Finding dense subgraphs of a large network is a fundamental problem in graph mining that has been studied extensively both for its theoretical richness and its many practical applications over the last five decades. However, most existing studies have focused on graphs with a single type of connection. In applications such as biological, social, and transportation networks, interactions between objects span multiple aspects, yielding multiplex graphs. Existing dense subgraph mining methods in multiplex graphs consider the same importance for different types of connections, while in real-world applications, one relation type can be noisy, insignificant, or irrelevant. Moreover, they are limited to the edge-density measure, unable to change the emphasis on larger/smaller degrees depending on the application. To this end, we define a new family of dense subgraph objectives, parametrized by two variables $p$ and $β$, that can (1) consider different importance weights for each relation type, and (2) change the emphasis on the larger/smaller degrees, depending on the application. Due to the NP-hardness of this problem, we first extend the FirmCore, $k$-core counterpart in multiplex graphs, to layer-weighted multiplex graphs, and based on it, we propose two polynomial-time approximation algorithms for the generalized densest subgraph problem, when $p \geq 1$ and the general case. Our experimental results show the importance of considering different weights for different relation types and the effectiveness and efficiency of our algorithms.