Positive radial solutions involving nonlinearities with zeros

In this paper we consider the quasilinear elliptic problem In this paper we consider the quasilinear elliptic problem $$ $$ -Delta_p u=lambda |x|^{delta} f(u) quad extrm{in }B_1(0)$$ $$ u=0 quad extrm{in }partial B_1(0), $$ where $f$ is nonnegative. We assume $f(0)=0$, that there is $U>0$ such that $f(U)=0$, and that $f$ is subcritical in $0$ and in $U$ with respect to Sobolev. We show that there is a positive $lambda^*$ such that for $lambda>lambda^*$ there are at least three positive radial solutions. The same conclusion is obtained replacing the subcriticality in $U$ with the subcriticality of $f$ for $u$ large. where $f$ is nonnegative. We assume $f(0)=0$, that there is $U>0$ such that $f(U)=0$, and that $f$ is subcritical in $0$ and in $U$ with respect to Sobolev. We show that there is a positive $lambda^*$ such that for $lambda>lambda^*$ there are at least three positive radial solutions. The same conclusion is obtained replacing the subcriticality in $U$ with the subcriticality of $f$ for $u$ large.