Context: The supernova remnant (SNR) G35.6-0.4 shows a non-thermal radio shell, however, no {gamma}-ray or X-ray counterparts have been found for it thus far. One TeV source, HESS J1858+020, was found near the SNR and this source is spatially associated with some clouds at 3.6 kpc. Aims: To attain a better understanding of the origin of HESS J1858+020, we further investigate the association between SNR cosmic rays (CRs) and the clouds through the Fermi-LAT analysis and hadronic modeling. Methods: We performed the Fermi-LAT analysis to explore the GeV emission in and around the SNR. We explored the SNR physics with previously observed multi-wavelength data. We built a hadronic model using runaway CRs of the SNR to explain the GeV-TeV observation. Results: We found a hard GeV source (SrcX2) that is spatially coincident with both HESS J1858+020 and a molecular cloud complex at 3.6 kpc. In addition, a soft GeV source (SrcX1) was found at the northern edge of the SNR. The GeV spectrum of SrcX2 connects well with the TeV spectrum of HESS J1858+020. The entire {gamma}-ray spectrum ranges from several GeV up to tens of TeV and it follows a power-law with an index of ~2.15. We discuss several pieces of observational evidence to support the middle-aged SNR argument. Using runaway CRs from the SNR, our hadronic model explains the GeV-TeV emission at HESS J1858+020, with a diffusion coefficient that is much lower than the Galactic value.
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