As long as you link dinamically to the LGPL-ed library you can do almoust anything with your program, includeing selling it or putting nagging screen to it. The apache license is much less strict then the LGPL. If you link statically to the LGPL-ed API you can still distribute it, but only with source code. You need to give the source to whoever buys your software. So you can't give your program without the source to anyone, thus a nagging screen is a non-sense, one can take it out and recompile it, a good example for this is FlowPlayer.
I recommend writing a new license if you want to use your own rules.
For Java there's no such thing as static or dynamic linkage, everything is linked somewhat dynamically, yet not fully. As the writer of the license, Dave Turner said, the linking method used by Java (the use of the 'import' keyword) falls under section 6 of the license. This means you must provide a way to the user to change the version of the library you used, also reverse engineer it to debug his changes. So you don't need to give away the source, neither you need to give description of the internals of your program, but you can not deny the users to reverse engineer it (which contradicts most proprietary licenses). You still need to provide the source of library. So basically you can sell your program to non-programmer people, or who are not smart enough to decompile your program and remove the nagging.
I'm not sure, but I think other's can also publish a modified version of your program, ex.: a coder takes your limited version, decompiles, removes every limitation, compiles it and publishes your work. Don't be fooled of the idea that no one will bother doing that, I did something like this (I wrote something like, because I modified and republished an LGPL-ed program). People can go far enough when there's about money.
You might also want to write your own license where you can specify what parts of the code can not be changed if reverse engineered (ex. The User is permitted to reverse engineer the Product, but can not modify sections in the code marked with *insert mark here*), this way blocking them to remove the nagging/limitations.