Existence of Smooth Solutions for a Class of Euclidean Bosonic Equations

We study a class of non-linear equations on Euclidean space building on previous works by the present authors. These equations emerge in the study of cosmology and string theory, see for instance Calcagni, Montobbio and Nardelli (2007) [3]; Calcagni, Montobbio and Nardelli (2008) [4] and they depend on operators of the form exp⁡(−cΔ), in which Δ is the Euclidean Laplace operator and c>0. We introduce an appropriate space of functions on which this operator is well defined; this domain is continuously embedded into the scale of Sobolev spaces, and this fact allows us to investigate weak and strong solutions. We prove that, for a wide class of non-linearities, there exists non-trivial smooth solutions. Our tools are classical fixed point theorems and variational calculus. We are able to prove existence of (strong) solutions using Schaeffer s fixed point theorem, and existence of radial and non-radial non-trivial strong solutions using the variational approach and the principle of symmetric criticality. © 2022 Elsevier Inc.