1. Bad Defending
Arsenal's defence is the weakest point, no doubt about it. Some might argue Arsenal's weakness is in goal, but Wojciech Szczesny is good enough, he is just affected by Arsenal's lack of coordination on the backline. It's an irony. Arsenal was very popular of "Boring-boring 1-0 Arsenal" before Arsene Wenger came to Arsenal, and they had a very strong defence, but now Arsenal's defence is far from strong, and even the worst from the top 6 teams.
Thomas Vermaelen's absence might be the factor why the backline was very vulnerable. Vermaelen was really outstanding last season, and Arsenal really miss Vermaelen's presence on the backline.
Koscielny could be a good player, but he is far from Vermaelen, lacking concentration, and still melts under the pressure. Squillaci is too fragile to deal with physically strong opponent, and also often makes silly mistakes. Djourou is good, but quite inconsistent.
Gael Clichy is lacking of coordination with his mates, and often, Arsenal concede from his "assist", breaking the offside trap. Bacary Sagna stands to be Arsenal's best defender last season, very consistent throughout season.
Arsenal might need to buy one world class central defender, a commanding central back, to pair with Thomas Vermaelen.
2. Giving Away a Victory
If one match only lasted for 80 minutes, Arsenal might have won the EPL. Arsenal usually took lead before equalized, or even lost. 4-4 against Newcastle and also Tottenham. 1-1 against Liverpool, and Sunderland. 1-2 against Bolton, 2-3 against arch-rival Tottenham. At least 12 points is wasted because of the late goals Arsenal concede, and Arsenal only at least got 5 points saved from scoring at final minutes. Lack of oncentration, and also composure, could be said like that. Again, the mentality is playing. This is the matter of mentality, you have to hold on if you have a lead, not giving away cheap goals!
3. Winning Mentality
Nah, it's something Arsenal don't have, they just don't show the real character a winner should have. Look at Barcelona, after they lost at Copa Del Rey final against Real Madrid, their mentality didn't drop. They even play better than before, and it's known that Barcelona beat Madrid in the semifinal of Champions League. They weren't too downhearted with the defeat in Copa Del Rey. However, Arsenal didn't show their mentality. After losing against Birmingham in the League Cup final, they just fell down, and tragically finished fourth at Premier League. Again, I emphasize the urgent of buying world class player(s) for Arsenal. One or two players could bring in the winning mentality for the young, potential players, that still need tutor from the more experienced players.
4. Second Half Season Poor Form
This has been Arsenal's chronic illness since the last 4 years, since 2007/2008, where they were leading 5 points ahead Manchester United and unbelievably they keep being held and losing, then the result was third place, losing the title painfully. Then 2008/2009, they still plays quite well, until mid-february if i'm not mistaken, and they fell down again. Last year was no different. English Premier League had a three horse race between the Gunners, MU, and also Chelsea, but after Fabregas was sidelined because of injury, everything went wrong. So did this season. Even mathematically they were on pole position following Manchester United two loss in a row, but they held by Blackburn and Sunderland, instead of winning to take the 1st position, and several draws and losses, then 4th place, Arsenal's second place was taken by Chelsea, and even Man City overtake the 3rd place, which means Arsenal need to start from qualifying of Champions League, not the automatic spot. The squad is a little bit lack of depth, with the bench players don't have same quality as the first team. Time to sign someone else, or at least rotating players more.