Fractional Euclidean bosonic equation via variational

In this paper, we study the existence of solutions for the following class of Euclidean bosonic equations with Liouville–Weyl fractional derivatives

$$\begin{aligned} {\left\{ \begin{array}{ll} {_{x}}D_{\infty }^{\beta }{_{-\infty }}D_{x}^{\beta }e^{C {_{x}}D_{\infty }^{\beta }{_{-\infty }}D_{x}^{\beta }}u = \lambda \omega (x)u+ Q(x)g(x,u)&{}\text{ in }\,\,{\mathbb {R}},\\ u\in \mathcal {H}_c^{\beta ,\infty } ({\mathbb {R}}), \end{array}\right. } \end{aligned}$$

where \(\beta \in (0,\frac{1}{2})\), \({_{-\infty }}D_{x}^{\beta }u(\cdot ), {_{x}}D_{\infty }^{\beta }u(\cdot )\) denote the left and right Liouville–Weyl fractional derivatives, \(\omega ,Q:{\mathbb {R}}\rightarrow {\mathbb {R}}\) is a positive function with \(\omega ,Q\in L^{\frac{1}{2\beta }} ({\mathbb {R}})\) and \(g: {\mathbb {R}}\rightarrow {\mathbb {R}}\) is a continuous function satisfying suitable conditions. Finally, an example is provided.