive normalized solution which is localised at l prescribed components {Omega(i)}(i=1)(l) for large lambda. The asymptotic profile of the solution is also analysed through taking the limit as lambda -> +infinity, and subsequently as rho -> +infinity in the L-2-subcritical case or rho -> 0(+) in the L-2-supercritical case. In particular, we find l-bump normalized solutions to the related Dirichlet problem {-Delta v + mu v = vertical bar v vertical bar(2 sigma)v, v is an element of H-0(1)(boolean OR(l)(i=1)Omega(i)), Sigma(l)(i=1) integral(Omega i) v(2) = rho, v vertical bar(Omega i) > 0 for i = 1, ..., l.