Love does exist. There are plenty of examples every where of how love and why love exists. There are some people that will willingly take a bullet for some one because they love them that much. There are some people that will give up basically every thing they have for one person, and that is love.
Though it depends upon what you mean by the word 'love'.
The word 'love' is actually a very inaccurate word because it implies a constant. But it's not a constant, it's always in flux. The way you fall in love at 15 is very different to the way you fall in love at 20, and different again at 30, 40, and so on. As your experiences are increased and you develop your own character more, so, too will the qualities you feel are necessary for 'love' to occur change. Not only that, but love is always increasing and decreasing. Sometimes we love our partners more on days when they have done something sweet than on day when they have done something disagreeable. Some people insist that love is a constant, but when a divorced couple are arguing over who gets the house, the car and the kids, where is the love? It's either constant or it isn't.
But all of the emotions and fuzzy feelings aren't necessarily 'love'. Affection, certainly, but not necessarily love. So what is love? Love is actions. To show true love to someone is to be extensional to them, to provide them with tools for thinking that increase their experiences, help them see the world in new and different ways and enhance their life. For example if I met a girl who was into physics, and if I knew more about physics than she did, and I gave her all of the information and tools she needed to increase her knowledge in that area, that is enhancing her life and so is showing true love. If you feed a hungry person, that is true love. Push someone from the path of an oncoming car is also true love. And so on and so forth.