Aims. Previous observations with the High Energy Stereoscopic System (H.E.S.S.) have revealed an extended very-high-energy (VHE; E> 100 GeV) γ-ray source, HESS J1834−087, coincident with the supernova remnant (SNR) W41. The origin of the γ-ray emission was investigated in more detail with the H.E.S.S. array and the Large Area Telescope (LAT) onboard the Fermi Gamma-ray Space Telescope. Methods. The γ-ray data provided by 61 h of observations with H.E.S.S., and four years with the Fermi LAT were analyzed, covering over five decades in energy from 1.8 GeV up to 30 TeV. The morphology and spectrum of the TeV and GeV sources were studied and multiwavelength data were used to investigate the origin of the γ-ray emission toward W41. Results. The TeV source can be modeled with a sum of two components: one point-like and one significantly extended (σTeV = 0.17° ± 0.01°), both centered on SNR W41 and exhibiting spectra described by a power law with index ΓTeV ≃ 2.6. The GeV source detected with Fermi LAT is extended (σGeV = 0.15° ± 0.03°) and morphologically matches the VHE emission. Its spectrum can be described by a power-law model with an index ΓGeV = 2.15 ± 0.12 and smoothly joins the spectrum of the whole TeV source. A break appears in the γ-ray spectra around 100 GeV. No pulsations were found in the GeV range. Conclusions. Two main scenarios are proposed to explain the observed emission: a pulsar wind nebula (PWN) or the interaction of SNR W41 with an associated molecular cloud. X-ray observations suggest the presence of a point-like source (a pulsar candidate) near the center of the remnant and nonthermal X-ray diffuse emission that could arise from the possibly associated PWN. The PWN scenario is supported by the compatible positions of the TeV and GeV sources with the putative pulsar. However, the spectral energy distribution from radio to γ-rays is reproduced by a one-zone leptonic model only if an excess of low-energy electrons is injected following a Maxwellian distribution by a pulsar with a high spin-down power (>1037 erg s-1). This additional low-energy component is not needed if we consider that 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 γ-ray sources, the detection of OH (1720 MHz) maser lines, and the hadronic modeling. ; H.E.S.S. Collaboration: A. Abramowski . P. deWilt . N. Maxted . G. Rowell . et al.