In this work, a dense recurrent convolutional neural network (DRCNN) was constructed to detect sleep disorders including arousal, apnea and hypopnea using available Polysomnography (PSG) measurement channels provided in the 2018 PhysioNet challenge database. Our model structure is composed of multiple dense convolutional units (DCU) followed by a bidirectional long-short term memory (LSTM) layer followed by a softmax output layer. The sleep events including sleep stages, arousal regions and multiple types of apnea and hypopnea are manually annotated by experts which enables us to train our proposed network using a multi-task learning mechanism. Three binary cross-entropy loss functions corresponding to sleep/wake, arousal presence/absence and apnea-hypopnea/normal detection tasks are summed up to generate our overall network loss function that is optimized using the Adam method. Our model performance was evaluated using two metrics: the area under the precision-recall curve (AUPRC) and the area under the receiver operating characteristic curve (AUROC). To measure our model generalization, 4-fold cross-validation was also performed. For training, our model was applied to full night recording data. Finally, the average AUPRC and AUROC values associated with the arousal detection task were 0.505 and 0.922, respectively on our testing dataset. An ensemble of four models trained on different data folds improved the AUPRC and AUROC to 0.543 and 0.931, respectively. Our proposed algorithm achieved the first place in the official stage of the 2018 PhysioNet challenge for detecting sleep arousals with AUPRC of $0.54$ on the blind testing dataset.