Convert PNG to GeoTiff using GDAL

If you don't know the extent in coordinates of the prj file, you have to georeference the file manually using ground control points. This is rather comfortable if you have QGIS installed, and you can guess some details from the image.

In some cases, you can build the extent manually if the filename follows a certain rule, like the one-degree-SRTM files do.

Or the file covers the whole world, but that would require a certain size ratio, like 2:1 for WGS84. With LCC projection this is not possible.

By the way: You will have to expand your paletted colours to RGB or RGBA using -expand rgb. A projected palette will look very weird, because GDAL will try to interpolate the palette values.