The price elasticity of supply does not have any great effect on general market demand. Supply and demand do not typically affect one another.
Price elasticity of supply tells us how much the quantity supplied of a product will change if the price of that good changes. Put differently, it tells us what the slope of the supply curve is. But the shape of the supply curve has no bearing on the demand curve.
While price elasticity of supply does not affect demand, it does affect what equilibrium quantity is sold when demand changes. As demand changes, the price of a product changes. But the price elasticity of supply determines how much the price will change. This is because the price elasticity of supply determines the slope of the supply curve.