In a medium bowl combine graham cracker crumbs, sugar, and butter. Press the mixture onto the bottom of a 9-inch springform pan. Refrigerate for 30 minutes.
In a blender combine strawberries and cornstarch. Cover and blend until smooth. Pour the mixture into a saucepan and bring to a boil. Boil and stir for 2 minutes. Set aside 1/3 cup of the strawberry sauce, allow to cool. Cover and refrigerate the remaining sauce before serving.
In a large mixing bowl beat cream cheese until light and fluffy. Gradually beat in the sweetened condensed milk. Add lemon juice and mix well. Add eggs one at a time, beating on low speed after each addition. Pour half of the cream cheese mixture over the crust. Drop half of the reserved strawberry mixture over the cream cheese mixture by ½ teaspoonfuls. Carefully spoon the remaining cream cheese mixture on top of that. Drop the remaining strawberry sauce by ½ teaspoonfuls over the top. With a knife cut through the top layer to swirl the strawberry sauce.
Place into the oven and bake at 300 degrees for 45-50 minutes or until the center is almost set. Cool on a wire rack for 10 minutes. Refrigerate overnight. Serve the cheesecake with the leftover strawberry sauce.