Abstract

We establish the Trudinger-Moser inequality on weighted Sobolev spaces in the whole space, and for a class of quasilinear elliptic operators in radial form of the type Lu:=−r−ξ(rα|uâ€Č(r)|ÎČuâ€Č(r))â€Č, where Ξ,ÎČ≄0 and α>0, are constants satisfying some existence conditions. It is worth emphasizing that these operators generalize the p-Laplacian and k-Hessian operators in the radial case. Our results involve fractional dimensions, a new weighted PĂłlya-Szegö principle, and a boundness value for the optimal constant in a Gagliardo-Nirenberg type inequality.

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