Deep learning (DL) in remote sensing has nowadays become an effective operative tool: it is largely used in applications, such as change detection, image restoration, segmentation, detection, and classification. With reference to the synthetic aperture radar (SAR) domain, the application of DL techniques is not straightforward due to the nontrivial interpretation of SAR images, especially caused by the presence of speckle. Several DL solutions for SAR despeckling have been proposed in the last few years. Most of these solutions focus on the definition of different network architectures with similar cost functions, not involving SAR image properties. In this article, a convolutional neural network (CNN) with a multi-objective cost function taking care of spatial and statistical properties of the SAR image is proposed. This is achieved by the definition of a peculiar loss function obtained by the weighted combination of three different terms. Each of these terms is dedicated mainly to one of the following SAR image characteristics: spatial details, speckle statistical properties, and strong scatterers identification. Their combination allows balancing these effects. Moreover, a specifically designed architecture is proposed to effectively extract distinctive features within the considered framework. Experiments on simulated and real SAR images show the accuracy of the proposed method compared with the state-of-art despeckling algorithms, both from a quantitative and qualitative point of view. The importance of considering such SAR properties in the cost function is crucial for correct noise rejection and details preservation in different underlined scenarios, such as homogeneous, heterogeneous, and extremely heterogeneous.