Generalized Euclidean Bosonic String Equations

We consider nonlinear equations of the form p(?)u = U(x, u(x)), in which p is a real-valued function satisfying some suitable technical conditions, and ? stands for the Laplacian operator.W e formulate a functional calculus appropriate for the study of such equations, and we establish results on the existence and regularity of solutions to the Euclidean bosonic string equation ? exp(-c ?) u = U(x, u(x)), and we introduce a functional calculus appropriate for the study of very general nonlinear equations depending on functions of the Laplace operator.W e also prove that under some further technical conditions, these "nonlocal" equations admit smooth, and even realanalytic, solutions.O ur motivation comes from recent developments in string theory and nonlocal cosmology. © Springer Basel 2012. All rights reserved.