This paper is devoted to the following boundary value problem with tempered fractional derivatives \(\begin{aligned} & {\mathbb {D}}_{b^-}^{\alpha , \sigma } ({^{C}}{\mathbb {D}}_{a^+}^{\alpha , \sigma }u(x)) = f(x,u),\, \, x\in (a,b)\\ & u(x)= 0\, \, \hbox { on}\ \mathbb {R}\setminus (a,b),\qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad {(P)} \end{aligned}\) where \(\alpha \in (0,\frac{1}{2})\) , \(\sigma >0\) , \({\mathbb {D}}_{b^-}^{\alpha , \sigma }u, {^{C}}{\mathbb {D}}_{a^+}^{\alpha , \sigma }u\) are the right Riemann-Liouville and left Caputo tempered fractional derivative respectively and \(f: [a,b] \times {\mathbb {R}} \rightarrow {\mathbb {R}}\) is a suitable Carathéodory function. In order to obtain at least one weak solution, we use minimal principle and Morse theory. To assure the (PS) conditions, a new compact embedding is presented. This work complements the previous works (Cuti Gutierrez et al. in J Appl Anal Comput 14:3496-3519, 2024; Torres Ledesma in Progr Fract Differ Appl 10:677-694, 2024, J Pseudo-Differ Oper Appl 14:62, 2023) considering the case \(\alpha \in (0, \frac{1}{2})\) .
