A note on quasilinear Schrödinger equations with singular or vanishing radial potentials

Abstract

. In this note we complete the study of [3], where we got existence results for the quasilinear elliptic equation N , with singular or vanishing continuous radial potentials V ( r ), K ( r ). In [3] we assumed, for technical reasons, that K ( r ) was vanishing as r → 0, while in the present paper we remove this obstruction. To face the problem we apply a suitable change of variables w = f ( u ) and we find existence of non negative solutions by the application of variational methods. Our solutions satisfy a weak formulations of the above equation, but they are in fact classical solutions in R N \ { 0 } . The nonlinearity g has a double-power behavior, whose standard example is g ( t ) = min { t q 1 − 1 , t q 2 − 1 } ( t > 0), recovering the usual case of a single-power behavior when q 1 = q 2 .