Linux executable can't find shared library in same folder
LD_LIBRARY_PATH is a quick ad-hoc hack to specify alternate library loading search paths. A more permanent and cleaner solution is to specify the specific sets of paths in which libraries shall be searched specific for your particular binary. This is called the rpath (Wikipedia article on it: https://en.wikipedia.org/wiki/Rpath). There are a number of "variables" that can be specified in the binary rpath that get substituted. In your case the rpath variable ${ORIGIN} would be the most interesting for you. ${ORIGIN} tells the dynamic linker to look for libraries within the very same directory in which also the binary resides.
The rpath can be set at link time with the -rpath linker option, i.e. when invoked through GCC the option would be -Wl,-rpath='${ORIGIN}', i.e.
gcc -o program_binary -Wl,-rpath='${ORIGIN}' -lSDL2_mixer a.o b.o …For an existing binary the rpath can be set post-hoc using the chrpath or patchelf tools; it's better to set it at link time proper, though.