In light of the failure of the LUX experiment to find dark matter I was wondering if the whole idea of dark matter possibly indicates some problem with GR rather than simply being as yet undiscovered. Is this likely?
The problem is that there is lots of evidence that dark matter (whatever we mean by that) exists. We have observations like the Bullet Cluster, where the dark matter can be detected by its lensing, and galaxies like Dragonfly 44 where 99.99% of its matter is dark. Trying to explain away such extreme examples by fiddling with theories of gravitation brings a whole new meaning to the word contrived.
On the other hand general relativity works extremely well in the low field limit where we've been able to test it. Gravitational lensing and gravitational time dilation agree spectacularly well with the predictions of GR.
It is possible to modify GR because it's just one of a range of theories collectively known as metric theories. In the context of dark matter the most obvious alternative is TeVeS since that was specifically designed as a covariant form of the MOND theory that purports to explain away dark matter. The problem is that any modified theories have to give the same predictions as GR in the regimes where we have experimental data, and it's hard to get them to agree with current experimental evidence for GR and still explain away dark matter.
So at the end of the day, unpalatable though it may be to some, dark matter remains the best explanation for the behaviour of the universe. It's not that physicists love the idea but that right now there aren't any convincing alternatives.
The MOND (Modified Newton Dynamic) theory as always been active and might be the closest to what you are questioning. The idea of particles that we can't detect is quite simple and beautiful but there is another way to solve the problem of the missing mass. You can modified the law of gravitation, or Newton's law of acceleration :
It is said here that the attempt to get rid of dark matter is not successful :
The most serious problem facing Milgrom's law is that it cannot completely eliminate the need for dark matter in all astrophysical systems: galaxy clusters show a residual mass discrepancy even when analysed using MOND. The fact that some form of unseen mass must exist in these systems detracts from the elegance of MOND as a solution to the missing mass problem, although the amount of extra mass required is 5 times less than in a Newtonian analysis, and there is no requirement that the missing mass be non-baryonic. It has been speculated that 2 eV neutrinos could account for the cluster observations in MOND while preserving the theory's successes at the galaxy scale
I have to say that there is still a lot of experiments looking for dark matter (like Xenon100 and maybe soon Xenon1T)
The problem is not with GR, it's with the current incorrect understanding of dark matter.
The notion of dark matter as a weakly interacting clump of stuff that travels with the matter is incorrect.
Dark matter fills ‘empty’ space, strongly interacts with matter and is displaced by matter.
The dark matter displaced by the Earth pushing back and exerting pressure toward the Earth is gravity.
Curved spacetime is the state of displacement of the dark matter.