On some Hénon equations involving supercritical nonlinearity

We prove the existence of a positive radial solution in a unit ball centered at the origin for some classes of Hénon equations involving supercritical nonlinearity. More precisely, we study how Hénon's weight impacts the variable supercritical exponent in the context of the work by do Ó, Ruf, and Ubilla. For this purpose, we combine variational methods with a new Sobolev–Hardy type embedding for radial functions into variable exponent Lebesgue spaces. The suitable use of the Radial Lemma allows us to arrive at the necessary estimate with fewer assumptions than those found in the existing literature.