Revisiting the sub- and super-solution method for the classical radial solutions of the mean curvature equation

Abstract

This paper focuses on the existence and the multiplicity of classical radially symmetric solutions of the mean curvature problem:−div∇v1+|∇v|2=f(x,v,∇v)inΩ,a0v+a1∂v∂ν=0on∂Ω,\left\{\begin{array}{ll}-\text{div}\left(\frac{\nabla v}{\sqrt{1+|\nabla v{|}^{2}}}\right)=f(x,v,\nabla v)& \text{in}\hspace{.5em}\text{Ω},\\ {a}_{0}v+{a}_{1}\tfrac{\partial v}{\partial \nu }=0& \text{on}\hspace{.5em}\partial \text{Ω},\end{array}\right.with Ω\text{Ω} an open ball in ℝN{{\mathbb{R}}}^{N}, in the presence of one or more couples of sub- and super-solutions, satisfying or not satisfying the standard ordering condition. The novel assumptions introduced on the function f allow us to complement or improve several results in the literature.

Keywords

• prescribed mean curvature equation
• dirichlet
• neumann
• robin boundary conditions
• radial symmetry
• classical solution
• sub- and super-solutions
• 35j62
• 35j93
• 35j25
• 34c25