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 \(\begin{aligned} {\left\{ \begin{array}{ll} {_{x}}D_{\infty }^{\beta }{_{-\infty }}D_{x}^{\beta }e^{C {_{x}}D_{\infty }^{\beta }{_{-\infty }}D_{x}^{\beta }}u = \lambda \omega (x)u+ Q(x)g(x,u)&{}\text{ in }\,\,{\mathbb {R}},\\ u\in \mathcal {H}_c^{\beta ,\infty } ({\mathbb {R}}), \end{array}\right. } \end{aligned}\) $$\begin{array}{c}\hfill \left\{\begin{array}{cc}{}_x{D}_{\infty }^{\beta }{}_{-\infty }{D}_x^{\beta }{e}^{C{}_x{D}_{\infty }^{\beta }{}_{-\infty }{D}_x^{\beta }}u=\lambda \omega (x)u+Q(x)g(x,u)\hfill & \text{in}\mathbb{R},\hfill \\ u\in {\mathcal{H}}_c^{\beta ,\infty }(\mathbb{R}),\hfill \end{array}\right)\end{array}$$ where \(\beta \in (0,\frac{1}{2})\) $\beta \in (0,\frac{1}{2})$ , \({_{-\infty }}D_{x}^{\beta }u(\cdot ), {_{x}}D_{\infty }^{\beta }u(\cdot )\) ${}_{-\infty }{D}_x^{\beta }u(·),{}_x{D}_{\infty }^{\beta }u(·)$ denote the left and right Liouville–Weyl fractional derivatives, \(\omega ,Q:{\mathbb {R}}\rightarrow {\mathbb {R}}\) $\omega ,Q:\mathbb{R}\to \mathbb{R}$ is a positive function with \(\omega ,Q\in L^{\frac{1}{2\beta }} ({\mathbb {R}})\) $\omega ,Q\in L^{\frac{1}{2\beta }}(\mathbb{R})$ and \(g: {\mathbb {R}}\rightarrow {\mathbb {R}}\) $g:\mathbb{R}\to \mathbb{R}$ is a continuous function satisfying suitable conditions. Finally, an example is provided.