On the subcriticality assumptions for the existence of ground states of quasilinear elliptic equations, Adv. Differential Equations 8

Abstract. We study conditions on f which ensure the existence of non-negative, nontrivial radial solutions vanishing at in¯nity of the quasilin-ear elliptic equation ¡¢pu = f(u) in Rn; with n> p: Both the behaviors of f at the origin and at in¯nity are important. We discuss several di®er-ent subcritical growth conditions at in¯nity, and we show that it is pos-sible to obtain existence of solutions also in some supercritical cases. We also show that, after an arbitrarily small Lq perturbation (1 · q < 1) on f, solutions can be obtained without any restrictions on the behavior at in¯nity. In our proofs we use techniques from calculus of variations and arguments from the theory of ordinary di®erential equations such as shooting methods and the Emden-Fowler inversion. 1