How to send multiple command without using delay to serial port?
The PIC32 UART receiver has an internal buffer which can store up to 8 characters. If your code cannot read the data out of this buffer fast enough it will 'overrun' and further received characters will be lost until the overrun error is cleared.

Without seeing your receive code I can't say exactly why this is happening, but since the first command is apparently received and executed correctly I suspect the serial port is not being read fast enough (or at all) while processing the command, and this is taking so long that the UART buffer overruns before it finishes. There are several ways you can fix this:-


Add a delay after sending each command, with a time period longer than the time it takes for the PIC to process the command (light LEDs etc.).
Continue receiving data while processing the command. The normal way to do this is have each received character generate an interrupt. The interrupt halts your program wherever it may be, saves the machine state and jumps into your interrupt handler, which reads characters out of the UART buffer and stores them in a RAM buffer. On exit the machine state is restored and your program continues as if nothing happened, except you now have some characters in the RAM buffer waiting to be processed.
After receiving and processing a command, send an 'acknowledge' character back to the Linux machine. The Linux machine is programmed to wait until until it has received this character before sending the next command. Different characters could be sent back depending on what the PIC wants to tell Linux about its status.
'Echo' back each character after it is received. The Linux machine waits for the echo and (if it matches the transmitted characters) knows that it has been received correctly. The PIC delays echoing back the last character until it has processed the command and is ready to receive the next command.   
Use a standard 'software' flow control protocol such as XON/XOFF, which uses two control codes to tell the transmitter when it can send data. 


Several of these techniques could be combined for enhanced security and reliability.