How does the couple overcome hardships in "The Gift of the Magi" by O. Henry?

Jim and Della face the challenge of buying Christmas presents for each other when they, as a couple, are poor and have very limited funds available for luxuries like Christmas gifts.

However, they each display resourcefulness, imagination, and determination to overcome lack of financial resources in order to purchase "something just a little bit near to being worthy of the honor of being owned by" the other. In the process, they also demonstrate the depth of their devotion and the length to which they are willing to make personal sacrifices in order to please the other.

Della sold her fabulously long hair, which she had probably been growing for most of her life, in order to obtain the money she needed to buy the watch fob for Jim. Jim sold his family heirloom pocket watch, inherited from his grandfather and father, in order to raise the funds to buy the set of tortoise shell hair combs for Della.

In "The Gift of the Magi," how does O. Henry portray the couple's poverty and its impact on their relationship?

In “The Gift of the Magi,” O. Henry shows us the poverty of the young couple—Della and Jim—rather then telling us that they are poor. Let's look at how he does that and then see how their poverty enhances the beauty of their relationship.

The story begins with “One dollar and eighty-seven cents.” This is all the money Della has to spend on a Christmas present for her husband, and it certainly isn't much. We can tell this because Della sits down on the “shabby little couch” and cries. Della and Jim are poor. They cannot afford nice furniture; what they have comes with their flat, which only costs eight dollars a week. Jim only earns twenty dollars a week, and as Della reflects, that “doesn't go far,” and their expenses always seem to be more than she has planned. Indeed, this young couple is quite poor.

Yet Della is determined to buy Jim a wonderful Christmas present, and she decides what she must do. She puts on her “old brown jacket” and “old brown hat” (for she cannot afford new clothes) and goes out. Then Della sells her hair for twenty dollars and buys her husband a wonderful watch fob chain to go on his heirloom watch. Yes, Della is forced to sell her hair to get the kind of present she wants for Jim.

The irony of the story comes when Jim reveals his present to Della. He has pawned his watch to get her hair combs. When the two young people realize what has happened, they decide to put their gifts aside for a bit and not use them. Yet their sacrifices in the midst of their poverty reveal how much they love each other. They are both willing to give up their most prized possession for the sake of their beloved, and this is truly beautiful. Poverty cannot hinder love.

In "The Gift of the Magi," how do the characters overcome challenges through love?

As you say there is a problem in the short story. The main two characters, Della and Jim are poor and they want to express their love to each other in the form of gifts for Christmas. However, both lack money as they are barely getting by. 

They both have one prized possession. Della has her beautiful hair and Jim has a gold pocket watch. They, of course, love their prized possessions, but they love each other more. So, Della cuts her hair and sells it. She buys a chain for Jim's watch. Jim, in the meantime, sells his watch and buys combs for Della's beautiful hair. 

In the end, they have gifts that they cannot use. However, through their act of sacrifice, they demonstrated incredible love and sacrifice. For this reason, O. Henry call them wise. It is better to give than to receive is the moral of the story. And consequently this is how they overcome with love.