Positive solutions for a nonlinear problem involving the bosonic operator

We study the existence and multiplicity of positive solutions for a class of nonlocal elliptic equations involving the bosonic operator $ -\Delta e^{-c\Delta} $, which arises in models from bosonic string theory and nonlocal cosmology.For the parametric problem with a nonlinearity that is sublinear at infinity,$ -\Delta e^{-c\Delta}u + u = \lambda a(x)g(u) \quad \text{in } \mathbb{R}^N, $we use tools tailored to the nonlocal framework, including sharp decay estimates for the associated kernel, a strong maximum principle, and the method of lower and upper solutions.Our main result identifies a critical parameter $ \lambda^*>0 $ such that the problem admits no positive solutions for $ \lambda\lambda^* $. One solution follows from coercivity arguments together with the method of lower and upper solutions, whereas the second is obtained through a nontrivial linking argument. This multiplicity pattern reveals that the problem exhibits a structure that can be viewed as a dual counterpart to the classical concave-convex type equations. A key ingredient in this analysis is a local minimizer equivalence principle between the natural energy space and a suitable weighted space with nonempty interior, which enables comparison and minimization techniques in the whole space.