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We shall show that T : C → C and is compact provided t0 is sufficiently small, namely 1/(m−1) tm ≤ ε. 0 M Indeed for v ∈ C we have t0 T [v] − α ∞ s ≤ 0 0 q(τ ) f (v(τ ))dτ q(s) 1/(m−1) ds ≤ 1 m 1/(m−1) t M ≤ε m 0 32 ´ P. PUCCI, M. GARC´IA-HUIDOBRO, R. MANASEVICH, AND J. SERRIN and in turn T [v] ∈ C. Hence T (C) ⊂ C. Let (vk )k be a sequence in C and let s, t be two points in [0, t0 ]. Then 2 ¯m 1/(m−1) t M |t − s|. |T [vk ](t) − T [vk ](s)| ≤ m By the Ascoli-Arzel`a theorem this means that T maps bounded sequences into relatively compact sequences with limit points in C, since C is closed.

Montefusco and P. Pucci, Existence of radial ground states for quasilinear elliptic equations, Advances in Diff. Equations, 6 (2001), 959-986. A. Peletier and J. Serrin, Uniqueness of positive solutions of semilinear equations, Arch. Rational Mech. , 81 (1983), 181-197. A. Peletier and J. Serrin, Uniqueness of non-negative solutions of semilinear equations, J. Diff. Equations, 61 (1986), 380-397.  P. Pucci and J. Serrin, Uniqueness of ground states for quasilinear elliptic operators, Indiana Univ.

If v ∈ C then v([0, t0 ]) ⊂ [α − ε, α + ε], and in turn 0 < f (v(t)) ≤ M . Therefore from (Q1), s q(τ ) f (v(τ ))dτ ≤ q(s) 0≤ 0 s 0 < s ≤ t0 , f (v(τ ))dτ, 0 where the last integral approaches 0 as s → 0 by (F 1). 6) is well defined. We shall show that T : C → C and is compact provided t0 is sufficiently small, namely 1/(m−1) tm ≤ ε. 0 M Indeed for v ∈ C we have t0 T [v] − α ∞ s ≤ 0 0 q(τ ) f (v(τ ))dτ q(s) 1/(m−1) ds ≤ 1 m 1/(m−1) t M ≤ε m 0 32 ´ P. PUCCI, M. GARC´IA-HUIDOBRO, R. MANASEVICH, AND J.