Ambrosetti-Prodi problem with degenerate potential and Neumann boundary condition

Abstract

We study the degenerate elliptic equation $$ -\hbox{div}(|x|^\alpha\nabla u) =f(u)+t\phi(x)+h(x) $$ in a bounded open set $\Omega$ with homogeneous Neumann boundary condition, where $\alpha\in(0,2)$ and f has a linear growth. The main result establishes the existence of real numbers $t_*$ and $t^*$ such that the problem has at least two solutions if $t\leq t_*$, there is at least one solution if $t_*t^*$. The proof combines a priori estimates with topological degree arguments.

Keywords

• Ambrosetti-Prodi problem
• degenerate potential
• topological degree
• anisotropic continuous media