  Motivated by the success of experimental manipulation of the band structure
through biaxial strain in Sr$_2$RuO$_4$ thin film grown on a mismatched
substrate, we investigate theoretically the effects of biaxial strain on the
electronic instabilities, such as superconductivity (SC) and spin density wave
(SDW), by functional renormalization group. According to the experiment, the
positive strain (from lattice expansion) causes charge transfer to the
$\gamma$-band and consequently Lifshitz reconstruction of the Fermi surface.
Our theoretical calculations show that within a limited range of positive
strain a p-wave superconducting order is realized. However, as the strain is
increased further the system develops into the SDW state well before the
Lifshitz transition is reached. We also consider the effect of negative strains
(from lattice constriction). As the strain increases, there is a transition
from p-wave SC state to nodal s-wave SC state. The theoretical results are
discussed in comparison to experiment and can be checked by further
experiments.
