A strategic and tactical conundrum.
1) The rank and file of the LDP selected Abe Shinzō as party president in late 2006 on the assumption that Abe was the candidate most likely to lead the party to electoral victory. Abe's main rival, former Foreign Minister Fukuda Yasuo dropped out of the leadership race early when LDP members voiced strong doubts about the public responding positively to the selection of a man of 70 as party leader.
Now that Abe has failed to lead the party to victory in this past election, the raison d'être of the Abe party presidency has evaporated for many LDP members, particularly for the 81 freshmen representatives (the so-called "Koizumi Children") elected in the September 2005 House of Representatives landslide. LDP members in tight electoral races are thinking of how they will survive Abe's leadership, not how they will ride his coattails to reelection.
2) The steep decline of the prime minister's and the LDP's popularity ratings began not with the pensions debacle or even the money scandals and verbal gaffes of the various members of the Cabinet. It began with the readmittance of 11 former LDP members Koizumi Junichiro had expelled from the LDP in August 2005 over their opposition to the reform of the Post Office. The public disliked Abe's reversal of the previous prime minister's main political program and resented the repudiation of the public's will as expressed in the results of the 2005 election. Readmitting the 11 "exiles" made sense to the members of the LDP leadership because they thought the 11 would provide crucial campaign support to certain LDP candidates in the 2007 House of Councillors election.
In the end, the move proved to be a fiasco. The readmitted exiles had little influence on the vote in the districts they were supposed to deliver to the LDP, and the stigma of “politics without principle” sunk the LDP all over the country.
From what has gone on in the first 10 months of his tenure, Abe apparently has no interest in or plans for the rural-urban divide, either for or against. He has no concept of the role of the state in the economy, feeling only that it should be sympathetic toward 30 year olds with dodgy work histories and companies with dodgy credit histories. His main contribution to Japanese diplomacy--the non-act of not visiting Yasukuni Shrine--could be accomplished just as competently by a tree or fire extinguisher ("Good Morning. This just in: Abe Shinzō, prime minister of Japan, still has not gone to Yasukuni.") He has not the slightest understanding of the United States outside of what can be experienced in four star hotel lobbies. His understanding of Asian countries has even less depth. As for the Japanese people, he has no particular love for them, especially after what happened on Sunday.
So what is the benefit of having Abe Shinzō carry on--for members of the LDP, that is?