On subcriticality assumptions for the existence of ground states of quasilinear elliptic equations

Abstract

We study conditions on $f$ which ensure the existence of nonnegative, nontrivial radial solutions vanishing at infinity of the quasilinear elliptic equation $-\Delta _{p}u=f(u)$ in $\mathbb{R}^{n},$ with $n>p.$ Both the behaviors of $f$ at the origin and at infinity are important. We discuss several different subcritical growth conditions at infinity, and we show that it is possible to obtain existence of solutions also in some supercritical cases. We also show that, after an arbitrarily small $L^{q}$ perturbation ($1\le q <\infty$) on $f $, solutions can be obtained without any restrictions on the behavior at infinity. In our proofs we use techniques from calculus of variations and arguments from the theory of ordinary differential equations such as shooting methods and the Emden-Fowler inversion.