Multiplicity of positive radial solutions of a quasilinear elliptic problem in a ball

Abstract

We prove the existence of infinitely many positive, radially symmetric and decreasing solutions of the quasilinear elliptic problem $$ \begin{cases} -{\hbox{div} (a(|\nabla u|^2) \, \nabla u )} = f(|\hbox{x}|,u), & \hbox{ in }\, B_R,\\ \quad u = 0, & \hbox{ on }\, \partial B_R, \end{cases} $$ where $B_R$ is a ball in ${\Bbb R}^N$. The nonlinear differential operator is rather general and includes, as a particular case, the $m$--Laplacian, with $m>1$. The assumptions on $f$ are placed only at $+\infty$ and require a sort of oscillatory behaviour of the primitive $\int_0^s f(|\hbox{x}|,\xi)\, d\xi$. The proof is based on time--mapping estimates, the upper and lower solutions method and a maximum--like principle. The results are new even in the case of the Laplacian $\Delta$. A counterexample is also constructed in order to show that the assumptions cannot be relaxed.