Previous observations with HESS have revealed the existence of an extended very-high-energy (VHE; E>100 GeV) gamma-ray source, HESS J1834-087, coincident with the SNR W41. The origin of the gamma-ray emission has been further investigated with HESS and the Fermi-LAT. The gamma-ray data provided by 61h (HESS) and 4 yrs (Fermi LAT) of observations cover over 5 decades in energy (1.8GeV - 30TeV). The morphology and spectrum of the TeV and GeV sources have been studied and multi-wavelength data have been used to investigate the origin of the observed emission. The TeV source can be modeled with a sum of two components: one point-like and one significantly extended (sig_TeV = 0.17{\deg}), both centered on SNR W41 and exhibiting spectra described by a power law of index 2.6. The GeV source detected with Fermi is extended (sig_GeV =0.15{\deg}) and morphologically matches the VHE emission. Its spectrum can be described by a power-law with index 2.15 and joins smoothly the one of the whole TeV source. A break appears in the spectra around 100 GeV. Two main scenarios are proposed to explain the emission: a pulsar wind nebula (PWN) or the interaction of SNR W41 with a molecular cloud. X-ray observations suggest the presence of a point-like source (pulsar candidate) near the center of the SNR and non-thermal X-ray diffuse emission which could arise from a potential PWN. The PWN scenario is supported by the match of of the TeV and GeV positions with the putative pulsar. However, the overall spectrum is reproduced by a 1-zone leptonic model only if an excess of low-energy electrons is injected by a high spin-down power pulsar. This low-energy component is not needed if the point-like TeV source is unrelated to the extended GeV and TeV sources. The interacting SNR scenario is supported by the spatial coincidence between the gamma-ray sources, the detection of OH maser lines and the hadronic modeling.