"persistent doubts" is a correct and usable phrase in written English.
It refers to doubts that continue to linger even after attempts to resolve or dismiss them. You can use it in formal or informal writing to describe a situation or individual that consistently has doubts about something. Example: Despite all the evidence presented, Karen still had persistent doubts about the company's financial stability.
However, there have been persistent doubts about claims made for Horse Hill's potential.
Secretary of Defense Donald H. Rumsfeld has expressed persistent doubts about putting American troops into peacekeeping missions abroad.
The Spaniard's future with Astana has been unclear for months amid persistent doubts about the Kazakh team's financial viability.
Despite the interest in Warner, there are persistent doubts about the health and investment value of large music companies.
At the same time, though, there were persistent doubts that these rosy numbers were capturing what was happening inside the shuttle program.
It showed how she was responding to an intensely competitive field and to persistent doubts about her initial support for the war in Iraq and her electability.
That narrowness sometimes reflects the tight focus of a professor's research, but it can also reflect a persistent doubt about the humanistic enterprise.
(K 1 80 87) In the first of the discussion's three phases, the nature of this persistent doubt is explained.
For all the euphoria and the undoubted bravery seen on the streets of Cairo and elsewhere, there remains a fundamental and persistent doubt amongst Arabs that democracy can work for them as free-thinking individuals.
Ultimately, persistent doubt over how much the U.S. is willing to support a group its other allies hate cannot diminish what the Kurds have already accomplished.
I have a persistent doubt however that something is not quite right with this argument and the issue leaves me pondering somewhat.
