。
She published the book at her own expense.
Literal
She [topic-は] her own expense [at-で] book [object-を] published.
自分の負担で ('at one's own expense, on one's own dime') — 負担 ('burden, responsibility, expense') paired with the で of means/scope marks who covers the cost. Standard phrasing for self-financed projects (vanity publishing, independent runs, or paying out-of-pocket for any institutional cost). 出版する ('to publish') is the verb of choice for any deliberate release of printed material.