Are there general ways for given rational coefficients $a,b,c$ (I am particularly interested in $a=3,b=1,c=8076$, but in general case too) to answer whether this equation has a rational solution or not?



Diophantine equations $ax^4+by^2=c$ in rational numbers
Your curve is a genus 1 curve, usually expressed as 
$$ y^2= -abx^4+bc$$
(just by multiplying everything by $b$ and changing $y$ by $by$). 


The curve has local points everywhere, but since it looks like has no rational points, you can try to do a 2-descent. This can be done easily with the TwoCoverDescent algorithm as explained in 


Nils Bruin and Michael Stoll. Two-cover descent on hyperelliptic curves. Math. Comp., 78:2347--2370, 2009. 


It is not guaranteed to succeed, but in the special case you are interested $a=3$, $b=1$ and $c=8076$ it does and the answer is that **it has no rational points**. 


All this has been implemented in MAGMA. You can use the following code



> 
> P:=PolynomialRing(Rationals());
>  H:=HyperellipticCurve((-3)\*P.1^4+8076);
>  Hk:=TwoCoverDescent(H);#Hk;
> 
> 
> 


It answers 0, so there is no 2-cover with rational points, so the original curve has no rational points. 


