Existence of Strong and Ground State Solutions for a Class of Euclidean Bosonic Equations Involving a Composite Fractional Operator

Abstract

ABSTRACT In this paper, we study nonlinear Euclidean equations involving the composite fractional pseudo‐differential operator , where , are positive parameters and stands for the identity operator on . We here look to the solvability of such equations. Thus, we first show that every weak solution , for which the nonlinearity belongs to , is in fact a strong solution. Then, under general assumptions on the nonlinearity, we establish the existence of strong solutions, making use of Schauder's fixed point theorem (case of at most linear growth) and Schaefer's fixed point theorem (sublinear case). Next, we establish the existence of ground states in the radial class and ground state solutions in the periodic setting by the mountain pass theorem and the Nehari manifold method.