The main purpose of this paper is to establish the existence of positive solutions to a class of quasilinear elliptic equations involving the ($p$-$q$)-Laplacian operator. We consider a nonlinearity that can be subcritical at infinity and supercritical at the origin; we also consider potential functions that can vanish at infinity. The approach is based on variational arguments dealing with the mountain-pass lemma and an adaptation of the penalization method. In order to overcome the lack of compactness, we modify the original problem and the associated energy functional. Finally, to show that the solution of the modified problem is also a solution of the original problem we use an estimate obtained by the Moser iteration scheme.
"Existence result for a class of quasilinear elliptic equations with ($p$-$q$)-Laplacian and vanishing potentials." Illinois J. Math. 59 (3) 545 - 575, Fall 2015. https://doi.org/10.1215/ijm/1475266397