I've read some articles about Viewports to solve the problem with different resolution. But in every example they used a smaller size for the texture than the resolution of the screen. But what happens when my texture size is for example 1920 x 1080 and my screen is just 800 x 400? Is there any kind of Viewport that can scale the texture? I didn't found anything for this problem.
Looks like you mean scaling and not compressing. Compressing is a technique to lower the file size of a file, like a .jpg does over a uncompressed .bmp. So if you mean scaling then it completely depends on what you want. If you have a 1920x1080 texture and want this to completely fit the 800x400 screen it needs to be stretched since 1920x1080 = 16x9 and 800x400 = 2:1 = 16:8.
To stretch the image you need to use StretchViewport. You would create the viewport with the world dimensions you want to show. So in this case the 1920x1080 image, or perhaps 16x9 and put that texture with there "world units) in your world. StretchViewport will make sure the viewport is stretched to whatever size you use.
If you want it to fit as much as possible but remain the aspect ratio of 16:9 then you can use the FitViewport. Like StretchViewport each viewport (except ScreenViewport) need to be supplied with world units, this is a important concept. World units are not meters nor pixels until you define your world units as such. Anyway, FitViewport will also make sure each device sees as much of your world as you supply it but your assets won't stretch/deform. Instead it will create a "black" bar, much like TV's do when the aspect ratio is not equal.
ScreenViewport, by default just takes the size of the screen. So a 1920x1080 pixel screen would show 1920x1080 world units and a 640x480 pixel screen would show 640x480 world units. Therefore, the higher resolution the screen has the more of the world is visible. The amount of world units a screen pixel represents can be changed.
There are a view others: FillViewport is like FitViewport it will remain aspect ratio but always fills the screen. And thus some parts may not be visible. For that reason I can't think of a use for this one over FitViewport. And there is ExtentVieport, a mix between FitViewport and ScreenViewport.