EIGENVALUE ASYMPTOTICS AND A NONLINEAR SCHRODINGER-EQUATION

The equation () -DELTAu+qu+f(x,u)=lambdau, u is-an-element-of W1,2(R( N)) is considered, where q is bounded below and q(x) --> infinity as A bsolute value of x --> infinity. Under appropriate conditions on the p erturbation term f(x,u) it is shown that given any r > 0, () has an i nfinite sequence (lambda(n,r))n is-an-element-of N of eigenvalues, eac h lambda(n,r) being associated with an eigenfunction u(n,r) which sati sfies integral(RN) \u(n,r)\2 = r2. Information about the behaviour of lambda(n,r) for large n is provided. The proofs rely on the compactnes s of the embedding of a certain weighted Sobolov space in an L(p) spac e; this is proved in sectional sign 2.