Arduino nano esp8266 software serial problems

I have a problem with my arduino nano Atmega328 and my esp8266 01 module. I have connected the esp8266 on my arduino board with out ftdi.

I have made the following connection: I an using a 5V Arduino Nano which of course is 5V. The Esp8266 is a 3.3v device and so I am using a voltage divider to bring the Arduinos 5V down to 3.3v. If you are using a 3.3v Arduino then you do not need the voltage divider.

The wiring is very similar to the FTDI. Arduino pin 3 to voltage divider and then to ESP8266 RX Arduino pin 2 to ESP8266 TX Arduino GND to ESP8266 GND Pull CH_PD HIGH with a 10K resistor to +3.3v +3.3V to vcc

You only need the voltage divider on the Arduino TX pin. The 5V Arduino will read 3.3v as HIGH so you can connect the ESP8266 TX pin directly to the Arduino RX pin.

The problem i had and i cant handle is that when i start up the Arduino tx,rx leds works continuously so i cant have access to my serial monitor of Arduino Ide. So with that problem i cant send or receive any command to the ide. Also when i try to upload my program on Arduino i must have disconnected the esp. If i don t i can t upload it.

Any ideas guys? I would be gratefull

**I also try to implement code with AltSoftSerial but i have libraries issues my laptop doesnot recognize it. ( i download it and i put it in the same folder with my code ) ** I ve also linked a picture with my circuit schematic .

  #include <SoftwareSerial.h>

    SoftwareSerial ESPserial(2, 3); // RX | TX
    void setup() 
    {    Serial.begin(9600);     // communication with the host         computer

     // Start the software serial for communication with the ESP8266
     ESPserial.begin(9600);  

     Serial.println("");
     Serial.println("Remember to to set Both NL & CR in the serial monitor.");
     Serial.println("Ready");
     Serial.println("");}

     void loop() 
      {
      // listen for communication from the ESP8266 and then write it to the serial monitor
       if ( ESPserial.available() )   {  Serial.write( ESPserial.read();}  / / listen for user input and send it to the ESP8266
      if ( Serial.available() )       {  ESPserial.write( Serial.read()  );  }}