9X^4 + 4Y^4 + 3Z^4 = W^4

Existence of solution for diophantine equation ax^4 + by^4 + cz^4 + dw^4 = 0 are known if abcd is square number.
So, we are curious about whether above equation has a solution or not if abcd is not square number.
In particular, when does this equation have infinitely many integer solutions?

Parametric solutions of x^4 + ay^4 = z^4 + bt^4 are given below.
x^4 + ay^4 = z^4 + bt^4

We show diophantine equation 9X^4 + 4Y^4 + 3Z^4 = W^4 has infinitely many integer solutions.
This equation is related to X^4 + 36Y^4 + 27Z^4 = 9W^4 and 36X^4 + Y^4 + 12Z^4 = 4W^4.

9X^4 + 4Y^4 + 3Z^4 = W^4..........................................................(1)

We use an identity 9(t+1)^4 + 4(t)^4 + 3(t^2+2t+3)^2 = 4(2t^2+3t+3)^2.............(2)

So, we look for the integer solutions {Z^2 = t^2+2t+3, W^2 = 4t^2+6t+6}...........(3)

By parameterizing the second equation and substituting the result to first equation, then we obtain quartic equation below.

u^2 = 51k^4-216k^3+390k^2-288k+79.................................................(4)

This quartic equation is birationally equivalent to an elliptic curve below.
y^2 = x^3 - 18x + 8.
Rank is 2 and generator is [-4 , -4], [7 , 15].

Hence we can obtain infinitely many integer solutions for equation (1).

Numerical example:

[X,Y,Z,W] = [41, 239, 283, 424], [32030839, 15231959, 39879561, 64471262].