What was Jonas's solution to his problem in The Giver?

Jonas’s problem is that his community is overly restrictive and after he begins his training as Receiver of Memory he realizes this and wants to do something about it, so he runs away.
The situation in Jonas’s community is not something that he is aware of until he begins his training as Receiver of Memory.  Like everyone else in his community, he thinks that his community is perfect.  Once he becomes the new Receiver and begins training, he realizes that his community has made a lot of compromises and trade-offs for this perfection.
Jonas’s community wants everyone to be happy.  In the pursuit of this goal, the community has instituted a policy of Sameness. This means that there are strict rules for every aspect of life, and most decisions are made for the citizens.  They have common property and relationships are created for people.
Jonas's training gives him, and only him, access to memories of how things used to be.  As Jonas continues his training, he feels more and more isolated from people.  He starts to understand that they are living a life that is outside of what humans are meant to experience.  They do not live a full life.

They have never known pain, he thought. The realization made him feel desperately lonely, and he rubbed his throbbing leg. He eventually slept. Again and again he dreamed of the anguish and the isolation on the forsaken hill. (Ch. 14) 

Jonas’s community does not know pain, but they also do not know love.  There is no romance, because the population is controlled and people take special pills to prevent puberty.  Because of this, there are also no families in the sense that we know them.  No one knows affection.  Jonas realizes that people are missing out on a large part of the human experience. 
When Jonas realizes that his community has an even darker side, he decides to take action.  This comes when he sees his father give a lethal injection to a newborn baby.  His community calls this release, and it is fairly common. 

He killed it! My father killed it! Jonassaid to himself, stunned at what he was realizing. He continued to stare at the screen numbly.
… Then he picked up a small carton that lay waiting on the floor, set it on the bed, and lifted the limp body into it. He placed the lid on tightly. (Ch. 19)

Jonas decides to address this problem by running away.  He has the help of his mentor, The Giver.  When Jonas runs away, the community's memories will return to them, and they will no longer be able to maintain the Sameness than they have been trapped in.