We consider a two-obstacle problem with measure data. For measures that do not charge sets of zero $p(\cdot)$-capacity, we obtain the existence and uniqueness of the solution. On the other hand, for the measure concentrated on a set with zero $p(\cdot)$-capacity, we prove a nonexistence result in the sense that when one looks for solutions via approximation, one cannot find a reasonable solution; see Theorem 2.3 and Remark 2.1 below.
LI, HONGTAO and CHAI, XIAOJUAN
"A two-obstacle problem with variable exponent and measure data,"
Turkish Journal of Mathematics: Vol. 41:
3, Article 21.
Available at: https://journals.tubitak.gov.tr/math/vol41/iss3/21