Hénon type equations with jumping nonlinearities involving critical growth

Abstract

In this paper, our goal is to study the following class of Hénon type problems \begin{equation*} \left\{\begin{array}{rclcl}\displaystyle -\Delta u & & = \lambda u+|x|^{\alpha}k(u_+)+ f(x) &\mbox{in}&B_1, \\ u & & = 0 & \mbox{on} & \partial B_1, \end{array}\right. \end{equation*} where $B_1$ is the unit ball in $\mathbb R^N$, $k(t)$ is a $C^1$ function in $[0,+\infty)$ which is assumed to be in the critical growth range with subcritical perturbation, $f$ is radially symmetric and belongs to $L^{\mu}(B_1)$ for suitable $\mu$ depending on $N\geq 3$. Under appropriate hypotheses on the constant $\lambda$, we prove existence of at least two radial solutions for this problem using variational methods.