Boundary value problems for semilinear Schrodinger equations with singular potentials and measure data

Abstract

We study boundary value problems with measure data in smooth bounded domains Omega, for semilinear equations. Specifically we consider problems of the form - L(V)u + f (u) = tau in Omega and tr(V)u = nu on partial derivative Omega, where L-V = Delta + V, f. is an element of C(R) is monotone increasingwith f (0) = 0 and tr V u denotes themeasure boundary trace of u associated with L-V. The potential V is an element of C-1(Omega) typically blows up at a set F subset of partial derivative Omega as dist (x, F)(-2). In general the above boundary value problem may not have a solution. We are interested in questions related to the concept of 'reduced measures', introduced in Brezis et al. (Ann Math Stud 163:55-109, 20072007) for V = 0. Our results extend results of [4] and Brezis and Ponce (J Funct Anal 229(1):95-120, 2005) and apply to a larger class of nonlinear terms f. In the case of signed measures, some of the present results are new even for V = 0.