How to tackle a multilabel classification problem
LSTMs, like any other neural net, implicitly support multi-label classification.  You should ensure that your output layer has $n$ neurons, one for each class, and you should use logistic activation rather than softmax activation (which is typically used for the final layer in multi-class problems).
Each neuron $N_i$ in the final layer will output a value between 0 and 1.  If the output of $N_i$ is greater than 0.5, then the example is a member of label $i$.
Here's a pretty good article that might help.