JavaFX design writes to static field from constructor
One alternative is to use the Borg Pattern. Allow creating multiple instances, all of which, simply share the same state. Most online references will be in Python (where it is real easy to code), but the idea can be transplanted to Java too.

The advantage over the Singleton pattern is that it hides the fact that there is shared state from clients of the class, especially those that are interested in creating instance.

Clients (in casu the JavaFX framework) can simply use the constructor. Your own code can also easily get the Application by creating a separate instance.

so some sample code :

public class LJGM extends Application {

    private static SharedState sharedState = new SharedState();

    private static class SharedState {

        private Object variable;

        private Object getVariable() {
            return variable;
        }
    }

    public LJGM() {
    }

    public Object getVariable() {
        return sharedState.getVariable();
    }

    @Override
    public void start(Stage primaryStage) throws Exception {
        // (add GUI to the primaryStage here)
    }

    public static void main(String[] args) {
        launch(args);
    }
}


All that being said, I don't see any reason why you could not simply use dependency injection to avoid having global state here. So rather than having low level components query the Application object, let the Application pass them the parameters they will need, by passing itself (preferably implementing an interface) or by passing in the variables themselves.

public class LJGM extends Application implements ApplicationContext {

    private Object variable;

    public LJGM() {
    }

    public Object getVariable() {
        return variable;
    }

    @Override
    public void start(Stage primaryStage) throws Exception {
        // (add GUI to the primaryStage here)
        new LowLevelComponent(this);
        new OtherLowLevelComponent(variable);
    }

    public static void main(String[] args) {
        launch(args);
    }
}