Se p 20 06 On Decay of Solutions to Nonlinear Scrödinger Equations

We present general results on exponential decay of finite energy solutions to stationary nonlinear Schrödinger equations. AMS Subject Classification (2000): 35J60, 35B40 In this note we consider the equation −∆u + V (x)u = f(x, u), x ∈ R (1) and, under rather general assumptions, derive exponential decay estimates for its solutions. We suppose that (i) The potential V belongs to Lloc(R ) and is bounded below, i.e. V (x) ≥ −c0 for some c0 ∈ R. Under assumption (i) the left hand side of equation (1) defines a self-adjoint operator in L(R) denoted by H . The operator H is bounded below. We suppose that (ii) The essential spectrum σess(H) of the operator H does not contain the point 0. Note, however, that 0 can be an eigenvalue of finite multiplicity. The nonlinearity f is supposed to satisfy the following assumption. (iii)The function f(x, u) is a Carathéodory function, i.e. it is Lebesgue measurable with respect to x ∈ R for all u ∈ R and continuous with respect to u ∈ R for almost all x ∈ R. Furthermore, |f(x, u)| ≤ c(1 + |u|) , x ∈ R u ∈ R , (2) with c > 0 and 2 ≤ p < 2, where